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1
Yang, Fangyan, Yongming Cao, Lijuan Chen e Qingdu Li. "Sequence of Routes to Chaos in a Lorenz-Type System". Discrete Dynamics in Nature and Society 2020 (23 gennaio 2020): 1–10. http://dx.doi.org/10.1155/2020/3162170.
Abstract (sommario):
This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.
2
van Kekem, Dirk L., e Alef E. Sterk. "Wave propagation in the Lorenz-96 model". Nonlinear Processes in Geophysics 25, n. 2 (27 aprile 2018): 301–14. http://dx.doi.org/10.5194/npg-25-301-2018.
Abstract (sommario):
Abstract. In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F < 0 and odd n, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with n. For F < 0 and even n, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether n has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing n by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values n and F.
3
Honeycutt, Andrew, e Tony L. Schmitz. "Experimental Validation of Period-n Bifurcations in Milling". Procedia Manufacturing 5 (2016): 362–74. http://dx.doi.org/10.1016/j.promfg.2016.08.031.
4
Liu, Yun, e Xijuan Liu. "Bifurcations and Structures of the Parameter Space of a Discrete-Time SIS Epidemic Model". Journal of Mathematics 2022 (23 aprile 2022): 1–14. http://dx.doi.org/10.1155/2022/2233452.
Abstract (sommario):
The dynamics of a discrete-time SIS epidemic model are reported in this paper. Three types of codimension one bifurcation, namely, transcritical, flip, Neimark–Sacker (N-S) bifurcations, and their intersection codimension two bifurcations including 1 : 2, 1 : 3, and 1 : 4 resonances are discussed. The necessary and sufficient conditions for detecting these types of bifurcation are derived using algebraic criterion methods. Numerical simulations are conducted not only to illustrate analytical results but also to exhibit complex behaviors which include period-doubling bifurcation in period − 2 , − 4 , − 8 , − 16 orbits, invariant closed cycles, and attracting chaotic sets. Especially, here we investigate the parameter space of the discrete model. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region. We identify period-adding, Farey sequence of periodic structures embedded in this quasiperiodic region.
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SRINIVASAN, K. "MULTIPLE PERIOD DOUBLING BIFURCATION ROUTE TO CHAOS IN PERIODICALLY PULSED MURALI–LAKSHMANAN–CHUA (MLC) CIRCUIT". International Journal of Bifurcation and Chaos 18, n. 02 (febbraio 2008): 541–55. http://dx.doi.org/10.1142/s021812740802046x.
Abstract (sommario):
In this paper, we study the effect of additional periodic forces in Murali–Lakshmanan–Chua (MLC) circuit. We show that the additional periodic forces of pulse type display novel dynamical features including multiple period doubling bifurcation route to chaos, followed by a rich variety of dynamical phenomena including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures and so on. For certain types of periodic pulses, the circuit admits transcritical bifurcations preceding the onset of multiple period doubling bifurcations. We have characterized these dynamical behaviors using Lyapunov exponents, correlation dimension, Kaplan–Yorke dimension and power spectrum, which are found to be in good agreement with the experimental observations. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses, which has immense potential applications in secure communication. The above study facilitates one to generate any desired n-period doubling bifurcation behavior by applying n-periodic pulses to a chaotic system. Further, controlling and synchronization of chaos in this periodically pulsed MLC circuit have also been achieved by using suitable methods.
6
CHEN, XIANWEI, XIANGLING FU e ZHUJUN JING. "COMPLEX DYNAMICS IN A PENDULUM EQUATION WITH A PHASE SHIFT". International Journal of Bifurcation and Chaos 22, n. 12 (dicembre 2012): 1250307. http://dx.doi.org/10.1142/s0218127412503075.
Abstract (sommario):
Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.
7
Paidoussis, M. P., G. X. Li e R. H. Rand. "Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment". Journal of Applied Mechanics 58, n. 2 (1 giugno 1991): 559–65. http://dx.doi.org/10.1115/1.2897220.
Abstract (sommario):
A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.
8
Kulenović, M. R. S., Connor O’Loughlin e E. Pilav. "The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation". Discrete Dynamics in Nature and Society 2021 (26 novembre 2021): 1–14. http://dx.doi.org/10.1155/2021/2092709.
Abstract (sommario):
We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.
9
Zhao, Huitao, Yiping Lin e Yunxian Dai. "A New Feigenbaum-Like Chaotic 3D System". Discrete Dynamics in Nature and Society 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/328143.
Abstract (sommario):
Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.
10
Xing, Siyuan, e Albert C. J. Luo. "On an origami structure of period-1 motions to homoclinic orbits in the Rössler system". Chaos: An Interdisciplinary Journal of Nonlinear Science 32, n. 12 (dicembre 2022): 123121. http://dx.doi.org/10.1063/5.0131970.
Abstract (sommario):
In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2 n motions [Formula: see text]. On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.
11
Yang, Zhiyan, Tao Jiang e Zhujun Jing. "Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms". International Journal of Bifurcation and Chaos 24, n. 03 (marzo 2014): 1430011. http://dx.doi.org/10.1142/s0218127414300110.
Abstract (sommario):
Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.
12
Qiu, Bo, e Shuiming Chen. "Interannual-to-Decadal Variability in the Bifurcation of the North Equatorial Current off the Philippines". Journal of Physical Oceanography 40, n. 11 (1 novembre 2010): 2525–38. http://dx.doi.org/10.1175/2010jpo4462.1.
