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Journal articles on the topic 'Attractor analysis'

Author: Grafiati

Published: 28 June 2021

Last updated: 5 February 2022

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1

Li, Fuzhi, Jie Xin, Hongyong Cui, and Peter E. Kloeden. "Local equi-attraction of pullback attractor sections." Journal of Mathematical Analysis and Applications 494, no. 2 (February 2021): 124657. http://dx.doi.org/10.1016/j.jmaa.2020.124657.

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2

Yang, Ting. "Dynamical Analysis on a Finance System with Nonconstant Elasticity of Demand." International Journal of Bifurcation and Chaos 30, no. 10 (August 2020): 2050148. http://dx.doi.org/10.1142/s0218127420501485.

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This paper investigates a finance system with nonconstant elasticity of demand. First, under some conditions, the system has invariant algebraic surfaces and the analytic expressions of the surfaces are given. Furthermore, when the two surfaces coincide and become one surface, the dynamics on the surface are analyzed and a globally stable equilibrium is found. Second, by using the normal form theory, the Hopf bifurcation is studied and the approximate expression and stability of the bifurcating periodic orbit are obtained. Third, the chaotic behaviors are investigated and the route to chaos is period-doubling bifurcations. Moreover, it is found that the system has coexisting attractors, including periodic attractor and periodic attractor, chaotic attractor and chaotic attractor. With the change of parameter, the two chaotic attractors coincide and then a symmetrical chaotic attractor arises.

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3

SATO, TADANOBU, and YOUHEI TANAKA. "MINOR DAMAGE DETECTION USING CHAOTIC EXCITATION AND RECURRENCE ANALYSIS." Journal of Earthquake and Tsunami 05, no. 03 (September 2011): 259–70. http://dx.doi.org/10.1142/s1793431111001054.

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In this paper, we propose a new attractor-based structural damage detection technique using chaotic excitation. Attractor is reconstructed using vibration response data and sensitive to the change of the system dynamics. By comparing the change of attractors from healthy and damaged structures, we detect and localize the damage. We use recurrence analysis to analyze the change of attractor. Numerical example demonstrates the robustness and sensitivity of the proposed method.

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4

Li, Y. Charles, and Hong Yang. "A Mathematical Model of Demand-Supply Dynamics with Collectability and Saturation Factors." International Journal of Bifurcation and Chaos 27, no. 01 (January 2017): 1750016. http://dx.doi.org/10.1142/s021812741750016x.

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We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is not approaching the chaos), instead a periodic attractor (of period-3 under the Poincaré period map) exists near the chaos, and coexists with another periodic attractor (of period-1 under the Poincaré period map) near the market equilibrium. Outside the basins of attraction of the two periodic attractors, the dynamics approaches infinity indicating market irrational exuberance or flash crash. The period-3 attractor represents the product’s market cycle of growth and recession, while period-1 attractor near the market equilibrium represents the regular fluctuation of the product’s market. Thus our model captures more market phenomena besides Marshall’s market equilibrium. When the fluctuation of the determinants of demand and supply is strong enough, a three leaf danger zone exists where the basins of attraction of all attractors intertwine and fractal basin boundaries are formed. Small perturbations in the danger zone can lead to very different attractors. That is, small perturbations in the danger zone can cause the market to experience oscillation near market equilibrium, large growth and recession cycle, and irrational exuberance or flash crash.

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5

Yang, Yingjuan, Guoyuan Qi, Jianbing Hu, and Philippe Faradja. "Finding Method and Analysis of Hidden Chaotic Attractors for Plasma Chaotic System From Physical and Mechanistic Perspectives." International Journal of Bifurcation and Chaos 30, no. 05 (April 2020): 2050072. http://dx.doi.org/10.1142/s0218127420500728.

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A method for finding hidden chaotic attractors in the plasma system is presented. Using the Routh–Hurwitz criterion, the stability distribution associated with two parameters is identified to find the region around the equilibrium points of the stable nodes, stable focus-nodes, saddles and saddle-foci for the purpose of investigating hidden chaos. A physical interpretation is provided of the stability distribution for each type of equilibrium point. The basin of attraction and parameter region of hidden chaos are identified by excluding the self-excited chaotic attractors of all equilibrium points. Homotopy and numerical continuation are also employed to check whether the basin of chaotic attraction intersects with the neighborhood of a saddle equilibrium. Bifurcation analysis, phase portrait analysis, and basins of different dynamical attraction are used as tools to distinguish visually the self-excited chaotic attractor and hidden chaotic attractor. The Casimir power reflects the error power between the dissipative energy and the energy supplied by the whistler field. It explains physically, analytically, and numerically the conditions that generate the different dynamics, such as sinks, periodic orbits, and chaos.

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6

YU, PEI, WEIGUANG YAO, and GUANRON CHEN. "ANALYSIS ON TOPOLOGICAL PROPERTIES OF THE LORENZ AND THE CHEN ATTRACTORS USING GCM." International Journal of Bifurcation and Chaos 17, no. 08 (August 2007): 2791–96. http://dx.doi.org/10.1142/s0218127407018762.

