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Journal articles on the topic 'Piecewise-smooth dynamics'

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Published: 11 March 2023

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1

ZHUSUBALIYEV, ZHANYBAI T., EVGENIY A. SOUKHOTERIN, and ERIK MOSEKILDE. "BORDER-COLLISION BIFURCATIONS AND CHAOTIC OSCILLATIONS IN A PIECEWISE-SMOOTH DYNAMICAL SYSTEM." International Journal of Bifurcation and Chaos 11, no. 12 (December 2001): 2977–3001. http://dx.doi.org/10.1142/s0218127401003991.

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Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.

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2

Li, Shuangbao, Wei Zhang, and Yuxin Hao. "Melnikov-Type Method for a Class of Discontinuous Planar Systems and Applications." International Journal of Bifurcation and Chaos 24, no. 02 (February 2014): 1450022. http://dx.doi.org/10.1142/s0218127414500229.

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In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed piecewise smooth planar system. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise smooth homoclinic solution transversally crossing the switching manifold. The Melnikov-type function is explicitly derived by using the Hamiltonian function to measure the distance of the perturbed stable and unstable manifolds. Finally, we apply the obtained results to study the chaotic dynamics of a concrete piecewise smooth system.

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3

Kumar, Aloke, Soumitro Banerjee, and Daniel P. Lathrop. "Dynamics of a piecewise smooth map with singularity." Physics Letters A 337, no. 1-2 (March 2005): 87–92. http://dx.doi.org/10.1016/j.physleta.2005.01.046.

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4

Novaes, Douglas D., and Mike R. Jeffrey. "Regularization of hidden dynamics in piecewise smooth flows." Journal of Differential Equations 259, no. 9 (November 2015): 4615–33. http://dx.doi.org/10.1016/j.jde.2015.06.005.

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5

de Simoi, Jacopo, and Dmitry Dolgopyat. "Dynamics of some piecewise smooth Fermi-Ulam models." Chaos: An Interdisciplinary Journal of Nonlinear Science 22, no. 2 (June 2012): 026124. http://dx.doi.org/10.1063/1.3695379.

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6

Li, Shuangbao, Wensai Ma, Wei Zhang, and Yuxin Hao. "Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems." International Journal of Bifurcation and Chaos 26, no. 02 (February 2016): 1650030. http://dx.doi.org/10.1142/s0218127416500309.

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In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

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7

Langer, Cameron K., and Bruce N. Miller. "Regular and chaotic dynamics of a piecewise smooth bouncer." Chaos: An Interdisciplinary Journal of Nonlinear Science 25, no. 7 (July 2015): 073114. http://dx.doi.org/10.1063/1.4923747.

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8

Roy, Indrava, Mahashweta Patra, and Soumitro Banerjee. "Shilnikov-type dynamics in three-dimensional piecewise smooth maps." Chaos, Solitons & Fractals 133 (April 2020): 109655. http://dx.doi.org/10.1016/j.chaos.2020.109655.

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9

Li, Denghui, Hebai Chen, and Jianhua Xie. "Smale Horseshoe in a Piecewise Smooth Map." International Journal of Bifurcation and Chaos 29, no. 04 (April 2019): 1950051. http://dx.doi.org/10.1142/s0218127419500512.

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We investigate the chaotic dynamics of a two-dimensional piecewise smooth map. The map represents the normal form of a discrete time representation of impact oscillators near grazing states. It is proved that, in certain region of the parameter space, the nonwandering set of the map is contained in a bounded region and that, restricted to the nonwandering set, the map is topologically conjugate to the two-sided shift map on two symbols.

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10

DONDE, VAIBHAV, and IAN A. HISKENS. "SHOOTING METHODS FOR LOCATING GRAZING PHENOMENA IN HYBRID SYSTEMS." International Journal of Bifurcation and Chaos 16, no. 03 (March 2006): 671–92. http://dx.doi.org/10.1142/s0218127406015040.

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Hybrid systems are typified by strong coupling between continuous dynamics and discrete events. For such piecewise smooth systems, event triggering generally has a significant influence over subsequent system behavior. Therefore, it is important to identify situations where a small change in parameter values alters the event triggering pattern. The bounding case, which separates regions of (generally) quite different dynamic behaviors, is referred to as grazing. At a grazing point, the system trajectory makes tangential contact with an event triggering hypersurface. The paper formulates conditions governing grazing points. Both transient and periodic behaviors are considered. The resulting boundary value problems are solved using shooting methods that are applicable for general nonlinear hybrid (piecewise smooth) dynamical systems. The grazing point formulation underlies the development of a continuation process for exploring parametric dependence. It also provides the basis for an optimization technique that finds the smallest parameter change necessary to induce grazing. Examples are drawn from power electronics, power systems and robotics, all of which involve intrinsic interactions between continuous dynamics and discrete events.