Abstract (sommario):
Abstract Satellite altimeter sea surface height (SSH) data from the past 17 yr are used to investigate the interannual-to-decadal changes in the bifurcation of the North Equatorial Current (NEC) along the Philippine coast. The NEC bifurcation latitude migrated quasi decadally between 10° and 15°N with northerly bifurcations observed in late 1992, 1997–98, and 2003–04 and southerly bifurcations in 1999–2000 and 2008–09. The observed NEC bifurcation latitude can be approximated well by the SSH anomalies in the 12°–14°N and 127°–130°E box east of the mean NEC bifurcation point. Using a 1 ½-layer reduced-gravity model forced by the ECMWF reanalysis wind stress data, the authors find that the SSH anomalies in this box can be simulated favorably to serve as a proxy for the observed NEC bifurcation. With the availability of the long-term reanalysis wind stress data, this helps to lengthen the NEC bifurcation time series back to 1962. Although quasi-decadal variability was prominent in the last two decades, the NEC bifurcation was dominated by changes with a 3–5-yr period during the 1980s and had low variance prior to the 1970s. These interdecadal modulations in the characteristics of the NEC bifurcation reflect similar interdecadal modulations in the wind forcing field over the western tropical North Pacific Ocean. Although the NEC bifurcation on interannual and longer time scales is generally related to the Niño-3.4 index with a positive (negative) index corresponding to a northerly (southerly) bifurcation, the exact location of bifurcation is determined by wind forcing in the 12°–14°N band that contains variability not fully representable by the Niño-3.4 index.
13
De, Sudipto K., e N. R. Aluru. "U -sequence in electrostatic microelectromechanical systems (MEMS)". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, n. 2075 (31 maggio 2006): 3435–64. http://dx.doi.org/10.1098/rspa.2006.1733.
Abstract (sommario):
In this paper, the presence of U (Universal)-sequence (a sequence of periodic windows that appear beyond the period doubling (PD) route to chaos) in electrostatic microelectromechanical systems (MEMS) is reported. The MEM system is first brought to a nonlinear steady state by the application of a large dc bias close to the dynamic pull-in voltage of the device. An ac voltage (the bifurcation parameter) is next applied to the system and increased gradually. A sequence of PD bifurcations leading to chaos is observed for resonant and superharmonic excitations (frequency of the ac voltage). On further increase in the ac voltage (beyond where chaos sets in), U -sequence is observed in the system. Under superharmonic excitation, the sequence is found to be a modified form of the U -sequence referred to as the ‘ UM -sequence’ in this paper. The appearance of a periodic window with K oscillations per period or K -cycles in the normal U -sequence is replaced by a corresponding periodic window with KM -cycles in the UM -sequence. M stands for the M th superharmonic frequency of excitation. The formation of the periodic windows from a chaotic state in the UM -sequence takes place through intermittent chaos as the ac voltage is gradually increased. On the other hand, the periodic states/cycles formed through intermittent chaos transform back into a chaotic state through the period doubling route. A sequence of period doubling bifurcations of the UM -sequence cycles result in the formation of -cycles in electrostatic MEMS. n corresponds to the n th period doubling bifurcation in the sequence. A simplified mass–spring–damper (MSD) model for MEMS is used to understand the physical mechanism that gives rise to these nonlinear dynamic properties in MEMS. The nonlinear nature of the electrostatic force acting on the MEM device is found to be responsible for the reported observations.
14
BAKER, B. M., M. E. KIDWELL, R. P. KLINE e I. POPOVICI. "ON TWO-DIMENSIONAL DYNAMICAL SYSTEMS ASSOCIATED WITH MEMBRANE KINETICS UNDERLYING CARDIAC ARRHYTHMIAS". International Journal of Bifurcation and Chaos 19, n. 05 (maggio 2009): 1709–32. http://dx.doi.org/10.1142/s0218127409023792.
Abstract (sommario):
We study the orbits, stability and coexistence of orbits in the two-dimensional dynamical system introduced by Kline and Baker to model cardiac rhythmic response to periodic stimulation — as a function of (a) kinetic parameters (two amplitudes, two rate constants) and (b) stimulus period. The original paper focused mostly on the one-dimensional version of this model (one amplitude, one rate constant), whose orbits, stability properties, and bifurcations were analyzed via the theory of skew-tent (hence unimodal) maps; the principal family of orbits were so-called "n-escalators", with n a positive integer. The two-dimensional analog (motivated by experimental results) has led to the current study of continuous, piecewise smooth maps of a polygonal planar region into itself, whose dynamical behavior includes the coexistence of stable orbits. Our principal results show (1) how the amplitude parameters control which escalators can come into existence, (2) escalator bifurcation behavior as the stimulus period is lowered — leading to a "1/n bifurcation law", and (3) the existence of basins of attraction via the coexistence of three orbits (two of them stable, one unstable) at the first (largest stimulus period) bifurcation. We consider the latter result our most important, as it is conjectured to be connected with arrhythmia.
15
QIN, ZHIYING, YUEJING ZHAO e JICHEN YANG. "NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP". International Journal of Bifurcation and Chaos 22, n. 05 (maggio 2012): 1250112. http://dx.doi.org/10.1142/s021812741250112x.
Abstract (sommario):
In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.
16
Waugh, Iain C., K. Kashinath e Matthew P. Juniper. "Matrix-free continuation of limit cycles and their bifurcations for a ducted premixed flame". Journal of Fluid Mechanics 759 (17 ottobre 2014): 1–27. http://dx.doi.org/10.1017/jfm.2014.549.
Abstract (sommario):
AbstractMany experimental studies have demonstrated that ducted premixed flames exhibit stable limit cycles in some regions of parameter space. Recent experiments have also shown that these (period-1) limit cycles subsequently bifurcate to period-$2^{n}$, quasiperiodic, multiperiodic or chaotic behaviour. These secondary bifurcations cannot be found computationally using most existing frequency domain methods, because these methods assume that the velocity and pressure signals are harmonic. In an earlier study we have shown that matrix-free continuation methods can efficiently calculate the limit cycles of large thermoacoustic systems. This paper demonstrates that these continuation methods can also efficiently calculate the bifurcations from the limit cycles. Furthermore, once these bifurcations are found, it is then possible to isolate the coupled flame–acoustic motion that causes the qualitative change in behaviour. This information is vital for techniques that use selective damping to move bifurcations to more favourable locations in the parameter space. The matrix-free methods are demonstrated on a model of a ducted axisymmetric premixed flame, using a kinematic $G$-equation solver. The methods find limit cycles and period-2 limit cycles, and fold, period-doubling and Neimark–Sacker bifurcations as a function of the location of the flame in the duct, and the aspect ratio of the steady flame.