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This letter reports a study on some topological properties of chaos using a generalized competitive mode (GCM). The Lorenz system and the Chen system are used as examples for comparison. It is shown that for typical parameter values used in the two systems, the Lorenz attractor has one pair of GCMs in competition, while the Chen attractor has two pairs of GCMs in competition. This explains why the two attractors are topologically different, and furthermore indicates that the Chen attractor is more complex than the Lorenz attractor from the dynamics point of view.

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7

The Anh, Cung, and Nguyen Dinh Binh. "Attractors for Nonautonomous Parabolic Equations without Uniqueness." International Journal of Differential Equations 2010 (2010): 1–17. http://dx.doi.org/10.1155/2010/103510.

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Using the theory of uniform global attractors of multivalued semiprocesses, we prove the existence of a uniform global attractor for a nonautonomous semilinear degenerate parabolic equation in which the conditions imposed on the nonlinearity provide the global existence of a weak solution, but not uniqueness. The Kneser property of solutions is also studied, and as a result we obtain the connectedness of the uniform global attractor.

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8

REGA, G., and A. SALVATORI. "BIFURCATION STRUCTURE AT 1/3-SUBHARMONIC RESONANCE IN AN ASYMMETRIC NONLINEAR ELASTIC OSCILLATOR." International Journal of Bifurcation and Chaos 06, no. 08 (August 1996): 1529–46. http://dx.doi.org/10.1142/s0218127496000904.

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The attractor-basin bifurcation structure in an asymmetric nonlinear oscillator representative of the planar finite forced dynamics of elastic structural systems with initial curvature is studied at the 1/3-subharmonic resonance regime. Local and global analyses are made by means of different computational tools to obtain frequency-response curves of coexisting regular solutions, bifurcation diagrams ensuing from different sets of initial conditions, manifolds structure of direct and inverse saddles corresponding to unstable periodic solutions, basins of attraction at different values of the control parameter. Deep insight into the global dynamics of the system and its evolution is achieved through the analysis of synthetic attractor-basin-manifold phase portraits. The topological mechanisms which entail onset and disappearance of various attractors, and the main and secondary evolutions to chaos, are identified. Special attention is devoted to the analysis of sudden bifurcational events characterizing the system global dynamics, associated with the topological behavior of the invariant manifolds of several direct and inverse saddles. Features of basin metamorphosis, attractor-basin accessibility, and window occurrence are examined. The approach followed, consisting in combined bifurcation analysis of the attractor-basin structure and of the manifold structure, is thought to be useful for a variety of dynamical systems.

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9

Danca, Marius-F., Michal Fĕckan, Nikolay Kuznetsov, and Guanrong Chen. "Attractor as a convex combination of a set of attractors." Communications in Nonlinear Science and Numerical Simulation 96 (May 2021): 105721. http://dx.doi.org/10.1016/j.cnsns.2021.105721.

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10

Vollmer, J., J. Peinke, and A. Okniński. "Dweiltime Analysis of Symmetry-Breaking Dynamical Systems." Zeitschrift für Naturforschung A 50, no. 12 (December 1, 1995): 1117–22. http://dx.doi.org/10.1515/zna-1995-1209.

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Abstract Dweiltime analysis is known to characterize saddles giving rise to chaotic scattering. In the present paper it is used to characterize the dependence on initial conditions of the attractor approached by a trajectory in dissipative systems described by one-dimensional, noninvertible mappings which show symmetry breaking. There may be symmetry-related attractors in these systems, and which attractor is approached may depend sensitively on the initial conditions. Dwell-time analysis is useful in this context because it allows to visualize in another way the repellers on the basin boundary which cause this sensitive dependence.

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11

Samprogna, Rodrigo, and Jacson Simsen. "A selected pullback attractor." Journal of Mathematical Analysis and Applications 468, no. 1 (December 2018): 364–75. http://dx.doi.org/10.1016/j.jmaa.2018.08.027.

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12

XU, WEI, QUN HE, TONG FANG, and HAIWU RONG. "GLOBAL ANALYSIS OF STOCHASTIC BIFURCATION IN DUFFING SYSTEM." International Journal of Bifurcation and Chaos 13, no. 10 (October 2003): 3115–23. http://dx.doi.org/10.1142/s021812740300848x.

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Stochastic bifurcation of a Duffing system subject to a combination of a deterministic harmonic excitation and a white noise excitation is studied in detail by the generalized cell mapping method using digraph. It is found that under certain conditions there exist two stable invariant sets in the phase space, associated with the randomly perturbed steady-state motions, which may be called stochastic attractors. Each attractor owns its attractive basin, and the attractive basins are separated by boundaries. Along with attractors there also exists an unstable invariant set, which might be called a stochastic saddle as well, and stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motions, or to purely deterministic motions. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence and evolution of stochastic bifurcation can be explored clearly and vividly.

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13

Cholewa, Jan W., and Tomasz Dlotko. "Global Attractor for Sectorial Evolutionary Equation." Journal of Differential Equations 125, no. 1 (February 1996): 27–39. http://dx.doi.org/10.1006/jdeq.1996.0023.

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14

Zhao, Caidi, Yongsheng Li, and Shengfan Zhou. "Regularity of trajectory attractor and upper semicontinuity of global attractor for a 2D non-Newtonian fluid." Journal of Differential Equations 247, no. 8 (October 2009): 2331–63. http://dx.doi.org/10.1016/j.jde.2009.07.031.