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11

Belykh, Igor, Rachel Kuske, Maurizio Porfiri, and David J. W. Simpson. "Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 1 (January 2023): 010402. http://dx.doi.org/10.1063/5.0138169.

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Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic “Bristol book” [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied.

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12

Zhusubaliyev, Z. T., D. S. Kuzmina, and O. O. Yanochkina. "Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form." Proceedings of the Southwest State University 24, no. 3 (December 6, 2020): 137–51. http://dx.doi.org/10.21869/2223-1560-2020-24-3-137-151.

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Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

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13

Ladino, Lilia M., Cristiana Mammana, Elisabetta Michetti, and Jose C. Valverde. "Border Collision Bifurcations in a Generalized Model of Population Dynamics." Discrete Dynamics in Nature and Society 2016 (2016): 1–13. http://dx.doi.org/10.1155/2016/9724139.

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We analyze the dynamics of a generalized discrete time population model of a two-stage species with recruitment and capture. This generalization, which is inspired by other approaches and real data that one can find in literature, consists in considering no restriction for the value of the two key parameters appearing in the model, that is, the natural death rate and the mortality rate due to fishing activity. In the more general case the feasibility of the system has been preserved by posing opportune formulas for the piecewise map defining the model. The resulting two-dimensional nonlinear map is not smooth, though continuous, as its definition changes as any border is crossed in the phase plane. Hence, techniques from the mathematical theory of piecewise smooth dynamical systems must be applied to show that, due to the existence of borders, abrupt changes in the dynamic behavior of population sizes and multistability emerge. The main novelty of the present contribution with respect to the previous ones is that, while using real data, richer dynamics are produced, such as fluctuations and multistability. Such new evidences are of great interest in biology since new strategies to preserve the survival of the species can be suggested.

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14

Pikulin, Dmitry. "Effects of Non-smooth Phenomena on the Dynamics of DC-DC Converters." Scientific Journal of Riga Technical University. Power and Electrical Engineering 29, no. 1 (January 1, 2011): 119–22. http://dx.doi.org/10.2478/v10144-011-0020-z.

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Effects of Non-smooth Phenomena on the Dynamics of DC-DC ConvertersThis paper provides the analysis of nonlinear phenomena in switch-mode power converters. In distinction to majority of known researches this paper presents novelty approach, allowing the complete bifurcation analysis, considering stable and various types of unstable behavior of nonlinear systems. Main results are illustrated on one of the most widely used switching converters - current controlled boost converter, for which the complete one-parametric bifurcation diagrams are constructed. The results include the detection of various types of rare attractors, smooth bifurcations and non-smooth phenomena, specific to piecewise linear dynamical systems.

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15

Zhang, Yunhu, and Pengfei Song. "Dynamics of the piecewise smooth epidemic model with nonlinear incidence." Chaos, Solitons & Fractals 146 (May 2021): 110903. http://dx.doi.org/10.1016/j.chaos.2021.110903.

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16

HOMBURG, ALE JAN. "Piecewise smooth interval maps with non-vanishing derivative." Ergodic Theory and Dynamical Systems 20, no. 3 (June 2000): 749–73. http://dx.doi.org/10.1017/s0143385700000407.

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We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

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17

Jiang, Fangfang, Zhicheng Ji, Qing-Guo Wang, and Jitao Sun. "Analysis of the Dynamics of Piecewise Linear Memristors." International Journal of Bifurcation and Chaos 26, no. 13 (December 15, 2016): 1650217. http://dx.doi.org/10.1142/s0218127416502175.

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In this paper, we consider a class of flux controlled memristive circuits with a piecewise linear memristor (i.e. the characteristic curve of the memristor is given by a piecewise linear function). The mathematical model is described by a discontinuous planar piecewise smooth differential system, which is defined on three zones separated by two parallel straight lines [Formula: see text] (called as discontinuity lines in discontinuous differential systems). We first investigate the stability of equilibrium points and the existence and uniqueness of a crossing limit cycle for the memristor-based circuit under self-excited oscillation. We then analyze the existence of periodic orbits of forced nonlinear oscillation for the memristive circuit with an external exciting source. Finally, we give numerical simulations to show good matches between our theoretical and simulation results.