17
Khan, A. Q., e T. Khalique. "Bifurcations and chaos control in a discrete-time biological model". International Journal of Biomathematics 13, n. 04 (20 marzo 2020): 2050022. http://dx.doi.org/10.1142/s1793524520500229.
Abstract (sommario):
In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.
18
JING, ZHUJUN, e JIANPING YANG. "COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS I". International Journal of Bifurcation and Chaos 16, n. 10 (ottobre 2006): 2887–902. http://dx.doi.org/10.1142/s0218127406016525.
Abstract (sommario):
Pendulum equation with parametric and external excitations is investigated in (I) and (II). In (I), by applying Melnikov's method, we prove the criterion of existence of chaos under periodic perturbation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors including the period-n (n = 2 to 6, 10, 15 and 20) orbits in different chaotic regions, interlocking periodic orbits, symmetry-breaking of periodic orbit, cascade of period-doubling bifurcations from period-5 and -10 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for a given external frequency or parametric frequency and chaos suddenly converting to periodic orbits, sudden jump in the size of attractors which is associated with the transverse intersection of stable and unstable manifolds of perturbed saddle, hopping behavior of chaos, transient chaos with complex periodic windows and interior crisis, varied chaotic attractors including the more than three-band and eight-band chaotic attractors, chaotic attractor after strange nonchaotic attractor. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping δ, spring constant α and frequency Ω of parametric excitation which can be considered as a control strategy. In (II), we will investigate the complex dynamics under quasi-periodic perturbation.
19
WANG, HANQING, e XIANG LI. "BIFURCATIONS IN A FREQUENCY-WEIGHTED KURAMOTO OSCILLATORS NETWORK". International Journal of Bifurcation and Chaos 22, n. 09 (settembre 2012): 1250230. http://dx.doi.org/10.1142/s0218127412502306.
Abstract (sommario):
This paper focuses on the origin of rich dynamics in a frequency-weighted Kuramoto-oscillator network, which embeds the periodical activity patterns of human beings to understand the collective behaviorial dynamics in human population. We present analytical results of the dynamics of the model by reducing the N-dimensional system to solvable low-dimensional equations. The bifurcation analysis reveals that there exist supercritical Hopf bifurcation and infinite-period saddle-node bifurcation, which explain the rich dynamics including the incoherent, oscillatory, and synchronous states observed in this model.
20
Barabash, N. V. "Period addition cascade in a model of neuron-glial interaction". Genes & Cells 18, n. 4 (15 dicembre 2023): 840–43. http://dx.doi.org/10.17816/gc623431.
Abstract (sommario):
We analyze a system of four differential equations that describe the dynamics of a neuron-glial network using the mean field approximation [1, 2]: (τ=–E+α ln(1+e1/α(JuxE+I0)), =(1–x)/τD–uxE,(1) =U(y)–u/τF+U(y)(1–u)E, =–y/τy+βσ(x), where E(t) is the average activity, x(t) is the fraction of available neurotransmitter released into the synaptic gap with a probability u(t); y(t) is the fraction of the gliatransmitter released by the astrocyte. Sigmoidal functions U(y) and σ(x) correspond to changes in the base probability level u(t) during the release of the gliatransmitter and activation of astrocytes during neurotransmitter release, respectively. The input inhibitory current corresponds to a bifurcation parameter with a negative value of I0 0, while the remaining parameters have positive and fixed values. The rest of the parameters are positive and fixed. For a detailed description of the model, including information on the type of functions and parameter values, refer to works [1, 2]. For a constant value of U(x)=const, the first three equations in system (1) represent the Tsodyks–Markram model, which explains the short-term synaptic plasticity phenomenon [1]. The model was enhanced with a fourth equation for y in [2], incorporating the influence of astrocytes via the concept of a tripartite synapse [3]. Model (1) illustrates a wide range of dynamic behavior including quiescence, regular tonic activity, and chaotic bursting activity. These behaviors correspond to various sets in the phase space, such as stable equilibrium states, limit cycles of period 1, limit cycles of any period n∈N, and chaotic attractors. Changing the I0 l parameter causes sets to bifurcate, resulting in the loss of stability of certain attractors and the emergence of others, leading to a shift in the dynamic regime. Therefore, in terms of dynamics, the conditions for bifurcation and the characteristics of the newly formed attractors are crucial. In this presentation, we have obtained a series of numerical bifurcations in system (1) that correspond to the shift from tonic activity to burst activity, resulting in subsequent modifications to the bursts. Specifically, our findings demonstrate that an increase in the number of spikes per burst is determined by a period adding cascade where the aforementioned limit cycle of period n becomes unstable, allowing for a previously established stable cycle of period n+1 to occupy the position of the “main” attractor. This process culminates in the vanishing of the orbit with an endless period due to the saddle-node bifurcation of cycles, followed by the creation of a dependable cycle with a period of 1. The main properties of the cascade were reproduced in our model one-dimensional piecewise-smooth map z¯=1−z6, forz0,μ−1−μ(z-1)6, forz0, where z∈R1, μ is a bifurcational parameter. The map’s results suggest that an increase in current I0 i in model (1) may lead to the emergence and disappearance of quasi-strange attractors (quasi-attractors), implying chaotic behavior in connection with burst variation.
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CONTOPOULOS, G., M. HARSOULA, R. DVORAK e F. FREISTETTER. "RECURRENCE OF ORDER IN CHAOS". International Journal of Bifurcation and Chaos 15, n. 09 (settembre 2005): 2865–82. http://dx.doi.org/10.1142/s021812740501371x.
Abstract (sommario):
The standard map x′ = x + y′, y′ = y + (K/2π) sin (2πx), where both x and y are given modulo 1, becomes mostly chaotic for K ≥ 8, but important islands of stability appear in a recurrent way for values of K near K = 2nπ (groups of islands I and II), and K = (2n + 1)π (group III), where n ≥ 1. The maximum areas of the islands and the intervals ΔK, where the islands appear, follow power laws. The changes of the areas of the islands around a maximum follow universal patterns. All islands surround stable periodic orbits. Most of the orbits are irregular, i.e. unrelated to the orbits of the unperturbed problem K = 0. The main periodic orbits of periods 1, 2 and 4 and their stability are derived analytically. As K increases these orbits become unstable and they are followed by infinite period-doubling bifurcations with a bifurcation ratio δ = 8.72. We find theoretically the connections between the various families and the extent of their stability. Numerical calculations verify the theoretical results.