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15

Cui, Li, Wenhui Luo, and Qingli Ou. "Analysis of basins of attraction of new coupled hidden attractor system." Chaos, Solitons & Fractals 146 (May 2021): 110913. http://dx.doi.org/10.1016/j.chaos.2021.110913.

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16

Zhao, Caidi, Shengfan Zhou, and Yongsheng Li. "Trajectory attractor and global attractor for a two-dimensional incompressible non-Newtonian fluid." Journal of Mathematical Analysis and Applications 325, no. 2 (January 2007): 1350–62. http://dx.doi.org/10.1016/j.jmaa.2006.02.069.

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17

Miller, Paul. "Analysis of Spike Statistics in Neuronal Systems with Continuous Attractors or Multiple, Discrete Attractor States." Neural Computation 18, no. 6 (June 2006): 1268–317. http://dx.doi.org/10.1162/neco.2006.18.6.1268.

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Attractor networks are likely to underlie working memory and integrator circuits in the brain. It is unknown whether continuous quantities are stored in an analog manner or discretized and stored in a set of discrete attractors. In order to investigate the important issue of how to differentiate the two systems, here we compare the neuronal spiking activity that arises from a continuous (line) attractor with that from a series of discrete attractors. Stochastic fluctuations cause the position of the system along its continuous attractor to vary as a random walk, whereas in a discrete attractor, noise causes spontaneous transitions to occur between discrete states at random intervals. We calculate the statistics of spike trains of neurons firing as a Poisson process with rates that vary according to the underlying attractor network. Since individual neurons fire spikes probabilistically and since the state of the network as a whole drifts randomly, the spike trains of individual neurons follow a doubly stochastic (Poisson) point process. We compare the series of spike trains from the two systems using the autocorrelation function, Fano factor, and interspike interval (ISI) distribution. Although the variation in rate can be dramatically different, especially for short time intervals, surprisingly both the autocorrelation functions and Fano factors are identical, given appropriate scaling of the noise terms. Since the range of firing rates is limited in neurons, we also investigate systems for which the variation in rate is bounded by either rigid limits or because of leak to a single attractor state, such as the Ornstein-Uhlenbeck process. In these cases, the time dependence of the variance in rate can be different between discrete and continuous systems, so that in principle, these processes can be distinguished using second-order spike statistics.

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18

Varadharajan, Manisekaran, Prakash Duraisamy, and Anitha Karthikeyan. "Route to Chaos and Bistability Analysis of Quasi-Periodically Excited Three-Leg Supporter with Shape Memory Alloy." Complexity 2020 (September 19, 2020): 1–10. http://dx.doi.org/10.1155/2020/7672303.

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In this paper, the effect of quasi-periodic excitation on a three-leg supporter configured with shape memory alloy is investigated. We derived the equation of motion for the system using the supporter configuration and polynomial constitutive model of the shape memory alloys (SMAs) based on Falk model. Two sets of parameters and symmetric initial conditions are used to analyze the system. The system responded with a chaotic attractor and a strange nonchaotic attractor. Coexistence of these attractors is studied and discussed with corresponding phase portrait, bifurcation plot, and cross section of basin of attraction. We confirm the quasi-periodic excitation results with generation of strange nonchaotic attractors as discussed in the literature. The special properties like symmetricity and bistability are revealed and the parameter ranges of existence of such behaviors are discussed. The system is analyzed for different phases and the existence of bistability in martensite phase and transition phase is explained. While the system enters into austenite phase, the bistability behavior vanishes. The results provide insight knowledge into dynamical response of a quasi-periodically excited SMA leg support system and will be useful for design improvements and controller design.

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19

Jia, Xiaoyao, and Xiaoquan Ding. "Random Attractors for Stochastic Retarded 2D-Navier-Stokes Equations with Additive Noise." Journal of Function Spaces 2018 (September 13, 2018): 1–14. http://dx.doi.org/10.1155/2018/3105239.

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In this paper, the existence and the upper semicontinuity of a pullback attractor for stochastic retarded 2D-Navier-Stokes equation on a bounded domain are obtained. We first transform the stochastic equation into a random equation and then obtain the existence of a random attractor for random equation. Then conjugation relation between two random dynamical systems implies the existence of a random attractor for the stochastic equation. At last, we get the upper semicontinuity of random attractor.

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20

Savin, A., B. Silvi, and F. Colonna. "Topological analysis of the electron localization function applied to delocalized bonds." Canadian Journal of Chemistry 74, no. 6 (June 1, 1996): 1088–96. http://dx.doi.org/10.1139/v96-122.

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What is a local viewpoint of delocalized bonds? We try to provide an answer to this paradoxical question by investigating representative conjugated organic molecules (namely, allyl cation, trans-butadiene, and benzene) together with reference nonconjugated systems (ethylene and propene) by means of topological analysis of the electron localization function ELF. The valence attractors of the ELF gradient field are classified according to their synaptic order (i.e., connections with core attractors). The basin populations [Formula: see text] (i.e., the integrated density over the attractor basins) and their standard deviation, σ, have been calculated and are discussed. The basin populations and their relative fluctuations, defined as [Formula: see text] are sensitive criteria of delocalization. In the case of well-localized C—C or C=C bonds, λ ~0.4, whereas for delocalized bonds λ increases to about 0.5. Another criterion of delocalization is provided by the basin hierarchy, which is defined from the reduction of the localization domains. For most systems, delocalization occurs not only for neighboring carbon-carbon disynaptic attractor basins, but also for nearest neighbor disynaptic protonated attractor basins. Key words: electron localization function, topological analysis, delocalization, population analysis.

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21

CHUA, LEON O., VALERY I. SBITNEV, and SOOK YOON. "A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART VI: FROM TIME-REVERSIBLE ATTRACTORS TO THE ARROW OF TIME." International Journal of Bifurcation and Chaos 16, no. 05 (May 2006): 1097–373. http://dx.doi.org/10.1142/s0218127406015544.

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This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,. [Formula: see text] or its inverse map. These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

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22

Wang, Zhonglin, Shijian Cang, Zenghui Wang, Wei Xue, and Zengqiang Chen. "A Strange Double-Deck Butterfly Chaotic Attractor from a Permanent Magnet Synchronous Motor with Smooth Air Gap: Numerical Analysis and Experimental Observation." Abstract and Applied Analysis 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/495126.

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A permanent magnet synchronous motor (PMSM) model with smooth air gap and an exogenous periodic input is introduced and analyzed in this paper. With a simple mathematical transformation, a new nonautonomous Lorenz-like system is derived from this PMSM model, and this new three-dimensional system can display the complicated dynamics such as the chaotic attractor and the multiperiodic orbits by adjusting the frequency and amplitude of the exogenous periodic inputs. Moreover, this new system shows a double-deck chaotic attractor that is completely different from the four-wing chaotic attractors on topological structures, although the phase portrait shapes of the new attractor and the four-wing chaotic attractors are similar. The exotic phenomenon has been well demonstrated and investigated by numerical simulations, bifurcation analysis, and electronic circuit implementation.

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23

White, D. "The major league baseball attractor." Journal of Interdisciplinary Mathematics 10, no. 2 (April 2007): 229–44. http://dx.doi.org/10.1080/09720502.2007.10700489.

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Li, Chunqiu, Desheng Li, and Jintao Wang. "A remark on attractor bifurcation." Dynamics of Partial Differential Equations 18, no. 2 (2021): 157–72. http://dx.doi.org/10.4310/dpde.2021.v18.n2.a4.

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25

GRISOUARD, NICOLAS, CHANTAL STAQUET, and IVANE PAIRAUD. "Numerical simulation of a two-dimensional internal wave attractor." Journal of Fluid Mechanics 614 (October 16, 2008): 1–14. http://dx.doi.org/10.1017/s002211200800325x.

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Internal (gravity) wave attractors may form in closed containers with boundaries non-parallel and non-normal to the gravity vector. Such attractors have been studied from a theoretical point of view, in laboratory experiments and using linear numerical computations. In the present paper two-dimensional numerical simulations of an internal wave attractor are reported, based upon the nonlinear and non-hydrostatic MIT-gcm numerical code. We first reproduce the laboratory experiment of a wave attractor performed by Hazewinkel et al. (J. Fluid Mech. Vol. 598, 2008 p. 373) and obtain very good agreement with the experimental data. We next propose simple ideas to model the thickness of the attractor. The model predicts that the thickness should scale as the 1/3 power of the non-dimensional parameter measuring the ratio of viscous to buoyancy effects. When the attractor is strongly focusing, the thickness should also scale as the 1/3 power of the spatial coordinate along the attractor. Analysis of the numerical data for two different attractors yields values of the exponent close to 1/3, within 30%. Finally, we study nonlinear effects induced by the attractor.

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QI, GUOYUAN, GUANRONG CHEN, SHAOWEN LI, and YUHUI ZHANG. "FOUR-WING ATTRACTORS: FROM PSEUDO TO REAL." International Journal of Bifurcation and Chaos 16, no. 04 (April 2006): 859–85. http://dx.doi.org/10.1142/s0218127406015180.

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Some basic dynamical behaviors and the compound structure of a new four-dimensional autonomous chaotic system with cubic nonlinearities are investigated. A four-wing chaotic attractor is observed numerically. This attractor, however, is shown to be an numerical artifact by further theoretical analysis and analog circuit experiment. The observed four-wing attractor actually has two coexisting (upper and lower) attractors, which appear simultaneously and are located arbitrarily closely in the phase space. By introducing a simple linear state-feedback control term, some symmetries of the system and similarities of the linearized characteristics can be destroyed, thereby leading to the appearance of some diagonal and anti-diagonal periodic orbits, through which the upper and lower attractors can indeed be merged together to form a truly single four-wing chaotic attractor. This four-wing attractor is real; it is further confirmed analytically, numerically, as well as electronically in the paper. Moreover, by introducing a sign-switching control function, the system orbit can be manipulated so as to switch between two equilibria or among four equilibria, generating two one-side double-wing attractors, which can also be merged to yield a real four-wing attractor.