22
Watanabe, Masahito, e Hiroaki Yoshimura. "Resonance, symmetry, and bifurcation of periodic orbits in perturbed Rayleigh–Bénard convection". Nonlinearity 36, n. 2 (6 gennaio 2023): 955–99. http://dx.doi.org/10.1088/1361-6544/aca73b.
Abstract (sommario):
Abstract This paper investigates the global structures of periodic orbits that appear in Rayleigh–Bénard convection, which is modelled by a two-dimensional perturbed Hamiltonian model, by focusing upon resonance, symmetry and bifurcation of the periodic orbits. First, we show the global structures of periodic orbits in the extended phase space by numerically detecting the associated periodic points on the Poincaré section. Then, we illustrate how resonant periodic orbits appear and specifically clarify that there exist some symmetric properties of such resonant periodic orbits which are projected on the phase space; namely, the period m and the winding number n become odd when an m-periodic orbit is symmetric with respect to the horizontal and vertical centre lines of a cell. Furthermore, the global structures of bifurcations of periodic orbits are depicted when the amplitude ɛ of the perturbation is varied, since in experiments the amplitude of the oscillation of the convection gradually increases when the Rayleigh number is raised.
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LEUNG, A. Y. T., ZHONGJIN GUO e H. X. YANG. "TRANSITION CURVES AND BIFURCATIONS OF A CLASS OF FRACTIONAL MATHIEU-TYPE EQUATIONS". International Journal of Bifurcation and Chaos 22, n. 11 (novembre 2012): 1250275. http://dx.doi.org/10.1142/s0218127412502756.
Abstract (sommario):
A general version of the fractional Mathieu equation and the corresponding fractional Mathieu–Duffing equation are established for the first time and investigated via the harmonic balance method. The approximate expressions for the transition curves separating the regions of stability are derived. It is shown that a change in the fractional derivative order remarkably affects the shape and location of the transition curves in the n = 1 tongue. However, the shape of the transition curve does not change very much for different fractional orders for the n = 0 tongue. The steady state approximate responses of the corresponding fractional Mathieu–Duffing equation are obtained by means of harmonic balance, polynomial homotopy continuation and technique of linearization. The curves with respect to fractional order versus response amplitude, driving amplitude versus response amplitude with different fractional orders are shown. It can be found that the bifurcation point and stability of branch solutions is different under different fractional orders of system. When the fractional order increases to some value, the symmetric breaking, saddle-node bifurcation as well as period-doubling bifurcation phenomena are found and exhibited analytically by taking the driving amplitude as the bifurcation parameter.
24
HUI, JING, e DE-MING ZHU. "DYNAMICS OF SEIS EPIDEMIC MODELS WITH VARYING POPULATION SIZE". International Journal of Bifurcation and Chaos 17, n. 05 (maggio 2007): 1513–29. http://dx.doi.org/10.1142/s0218127407017902.
Abstract (sommario):
In this paper, SEIS epidemic models with varying population size are considered. Firstly, we consider the case when births of population are throughout the year. A threshold σ is identified, which determines the outcome of disease, that is, when σ < 1, the disease dies out; whereas when σ > 1, the disease persists and the unique endemic equilibrium is globally asymptotically stable; when σ = 1, bifurcation occurs and leads to "the change of stability". Two other thresholds σ′ and [Formula: see text] are also identified, which determine the dynamics of epidemic model with varying population size, when the disease dies out or it is endemic. Secondly, we consider the other case, birth pulse. The population density is increased by an amount B(N)N at the discrete time nτ, where n is any non-negative integer and τ is a positive constant, B(N) is density-dependent birth rate. By applying the corresponding stroboscopic map, we obtain the existence of infection-free periodic solution with period τ. Lastly, through numerical simulations, we show the dynamic complexities of SEIS epidemic models with varying population size, there is a sequence of bifurcations, leading to chaotic strange attractors. Non-unique attractors also appear, which implies that the dynamics of SEIS epidemic models with varying population size can be very complex.
25
Zhou, Zhong. "Super-stable Kneading Sequences with Double Cycles in 1D Bimodal Maps". Journal of Computing and Electronic Information Management 11, n. 3 (24 novembre 2023): 42–45. http://dx.doi.org/10.54097/jceim.v11i3.10.
Abstract (sommario):
It is well known that a super-stable kneading sequence (SSKS) is an important concept, all SSKSs in bimodal maps forms joints in the corresponding symbolic dynamics, it decides the multiplication table of star products, which the n-tupling bifurcations to chaos can be investigated and Feigenbaum’s metric universalities can be measured and reconstructed, this SSKSs have form which are periodic with single cycle. However, in fact, the SSKSs in bimodal maps have another form with double cycles which are little mentioned and researched, they have the same position and significance as the single cycle SSKS. In the paper, we presented the number of admissible SSKSs with period-n and the joints graph on the parameter plane.
26
Khyat, Toufik, e M. R. S. Kulenović. "Global Dynamics of Delayed Sigmoid Beverton–Holt Equation". Discrete Dynamics in Nature and Society 2020 (26 maggio 2020): 1–15. http://dx.doi.org/10.1155/2020/1364282.
Abstract (sommario):
In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.
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Gilbert, Joel, Sylvain Maugeais e Christophe Vergez. "Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation". Acta Acustica 4, n. 6 (2020): 27. http://dx.doi.org/10.1051/aacus/2020026.
Abstract (sommario):
A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.
28
GALLAS, JASON A. C. "THE ROLE OF CODIMENSION IN DYNAMICAL SYSTEMS". International Journal of Modern Physics C 03, n. 06 (dicembre 1992): 1295–321. http://dx.doi.org/10.1142/s0129183192000890.
Abstract (sommario):
Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The Pk=k×2n, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.