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Khalaf, Abdul Jalil M., Tomasz Kapitaniak, Karthikeyan Rajagopal, Ahmed Alsaedi, Tasawar Hayat, and Viet–Thanh Pham. "A new three-dimensional chaotic flow with one stable equilibrium: dynamical properties and complexity analysis." Open Physics 16, no. 1 (May 24, 2018): 260–65. http://dx.doi.org/10.1515/phys-2018-0037.

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Abstract This paper proposes a new three-dimensional chaotic flow with one stable equilibrium. Dynamical properties of this system are investigated. The system has a chaotic attractor coexisting with a stable equilibrium. Thus the chaotic attractor is hidden. Basin of attractions shows the tangle of different attractors. Also, some complexity measures of the system such as Lyapunov exponent and entropy will are analyzed. We show that the Kolmogorov-Sinai Entropy shows more accurate results in comparison with Shanon Entropy.

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Danca, Marius-F., Paul Bourke, and Nikolay Kuznetsov. "Graphical Structure of Attraction Basins of Hidden Chaotic Attractors: The Rabinovich–Fabrikant System." International Journal of Bifurcation and Chaos 29, no. 01 (January 2019): 1930001. http://dx.doi.org/10.1142/s0218127419300015.

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The attraction basin of hidden attractors does not intersect with small neighborhoods of any equilibrium point. To the best of our knowledge this property has not been explored using realtime interactive three-dimensions graphics. Aided by advanced computer graphic analysis, in this paper, we explore this characteristic of a particular nonlinear system with very rich and unusual dynamics, the Rabinovich–Fabrikant system. It is shown that there exists a neighborhood of one of the unstable equilibria within which the initial conditions do not lead to the considered hidden chaotic attractor, but to one of the stable equilibria or are divergent. The trajectories starting from any neighborhood of the other unstable equilibria are attracted either by the stable equilibria, or are divergent.

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Rosalie, Martin, and Christophe Letellier. "Toward a General Procedure for Extracting Templates from Chaotic Attractors Bounded by High Genus Torus." International Journal of Bifurcation and Chaos 24, no. 04 (April 2014): 1450045. http://dx.doi.org/10.1142/s021812741450045x.

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The topological analysis of chaotic attractor by means of template is rather well established for simple attractors as solution to the Rössler system. Lorenz-like attractors are already slightly more complicated because they are bounded by a genus-3 bounding torus, implying the necessity to use a two-component Poincaré section. In this paper, we enriched the concept of linking matrix to correctly describe an algebraic template for an attractor with (g - 1) components of Poincaré section and whose bounding torus has g interior holes aligned. An example with g = 5 — a multispiral attractor — is explicitly treated.

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SHUKLA, RAVI PRAKASH, SANDIPAN MUKHERJEE, and ASHOK KUMAR MITTAL. "COMPARISON OF GENERALIZED COMPETITIVE MODES AND RETURN MAPS FOR CHARACTERIZING DIFFERENT TYPES OF CHAOTIC ATTRACTORS IN CHEN SYSTEM." International Journal of Bifurcation and Chaos 20, no. 03 (March 2010): 735–48. http://dx.doi.org/10.1142/s0218127410026022.

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The Chen system of equations exhibits Lorenz, Transition, Chen and Transverse 8 type of chaotic attractors depending on the system parameters. Some authors have proposed a generalized competitive mode (GCM) technique to explain the topological difference between the Lorenz attractor and the Chen attractor. In this paper, we show a range of parameter values for which the nature of the topological attractor for the Chen system is not in accordance with that expected from GCM analysis. Instead, we find that return maps can be used to characterize the transition between different types of attractors more reliably.

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Wan, Li, and Qinghua Zhou. "Attractor Analysis of Cohen–Grossberg Neural Networks with Multiple Time-Varying Delays." International Journal of Bifurcation and Chaos 31, no. 02 (February 2021): 2150022. http://dx.doi.org/10.1142/s021812742150022x.

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This paper investigates the pullback attractor of Cohen–Grossberg neural networks with multiple time-varying delays. Compared with the existing references, the networks considered here are more general and cannot be expressed in the vector-matrix form due to multiple time-varying delays. After constructing a proper Lyapunov–Krasovskii functional and eliminating the terms involving multiple time-varying delays, two sets of new sufficient criteria on the existence of the pullback attractor are derived based on the theory of pullback attractors. In the end, two examples are given to demonstrate the effectiveness of our theoretical results.

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Bianchini, M., S. Fanelli, M. Gori, and M. Maggini. "Terminal attractor algorithms: A critical analysis." Neurocomputing 15, no. 1 (April 1997): 3–13. http://dx.doi.org/10.1016/s0925-2312(96)00045-8.

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Balibrea, Francisco, M. Victoria Caballero, and Lourdes Molera. "Recurrence quantification analysis in Liu’s attractor." Chaos, Solitons & Fractals 36, no. 3 (May 2008): 664–70. http://dx.doi.org/10.1016/j.chaos.2006.06.107.

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LÜ, JINHU, GUANRONG CHEN, and SUOCHUN ZHANG. "DYNAMICAL ANALYSIS OF A NEW CHAOTIC ATTRACTOR." International Journal of Bifurcation and Chaos 12, no. 05 (May 2002): 1001–15. http://dx.doi.org/10.1142/s0218127402004851.

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Dynamical behaviors of a new chaotic attractor is investigated in this paper. Some basic properties, bifurcations, routes to chaos, and periodic windows of the new system are studied either analytically or numerically. Meanwhile, the transition between the Lorenz attractor and Chen's attractor through the new system is explored.

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Li, Yin, Ruiying Wei, and Donghong Cai. "Hausdorff Dimension of a Random Attractor for Stochastic Boussinesq Equations with Double Multiplicative White Noises." Journal of Function Spaces 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/1832840.

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This paper investigates the existence of random attractor for stochastic Boussinesq equations driven by multiplicative white noises in both the velocity and temperature equations and estimates the Hausdorff dimension of the random attractor.

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Morales, C. A. "Lorenz attractor through saddle-node bifurcations." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 13, no. 5 (September 1996): 589–617. http://dx.doi.org/10.1016/s0294-1449(16)30116-0.

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Wang, Jinquan, Jian Cheng, Chao Zhang, and Xiaojun Li. "Cardioprotection Effects of Sevoflurane by Regulating the Pathway of Neuroactive Ligand-Receptor Interaction in Patients Undergoing Coronary Artery Bypass Graft Surgery." Computational and Mathematical Methods in Medicine 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/3618213.

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This study was designed to identify attractor modules and further reveal the potential biological processes involving in sevoflurane-induced anesthesia in patients treated with coronary artery bypass graft (CABG) surgery. Microarray profile data (ID: E-GEOD-4386) on atrial samples obtained from patients receiving anesthetic gas sevoflurane prior to and following CABG procedure were downloaded from EMBL-EBI database for further analysis. Protein-protein interaction (PPI) networks of baseline and sevoflurane groups were inferred and reweighted according to Spearman correlation coefficient (SCC), followed by systematic modules inference using clique-merging approach. Subsequently, attract method was utilized to explore attractor modules. Finally, pathway enrichment analyses for genes in the attractor modules were implemented to illuminate the biological processes in sevoflurane group. Using clique-merging approach, 27 and 36 modules were obtained from the PPI networks of baseline and sevoflurane-treated samples, respectively. By comparing with the baseline condition, 5 module pairs with the same gene composition were identified. Subsequently, 1 out of 5 modules was identified as an attractor based on attract method. Additionally, pathway analysis indicated that genes in the attractor module were associated with neuroactive ligand-receptor interaction. Accordingly, sevoflurane might exert important functions in cardioprotection in patients following CABG, partially through regulating the pathway of neuroactive ligand-receptor interaction.

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Li, Qin, Anders Wennborg, Erik Aurell, Erez Dekel, Jie-Zhi Zou, Yuting Xu, Sui Huang, and Ingemar Ernberg. "Dynamics inside the cancer cell attractor reveal cell heterogeneity, limits of stability, and escape." Proceedings of the National Academy of Sciences 113, no. 10 (February 29, 2016): 2672–77. http://dx.doi.org/10.1073/pnas.1519210113.

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The observed intercellular heterogeneity within a clonal cell population can be mapped as dynamical states clustered around an attractor point in gene expression space, owing to a balance between homeostatic forces and stochastic fluctuations. These dynamics have led to the cancer cell attractor conceptual model, with implications for both carcinogenesis and new therapeutic concepts. Immortalized and malignant EBV-carrying B-cell lines were used to explore this model and characterize the detailed structure of cell attractors. Any subpopulation selected from a population of cells repopulated the whole original basin of attraction within days to weeks. Cells at the basin edges were unstable and prone to apoptosis. Cells continuously changed states within their own attractor, thus driving the repopulation, as shown by fluorescent dye tracing. Perturbations of key regulatory genes induced a jump to a nearby attractor. Using the Fokker–Planck equation, this cell population behavior could be described as two virtual, opposing influences on the cells: one attracting toward the center and the other promoting diffusion in state space (noise). Transcriptome analysis suggests that these forces result from high-dimensional dynamics of the gene regulatory network. We propose that they can be generalized to all cancer cell populations and represent intrinsic behaviors of tumors, offering a previously unidentified characteristic for studying cancer.

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Karimi, Sohrab, and F. H. Ghane. "Analysis of Coexistence and Extinction in a Two-Species Competition Model." International Journal of Bifurcation and Chaos 30, no. 16 (December 28, 2020): 2050248. http://dx.doi.org/10.1142/s021812742050248x.

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We study a competition model of two competing species in population biology having exponential and rational growth functions described by Alexander et al. [1992]. They observed that, for some choice of parameters, the competition model has a chaotic attractor [Formula: see text] for which the basin of attraction is riddled. Here, we give a detailed analysis to illustrate what happens when the normal parameter in this model changes. In fact, by varying the normal parameter, we discuss how the geometry of the basin of attraction of [Formula: see text], the region of coexistence or extinction, changes and illustrate the transitions between the set [Formula: see text] being an asymptotically stable attractor (extinction of rational species), a locally riddled basin attractor and a normally repelling chaotic saddle (extinction of exponential species). Additionally, we show that the riddling and the blowout bifurcation occur. Numerical simulations are presented graphically to confirm the validity of our results. In particular, we verify the occurrence of synchronization for some values of parameters. Finally, we apply the uncertainty exponent and the stability index to quantify the degree of riddling basin. Our observation indicates that the stability index is positive for Lebesgue for almost all points whenever the riddling occurs.