29
Tung, P. C., e S. W. Shaw. "The Dynamics of an Impact Print Hammer". Journal of Vibration and Acoustics 110, n. 2 (1 aprile 1988): 193–200. http://dx.doi.org/10.1115/1.3269498.
Abstract (sommario):
A mathematical model is developed to describe the characteristic behavior of an impact print hammer of the stored energy type. The armature of the impact print hammer is represented by a rigid mass held against a backstop by a preloaded linear spring with negative stiffness which characterizes the net effect of a permanent magnet and a prestressed flexible beam acting on the armature. Periodic sine pulses are adopted to represent currents which release the armature to strike the ribbon and paper which is represented by a linear spring and a linear viscous dashpot. A coefficient of restitution is employed to characterize the instantaneous behavior of impact and rebound at the backstop. In this paper, periodic motions with n impacts against the backstop per forcing cycle, period doubling bifurcations, and chaotic motions are found. The stability of the periodic motions is investigated as is the influence of various parameters on the performance of the impact print hammer. With this simple model we can predict much of the qualitative behavior of the actual physical system.
30
Goswami, Binoy Krishna. "The Role of Period Tripling in the Development of a Self Similar Bifurcation Structure". International Journal of Bifurcation and Chaos 07, n. 12 (dicembre 1997): 2691–706. http://dx.doi.org/10.1142/s0218127497001813.
Abstract (sommario):
We study theoretically the period tripling phenomenon and its role in the development of the bifurcation structure of the periodically forced Toda oscillator. The analytical studies reveal some interesting features in the context of previously known period tripling in one-parameter systems where the loss of stability of the original equilibrium used to be associated with the birth of a bifurcating saddle. In the case of Toda oscillator, a period-tripled saddlenode is born around the stable original period where the saddle approaches the original period as the dissipativity is reduced. In the conservative limit the saddle undergoes a tangential interaction with the original centre. The recurrent appearance of period tripling and doubling suggest a new self similar feature of the associated bifurcation structure. The revelation of the underlying period tripling phenomena also helps to sequentially characterize each period, cascade and substructure, and to make some qualitative predictions regarding their locations in the phase and parameter space. The period tripling may give birth to some "degenerate period n orbits" (i.e. a certain class of period n orbits with identical n). However, they are distinctly characterized on the basis of concept of period tripling and we also predict the "degeneracy" (the number of such period n orbits for a given n) analytically. The same concept is extended in the case of degenerate cascades and substructures.
31
AVRUTIN, VIKTOR, MICHAEL SCHANZ e LAURA GARDINI. "CALCULATION OF BIFURCATION CURVES BY MAP REPLACEMENT". International Journal of Bifurcation and Chaos 20, n. 10 (ottobre 2010): 3105–35. http://dx.doi.org/10.1142/s0218127410027581.
Abstract (sommario):
The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.
32
Sun, Dan, Yunfei Gao, Linping Peng e Li Fu. "Limit cycles in piecewise smooth perturbations of a class of cubic differential systems". Electronic Journal of Qualitative Theory of Differential Equations, n. 49 (2023): 1–26. http://dx.doi.org/10.14232/ejqtde.2023.1.49.
Abstract (sommario):
In this paper, we study the bifurcation of limit cycles from a class of cubic integrable non-Hamiltonian systems under arbitrarily small piecewise smooth perturbations of degree n . By using the averaging theory and complex method, the lower and upper bounds for the maximum number of limit cycles bifurcating from the period annulus of the unperturbed systems are given at first order in ε . It is also shown that in this case, the maximum number of limit cycles produced by piecewise smooth perturbations is almost twice the upper bound of the maximum number of limit cycles produced by smooth perturbations for the considered systems.
33
Tan, Wei, Jianguo Gao e Wenjun Fan. "Bifurcation Analysis and Chaos Control in a Discrete Epidemic System". Discrete Dynamics in Nature and Society 2015 (2015): 1–13. http://dx.doi.org/10.1155/2015/974868.
Abstract (sommario):
The dynamics of discreteSIepidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interiorR+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the systemx→x+δ[rN(1-N/K)-βxy/N-(μ+m)x],y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.
34
Goswami, Binoy Krishna. "Self-similarity in the bifurcation structure involving period tripling, and a suggested generalization to period n-tupling". Physics Letters A 245, n. 1-2 (agosto 1998): 97–109. http://dx.doi.org/10.1016/s0375-9601(98)00338-7.
35
Ma, Junhai, e Hui Jiang. "Dynamics of a nonlinear differential advertising model with single parameter sales promotion strategy". Mathematical Biosciences and Engineering 30, n. 4 (2022): 1142–57. http://dx.doi.org/10.3934/era.2022061.
Abstract (sommario):
<abstract><p>Advertising and sales promotion are two important specific marketing communications tools. In this paper, nonlinear differential equation and single parameter sales promotion strategy are introduced into an advertising model and investigated quantitatively. The existence and stability of period-$ nT $ (n = 1, 2, 4, 8) solutions are investigated. Interestingly, both period doubling bifurcation and inverse flip bifurcation occur at different parameter values in the same advertising model. The results show that the system enters into chaos from stable state through flip bifurcation and enters into stable state from chaos through inverse flip bifurcation. An effective control strategy, which suppresses flip bifurcation and promotes inverse flip bifurcation, is proposed to eliminate chaos. These results have some significant theoretical and practical value in related markets.</p></abstract>
36
Kazantsev, A. N., K. P. Chernykh, N. E. Zarkua, R. Yu Leader, K. G. Kubachev, G. Sh Bagdavadze, E. Yu Kalinin, T. E. Zaitseva, A. E. Chikin e Yu P. Linets. "Carotid endarterectomy with extended lesion: formation of a new bifurcation according to A.V.Pokrovsky or autoarterial reconstruction according to A.A.Karpenko?" Research and Practical Medicine Journal 7, n. 3 (12 settembre 2020): 33–42. http://dx.doi.org/10.17709/2409-2231-2020-7-3-3.