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Boling, Guo, and Li Yongsheng. "Attractor for Dissipative Klein–Gordon–Schrödinger Equations inR3." Journal of Differential Equations 136, no. 2 (May 1997): 356–77. http://dx.doi.org/10.1006/jdeq.1996.3242.

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Feng, Jianfeng, and David Brown. "Fixed-Point Attractor Analysis for a Class of Neurodynamics." Neural Computation 10, no. 1 (January 1, 1998): 189–213. http://dx.doi.org/10.1162/089976698300017944.

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Nearly all models in neural networks start from the assumption that the input-output characteristic is a sigmoidal function. On parameter space, we present a systematic and feasible method for analyzing the whole spectrum of attractors—all-saturated, all-but-one-saturated, all-but-twosaturated, and so on—of a neurodynamical system with a saturated sigmoidal function as its input-output characteristic. We present an argument that claims, under a mild condition, that only all-saturated or all but-one-saturated attractors are observable for the neurodynamics. For any given all-saturated configuration [Formula: see text] (all-but-one-saturated configuration [Formula: see text]) the article shows how to construct an exact parameter region R([Formula: see text])([Formula: see text]([Formula: see text])) such that if and only if the parameters fall within R([Formula: see text])([Formula: see text]([Formula: see text])), then [Formula: see text]([Formula: see text]) is an attractor (a fixed point) of the dynamics. The parameter region for an all-saturated fixed-point attractor is independent of the specific choice of a saturated sigmoidal function, whereas for an all-but-one-saturated fixed point, it is sensitive to the input-output characteristic. Based on a similar idea, the role of weight normalization realized by a saturated sigmoidal function in competitive learning is discussed. A necessary and sufficient condition is provided to distinguish two kinds of competitive learning: stable competitive learning with the weight vectors representing extremes of input space and being fixed-point attractors, and unstable competitive learning. We apply our results to Linsker's model and (using extreme value theory in statistics) the Hopfield model and obtain some novel results on these two models.

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Vahedi, Shahed, and Mohd Salmi Md Noorani. "Analysis of a New Quadratic 3D Chaotic Attractor." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/540769.

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A new three-dimensional chaotic system is introduced. Basic properties of this system show that its corresponding attractor is topologically different from some well-known systems. Next, detailed information on dynamic of this system is obtained numerically by means of Lyapunov exponents spectrum, bifurcation diagrams, and 0-1 chaos indicator test. We finally prove existence of this chaotic attractor theoretically using Shil’nikov theorem and undetermined coefficient method.

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Ma, Yian, Qijun Tan, Ruoshi Yuan, Bo Yuan, and Ping Ao. "Potential Function in a Continuous Dissipative Chaotic System: Decomposition Scheme and Role of Strange Attractor." International Journal of Bifurcation and Chaos 24, no. 02 (February 2014): 1450015. http://dx.doi.org/10.1142/s0218127414500151.

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We demonstrate, first in literature, that potential functions can be constructed in a continuous dissipative chaotic system and can be used to reveal its dynamical properties. To attain this aim, a Lorenz-like system is proposed and rigorously proved chaotic for exemplified analysis. We explicitly construct a potential function monotonically decreasing along the system's dynamics, revealing the structure of the chaotic strange attractor. The potential function is not unique for a deterministic system. We also decompose the dynamical system corresponding to a curl-free structure and a divergence-free structure, explaining for the different origins of chaotic attractor and strange attractor. Consequently, reasons for the existence of both chaotic nonstrange attractors and nonchaotic strange attractors are discussed within current decomposition framework.

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LI, CHANGPIN, and WEIHUA DENG. "SCALING CHEN'S ATTRACTOR." Modern Physics Letters B 20, no. 11 (May 10, 2006): 633–39. http://dx.doi.org/10.1142/s0217984906010913.

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In this paper, Chen's attractor is scaled via the one-way coupling approach and the adaptive control approach. The proposed method is theoretically analyzed and numerically studied. And the numerical simulations are in line with the theoretical analysis.

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Xiong, Tianhong, Yipin Lv, and Wenjun Yi. "Analysis on Multistable Motion Characteristics of Supercavitating Vehicle." Shock and Vibration 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/9712687.

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Due to complex underwater environment, when the initial condition of launching is subjected to low external disturbance, the motion trace of a supercavitating vehicle might display many different motion states during underwater navigation. With the aim of addressing this problem, based on the dynamic map, in the present work the multistable phenomena of attractor coexistence of the supercavitating vehicle system under various initial conditions were analyzed and the initial condition effects on the multistable motion characteristics were investigated through the domains of attraction, time, and frequency. The results demonstrated that, unlike the ordinary dynamic systems, a supercavitating vehicle demonstrates multistable phenomena, such as the coexistence of the stable equilibrium point and the limit cycle and the coexistence of the limit cycle and the chaotic attractor, along with the coexistence of diversified limit cycles; under fixed system parameters, as the initial condition of launching varied, the vehicle displayed various motion states; in engineering practices, the initial condition of launching could be adjusted according to the domain of attraction, in order for the vehicle motion stability to be enhanced.