Abstract (sommario):
Purpose of the study. Comparison of hospital and long-term results of autoarterial reconstruction of carotid artery bifurcation and the formation of a new bifurcation with an extended atherosclerotic lesion of the internal carotid artery (ICA). Materials and methods. In the period from January 2018 to May 2020, this cohort, comparative, prospective, open-label study included 279 patients with an extended atherosclerotic lesion of the ICA operated on in the Alexandr Hospital. Depending on the implemented strategy of surgical correction, all patients were divided into two groups: group 1 (n=132) — autoarterial reconstruction of bifurcation of the carotid arteries; Group 2 (n=147) — the formation of a new bifurcation. Complications were recorded in the hospital and long-term postoperative periods. The total follow-up period was 16.4±9.3 months. The endpoints of the study were such adverse cardiovascular events as death, myocardial infarction (MI), stroke, thrombosis / restenosis of the anastomosis zone, combined endpoint (death from stroke / IM + IM + stroke). Results. The ICA clamping time in group 1 was 32.6±3.3 minutes, in group 2 – 31.7±3.5 minutes, which did not receive statistically significant differences (р=0.81). In the hospital postoperative period, adverse cardiovascular events were not recorded. In the long-term follow-up, the groups were comparable in the frequency of all complications. Identified lethal outcomes developed as a result of the formation of MI in patients with multiple lesions of the coronary arteries and a history of myocardial revascularization. The likely cause was shunt / stent thrombosis with subsequent coronary insufficiency and an increase in ischemic heart damage. The causes of stroke, recorded in each group in isolated cases, were the presence of atrial fibrillation. Patients did not comply with the recommended regimen of anticoagulant therapy, which provoked the development of cerebral catastrophe. In turn, the identified restenoses of the reconstruction zone were asymptomatic and were also observed in isolated cases in each group in the period 12 months after CEE. Conclusion. Autoarterial reconstruction of carotid bifurcation and the formation of a new bifurcation are comparable in safety and effectiveness methods of surgical treatment of an extended atherosclerotic lesion of the ICA. Operation techniques differ in the choice of an artery that is cut off from bifurcation — the external carotid artery or ICA. Further, the reconstruction progress is absolutely identical. Hospital and long-term follow-up results showed minimal indicators of the development of cardiovascular and hemodynamic changes due to the type of operation. Thus, both reconstruction techniques can be the operation of choice for an extended ICA lesion.
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Bassanelli, Giovanni, e François Berteloot. "Distribution of polynomials with cycles of a given multiplier". Nagoya Mathematical Journal 201 (marzo 2011): 23–43. http://dx.doi.org/10.1215/00277630-2010-016.
Abstract (sommario):
AbstractIn the space of degree d polynomials, the hypersurfaces defined by the existence of a cycle of period n and multiplier eiθ are known to be contained in the bifurcation locus. We prove that these hypersurfaces equidistribute the bifurcation current. This is a new result, even for the space of quadratic polynomials.
38
MA, SUQI, XIAOHUI WANG, JINZHI LEI e ZHAOSHENG FENG. "DYNAMICS OF THE DELAY HEMATOLOGICAL CELL MODEL". International Journal of Biomathematics 03, n. 01 (marzo 2010): 105–25. http://dx.doi.org/10.1142/s1793524510000829.
Abstract (sommario):
In this paper, complex dynamics of a two-compartment model of production and regulation of the circulating blood neutrophil number are investigated. It is shown that the proliferative disorders may be possible due to factors of the apoptosis rate rsof the haematopoietic stem cell and the cell cycle duration τs. Applying a recent geometrical criterion for the Hopf bifurcation and transient behaviors of delay systems to this model, we separate the stable regime from the unstable regime on the rs- τsplane. Numerically, regimes of patterned periodic oscillations with low periodicity in the number of circulating blood cells appear on the rs- τsplane. It is found that the dominated period-adding bifurcation mechanism leads transitions from period-n to period-(n + 1), eventually changes to the complex attractor with high-periodicity or chaos.
39
Liu, Alexander G., Jack J. Matthews, Latha R. Menon, Duncan McIlroy e Martin D. Brasier. "Haootia quadriformis n. gen., n. sp., interpreted as a muscular cnidarian impression from the Late Ediacaran period (approx. 560 Ma)". Proceedings of the Royal Society B: Biological Sciences 281, n. 1793 (22 ottobre 2014): 20141202. http://dx.doi.org/10.1098/rspb.2014.1202.
Abstract (sommario):
Muscle tissue is a fundamentally eumetazoan attribute. The oldest evidence for fossilized muscular tissue before the Early Cambrian has hitherto remained moot, being reliant upon indirect evidence in the form of Late Ediacaran ichnofossils. We here report a candidate muscle-bearing organism, Haootia quadriformis n. gen., n. sp., from approximately 560 Ma strata in Newfoundland, Canada. This taxon exhibits sediment moulds of twisted, superimposed fibrous bundles arranged quadrilaterally, extending into four prominent bifurcating corner branches. Haootia is distinct from all previously published contemporaneous Ediacaran macrofossils in its symmetrically fibrous, rather than frondose, architecture. Its bundled fibres, morphology, and taphonomy compare well with the muscle fibres of fossil and extant Cnidaria, particularly the benthic Staurozoa. Haootia quadriformis thus potentially provides the earliest body fossil evidence for both metazoan musculature, and for Eumetazoa, in the geological record.
40
Lim, Yongwhan, Min Chul Kim, Youngkeun Ahn, Doo Sun Sim, Young Joon Hong, Ju Han Kim, Myung Ho Jeong et al. "Effect of Stenting Strategy on the Outcome in Patients with Non-Left Main Bifurcation Lesions". Journal of Clinical Medicine 11, n. 19 (26 settembre 2022): 5658. http://dx.doi.org/10.3390/jcm11195658.