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Ju, Xuewei, Hongli Wang, Desheng Li, and Jinqiao Duan. "Global Mild Solutions and Attractors for Stochastic Viscous Cahn-Hilliard Equation." Abstract and Applied Analysis 2011 (2011): 1–22. http://dx.doi.org/10.1155/2011/670786.

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This paper is devoted to the study of mild solutions for the initial and boundary value problem of stochastic viscous Cahn-Hilliard equation driven by white noise. Under reasonable assumptions we first prove the existence and uniqueness result. Then, we show that the existence of a stochastic global attractor which pullback attracts each bounded set in appropriate phase spaces.

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Funabashi, Masatoshi. "Synthetic Modeling of Autonomous Learning with a Chaotic Neural Network." International Journal of Bifurcation and Chaos 25, no. 04 (April 2015): 1550054. http://dx.doi.org/10.1142/s0218127415500546.

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We investigate the possible role of intermittent chaotic dynamics called chaotic itinerancy, in interaction with nonsupervised learnings that reinforce and weaken the neural connection depending on the dynamics itself. We first performed hierarchical stability analysis of the Chaotic Neural Network model (CNN) according to the structure of invariant subspaces. Irregular transition between two attractor ruins with positive maximum Lyapunov exponent was triggered by the blowout bifurcation of the attractor spaces, and was associated with riddled basins structure. We secondly modeled two autonomous learnings, Hebbian learning and spike-timing-dependent plasticity (STDP) rule, and simulated the effect on the chaotic itinerancy state of CNN. Hebbian learning increased the residence time on attractor ruins, and produced novel attractors in the minimum higher-dimensional subspace. It also augmented the neuronal synchrony and established the uniform modularity in chaotic itinerancy. STDP rule reduced the residence time on attractor ruins, and brought a wide range of periodicity in emerged attractors, possibly including strange attractors. Both learning rules selectively destroyed and preserved the specific invariant subspaces, depending on the neuron synchrony of the subspace where the orbits are situated. Computational rationale of the autonomous learning is discussed in connectionist perspective.

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Jacob, Rinku, K. P. Harikrishnan, R. Misra, and G. Ambika. "Weighted recurrence networks for the analysis of time-series data." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2221 (January 2019): 20180256. http://dx.doi.org/10.1098/rspa.2018.0256.

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Recurrence networks (RNs) have become very popular tools for the nonlinear analysis of time-series data. They are unweighted and undirected complex networks constructed with specific criteria from time series. In this work, we propose a method to construct a ‘weighted recurrence network’ from a time series and show that it can reveal useful information regarding the structure of a chaotic attractor which the usual unweighted RN cannot provide. Especially, a network measure, the node strength distribution, from every chaotic attractor follows a power law (with exponential cut off at the tail) with an index characteristic to the fractal structure of the attractor. This provides a new class among complex networks to which networks from all standard chaotic attractors are found to belong. Two other prominent network measures, clustering coefficient and characteristic path length, are generalized and their utility in discriminating chaotic dynamics from noise is highlighted. As an application of the proposed measure, we present an analysis of variable star light curves whose behaviour has been reported to be strange non-chaotic in a recent study. Our numerical results indicate that the weighted recurrence network and the associated measures can become potentially important tools for the analysis of short and noisy time series from the real world.

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CUTLAND, NIGEL J., and H. JEROME KEISLER. "ATTRACTORS AND NEO-ATTRACTORS FOR 3D STOCHASTIC NAVIER–STOKES EQUATIONS." Stochastics and Dynamics 05, no. 04 (December 2005): 487–533. http://dx.doi.org/10.1142/s0219493705001559.

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In [14] nonstandard analysis was used to construct a (standard) global attractor for the 3D stochastic Navier–Stokes equations with general multiplicative noise, living on a Loeb space, using Sell's approach [26]. The attractor had somewhat ad hoc attracting and compactness properties. We strengthen this result by showing that the attractor has stronger properties making it a neo-attractor — a notion introduced here that arises naturally from the Keisler–Fajardo theory of neometric spaces [18]. To set this result in context we first survey the use of Loeb space and nonstandard techniques in the study of attractors, with special emphasis on results obtained for the Navier–Stokes equations both deterministic and stochastic, showing that such methods are well-suited to this enterprise.

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He, Jian-Jun, and Bang-Cheng Lai. "A novel 4D chaotic system with multistability: Dynamical analysis, circuit implementation, control design." Modern Physics Letters B 33, no. 21 (July 30, 2019): 1950240. http://dx.doi.org/10.1142/s0217984919502403.

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The purpose of this work is to introduce a novel 4D chaotic system and investigate its multistability. The novel system has an unstable origin and two stable symmetrical hyperbolic equilibria. When the parameter increases across a critical value, the equilibria lose their stability and double Hopf bifurcations occur with the appearance of limit cycles. A pair of point, periodic, chaotic attractors are observed in the system from different initial values for given parameters. The chaos of the system is yielded via period-doubling bifurcation. A double-scroll chaotic attractor is numerically observed as well. By using the electronic circuit, the chaotic attractor of the system is realized. The control problem of the system is reported. An effective controller is designed to stabilize the system.

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