Abstract (sommario):
Previous studies have not compared outcomes between different percutaneous coronary intervention (PCI) strategies and lesion locations in non-left main (LM) bifurcation lesions. We enrolled 2044 patients from a multicenter registry with an LAD bifurcation lesion (n = 1551) or non-LAD bifurcation lesion (n = 493). The primary outcome was target lesion failure (TLF), a composite of cardiac death, myocardial infarction, and target lesion revascularization (TLR). During a median follow-up period of 38 months, non-LAD bifurcation lesions treated with the two-stent strategy, compared with the one-stent strategy, were associated with more frequent TLF (20.7% vs. 6.3%, p < 0.01), TLR (16.7% vs. 4.7%, p < 0.01), and target vessel revascularization (TVR; 18.2% vs. 6.3%, p < 0.01). There was no significant difference in outcome among LAD bifurcation lesions treated with different PCI strategies. The two-stent strategy was associated with a higher risk of TLF (adjusted HR 4.34, CI 1.93–9.76, p < 0.01), TLR (adjusted HR 4.30, CI 1.64–11.27, p < 0.01), and TVR (adjusted HR 5.07, CI 1.69–9.74, p < 0.01) in the non-LAD bifurcation lesions. The planned one-stent strategy is preferable to the two-stent strategy for the treatment of non-LAD bifurcation lesions.
41
LUO, ALBERT C. J., e JIANZHE HUANG. "ANALYTICAL DYNAMICS OF PERIOD-m FLOWS AND CHAOS IN NONLINEAR SYSTEMS". International Journal of Bifurcation and Chaos 22, n. 04 (aprile 2012): 1250093. http://dx.doi.org/10.1142/s0218127412500939.
Abstract (sommario):
In this paper, the analytical solutions for period-m flows and chaos in nonlinear dynamical systems are presented through the generalized harmonic balance method. The nonlinear damping, periodically forced, Duffing oscillator was investigated as an example to demonstrate the analytical solutions of periodic motions and chaos. Through this investigation, the mechanism for a period-m motion jumping to another period-n motion in numerical computation is found. In this problem, the Hopf bifurcation of periodic motions is equivalent to the period-doubling bifurcation via Poincare mappings of dynamical systems. The stable and unstable period-m motions can be obtained analytically. Even more, the stable and unstable chaotic motions can be achieved analytically. The methodology presented in this paper can be applied to other nonlinear vibration systems, which is independent of small parameters.
42
Zhao, Yanqing, Kondo H. Adjallah, Alexandre Sava, Lyu Chang, Lichao Xu e Yong Chen. "Digital synchronous decomposition and period-N bifurcation size identification in dynamic systems: application to a milling process". Chaos, Solitons & Fractals 173 (agosto 2023): 113714. http://dx.doi.org/10.1016/j.chaos.2023.113714.
43
Geng, Fengjie, e Hairong Lian. "Bifurcation of Limit Cycles from a Quasi-Homogeneous Degenerate Center". International Journal of Bifurcation and Chaos 25, n. 01 (gennaio 2015): 1550007. http://dx.doi.org/10.1142/s0218127415500078.
Abstract (sommario):
In this paper, we deal with the following differential system [Formula: see text] where p, q are positive integers, and P(x, y), Q(x, y) are real polynomials of degree n, we obtain an upper bound for the maximum number of limit cycles bifurcating from the period annulus of a quasi-homogeneous center, that is (n - 1)p1 + (t + 1)q - 1 + 2rp1q1(q + 3) + 2tqrp1q1, where t = [n/2q] + 2, (p, q) = r(p1, q1), p1 and q1 are coprime.
44
LI, P., Y. Z. LIU, K. L. HU, B. H. WANG e H. J. QUAN. "THE CHAOTIC CONTROL ON THE OCCASIONAL NONLINEAR TIME-DELAYED FEEDBACK". International Journal of Modern Physics B 18, n. 17n19 (30 luglio 2004): 2680–85. http://dx.doi.org/10.1142/s0217979204025907.
Abstract (sommario):
The method of controlling chaos by occasional nonlinear time-delayed feedback is proposed. Through the numerical analysis of bifurcation diagram and Lyapunov exponent, we found that the systematic chaos can be controlled effectively by the nonlinear time-delayed feedback as the form of u(xn,xn-k)=cxnxn-k. Under the proper feedback coefficient c, time-delayed coefficient k and occasional feedback period N, the system could be controlled from chaos to the steady periodic orbit, and also the steady period is the integral multiple of the occasional feedback period N.
45
Atobe, Takashi. "Lagrangian Chaos in the Stokes Flow Between Two Eccentric Rotating Cylinders". International Journal of Bifurcation and Chaos 07, n. 05 (maggio 1997): 1007–23. http://dx.doi.org/10.1142/s0218127497000820.
Abstract (sommario):
Chaotic motion of the fluid particles in the Stokes flow between two eccentric cylinders rotating alternately is investigated numerically, analytically and experimentally. We examine the dependence of the motion of the fluid particles on the eccentricity ε, focusing on an equilibrium point of the Poincaré plot. When the bifurcation of the equilibrium point from the elliptic to the hyperbolic type occurs at ε = εb, the area of the chaotic region takes a maximum around εb. The results from the perturbation analysis show good agreement with the numerical results. The orbital instability of the motion of the fluid particles is also investigated experimentally. The orbital instability is visualized by injected dye in the "return experiment", in which the two cylinders are rotated alternately by N periods in the first half, and then rotated in its time reversal way for N periods in the second half. The dye starting from the regular region of the numerically computed Poincaré plot of particle positions after every period returns well to its initial position even for large N. However, the deviation of the dye starting from the chaotic region of the Poincaré plot from its initial position is large and rapidly increases with N.
46
JING, ZHUJUN, e JIANPING YANG. "COMPLEX DYNAMICS IN PENDULUM EQUATION WITH PARAMETRIC AND EXTERNAL EXCITATIONS II". International Journal of Bifurcation and Chaos 16, n. 10 (ottobre 2006): 3053–78. http://dx.doi.org/10.1142/s0218127406016653.
Abstract (sommario):
This paper (II) is a continuation of "Complex dynamics in pendulum equation with parametric and external excitations (I)." By applying second-order averaging method and Melnikov's method, we obtain the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 1, 2, 4 and cannot prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5–15 by Melnikov's method, where ν is not rational to ω. However, we show the occurrence of chaos in the averaged and original systems under quasi-periodic perturbation for Ω = nω + ∊ν, n = 3, 5 by numerical simulation. The numerical simulations, include the bifurcation diagram of fixed points, bifurcation diagrams in three- and two-dimensional spaces, homoclinic bifurcation surface, maximum Lyapunov exponent, phase portraits, Poincaré map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors, including period-3 orbits in different chaotic regions, interleaving occurrence of chaotic behaviors and quasi-periodic behaviors, a different kind of interior crisis, jumping behavior of quasi-periodic sets, different nice quasi-periodic attractors, nonchaotic attractors and chaotic attractors, coexistence of three quasi-periodic sets, onset of chaos which occurs more than once for a given external frequency or amplitudes, and quasi-periodic route to chaos. We do not find the period-doubling cascade. The dynamical behaviors under quasi-periodic perturbation are different from that of periodic perturbation.
47
Rana, Sarker Md Sohel, e Md Jasim Uddin. "Dynamics and Control of a Discrete Predator–Prey Model with Prey Refuge: Holling Type I Functional Response". Mathematical Problems in Engineering 2023 (27 novembre 2023): 1–14. http://dx.doi.org/10.1155/2023/5537632.
Abstract (sommario):
In this study, we examine the dynamics of a discrete-time predator–prey system with prey refuge. We discuss the stability prerequisite for effective fixed points. The existence criteria for period-doubling (PD) bifurcation and Neimark–Sacker (N–S) bifurcation are derived from the center manifold theorem and bifurcation theory. Examples of numerical simulations that demonstrate the validity of theoretical analysis, as well as complex dynamical behaviors and biological processes, include bifurcation diagrams, maximal Lyapunov exponents, fractal dimensions (FDs), and phase portraits, respectively. From a biological perspective, this suggests that the system can be stabilized into a locally stable coexistence by the tiny integral step size. However, the system might become unstable because of the large integral step size, resulting in richer and more complex dynamics. It has been discovered that the parameter values have a substantial impact on the dynamic behavior of the discrete prey–predator model. Finally, to control the chaotic trajectories that arise in the system, we employ a feedback control technique.
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Shimizu, Toshihiro, e Nozomi Morioka. "Chaos Induced Transition". International Journal of Bifurcation and Chaos 07, n. 04 (aprile 1997): 855–67. http://dx.doi.org/10.1142/s0218127497000650.
Abstract (sommario):
To study the coherent nature of chaos, two models are proposed. Model 1 is a simple nonlinear system [Formula: see text] and Model 2 is a linear harmonic oscillator [Formula: see text], which are driven by a chaotic force f(t). The chaotic force f(t) is defined by [Formula: see text] for nτ < t ≤ (n + 1)τ(n = 0, 1, 2, …), where yn+1 is a chaotic sequence of a map F(y; r) with the bifurcation parameter r: yn+1 = F(yn; r) (-0.5 ≤ yn ≤ 0.5) and ŷn = yn - < y0>. In Model 1 the relaxation process of this system and the τ- and r-dependence of the stationary distribution of x are discussed. It is shown that the small change of the bifurcation parameter r causes the drastic change of the stationary distribution. In Model 2, resonance phenomena are investigated near the period 3 window of the logistic map, in particular, in the intermittent chaos region and the period doubling region. The theoretical results are shown to be in a good agreement with numerical ones, which have been done for the logistic map as F(y; r).
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Stromp, Mark, Alexandra Farkas, Balázs Kretzer, Dénes Száz, András Barta e Gábor Horváth. "How realistic are painted lightnings? Quantitative comparison of the morphology of painted and real lightnings: a psychophysical approach". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, n. 2214 (giugno 2018): 20170859. http://dx.doi.org/10.1098/rspa.2017.0859.
Abstract (sommario):
Inspired by the pioneer work of the nineteenth century photographer, William Nicholson Jennings, we studied quantitatively how realistic painted lightnings are. In order to answer this question, we examined 100 paintings and 400 photographs of lightnings. We used our software package to process and evaluate the morphology of lightnings. Three morphological parameters of the main lightning branch were analysed: (i) number of branches N b , (ii) relative length r , and (iii) number of local maxima (peaks) N p of the turning angle distribution. We concluded: (i) Painted lightnings differ from real ones in N b and N p . (ii) The r -values of painted and real lightnings vary in the same range. (iii) 67 and 22% of the studied painted and real lightnings were non-bifurcating ( N b = 1, meaning only the main branch), the maximum of N b of painted and real lightnings is 11 and 51, respectively, and painted bifurcating lightnings possess mostly 2–4 branches, while real lightnings have mostly 2–10 branches. To understand these findings, we performed two psychophysical experiments with 10 test persons, whose task was to guess N b on photographs of real lightnings which were flashed for short time periods Δ t = 0.5, 0.75 and 1 s (characteristic to lightnings) on a monitor. We obtained that (i) test persons can estimate the number of lightning branches quite correctly if N b ≤ 11. (ii) If N b > 11, its value is strongly underestimated with exponentially increasing difference between the real and estimated numbers. (iii) The estimation is independent of the flashing period Δ t of lightning photos/pictures. (iv) The estimation is more accurate, if skeletonized lightning pictures are flashed, rather than real lightning photos. These findings explain why artists usually illustrate lightnings with branches not larger than 11.
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WU, STEVEN. "DEFECT-LINE DYNAMICS IN OSCILLATORY SPIRAL WAVES". Fluctuation and Noise Letters 06, n. 04 (dicembre 2006): L379—L386. http://dx.doi.org/10.1142/s0219477506003537.
Abstract (sommario):
We study defect-line dynamics in a 2-D spiral-wave pair in the Rössler model for its underlying local dynamics in period-N and chaotic regimes with a single bifurcation parameter κ. We find that a spiral wave pair is always stable across the period-doubling cascade and in the chaotic regime. When N ≥ 2 defect lines appear spontaneously and a loop exchange occurs across the defect line. There exists a "critical point" κ c below and above which the time-averaged total length of defect lines L converges to almost constant but different values L1 and L2. When κ > κ c defect lines show large fluctuations due to creation and annihilation processes.