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18

Simpson, David John Warwick, and Rachel Kuske. "The influence of localized randomness on regular grazing bifurcations with applications to impacting dynamics." Journal of Vibration and Control 24, no. 2 (April 12, 2016): 407–26. http://dx.doi.org/10.1177/1077546316642054.

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This paper concerns stochastic perturbations of piecewise-smooth ODE systems relevant for vibro-impacting dynamics, where impact events constitute the primary source of randomness. Such systems are characterized by the existence of switching manifolds that divide the phase space into regions where the system is smooth. The initiation of impacts is captured by a grazing bifurcation, at which a periodic orbit describing motion without impacts develops a tangential intersection with a switching manifold. Oscillatory dynamics near regular grazing bifurcations are described by piecewise-smooth maps involving a square-root singularity, known as Nordmark maps. We consider three scenarios where colored noise only affects impacting dynamics, and derive three two-dimensional stochastic Nordmark maps with the noise appearing in different nonlinear or multiplicative ways, depending on the source of the noise. Consequently the stochastic dynamics differs between the three noise sources, and is fundamentally different to that of a Nordmark map with additive noise. This critical dependence on the nature of the noise is illustrated with a prototypical one-degree-of-freedom impact oscillator.

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19

Aharonov, D., R. L. Devaney, and U. Elias. "The Dynamics of a Piecewise Linear Map and its Smooth Approximation." International Journal of Bifurcation and Chaos 07, no. 02 (February 1997): 351–72. http://dx.doi.org/10.1142/s0218127497000236.

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The paper describes the dynamics of a piecewise linear area preserving map of the plane, F: (x, y) → (1 - y - |x|, x), as well as that portion of the dynamics that persists when the map is approximated by the real analytic map Fε: (x, y) → (1 - y - fε(x), x), where fε(x) is real analytic and close to |x| for small values of ε. Our goal in this paper is to describe in detail the island structure and the chaotic behavior of the piecewise linear map F. Then we will show that these islands do indeed persist and contain infinitely many invariant curves for Fε, provided that ε is small.

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20

Zhusubaliyev, Z. T., V. G. Rubanov, and Yu A. Gol’tsov. "Calculation of Invariant Manifolds of Piecewise-Smooth Maps." Proceedings of the Southwest State University 24, no. 3 (December 6, 2020): 166–82. http://dx.doi.org/10.21869/2223-1560-2020-24-3-166-182.

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Purpose of reseach is of the work is to develop an algorithm for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps. Method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. Results. A method for calculating stable invariant manifolds of saddle periodic orbits of piecewise smooth maps is developed. The main result is formulated as a statement. The method is based on an original approach to finding the inverse function, the idea of which is to reduce the problem to a nonlinear first-order equation. Conclusion. A numerical method is described for calculating stable invariant manifolds of piecewise smooth maps that simulate impulse automatic control systems. The method is based on iterating the fundamental domain along a stable subspace of eigenvectors of the Jacobi matrix calculated at a saddle periodic fixed point. The method is based on an original approach to finding the inverse function, which consists in reducing the problem to solving a nonlinear first-order equation. This approach eliminates the need to solve systems of nonlinear equations to determine the inverse function and overcome the accompanying computational problems. Examples of studying the global dynamics of piecewise-smooth mappings with multistable behavior are given.

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21

PARK, Y., K. M. SHAW, H. J. CHIEL, and P. J. THOMAS. "The infinitesimal phase response curves of oscillators in piecewise smooth dynamical systems." European Journal of Applied Mathematics 29, no. 5 (April 2, 2018): 905–40. http://dx.doi.org/10.1017/s0956792518000128.

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The asymptotic phase θ of an initial point x in the stable manifold of a limit cycle (LC) identifies the phase of the point on the LC to which the flow φt(x) converges as t → ∞. The infinitesimal phase response curve (iPRC) quantifies the change in timing due to a small perturbation of a LC trajectory. For a stable LC in a smooth dynamical system, the iPRC is the gradient ∇x(θ) of the phase function, which can be obtained via the adjoint of the variational equation. For systems with discontinuous dynamics, the standard approach to obtaining the iPRC fails. We derive a formula for the iPRCs of LCs occurring in piecewise smooth (Filippov) dynamical systems of arbitrary dimension, subject to a transverse flow condition. Discontinuous jumps in the iPRC can occur at the boundaries separating subdomains, and are captured by a linear matching condition. The matching matrix, M, can be derived from the saltation matrix arising in the associated variational problem. For the special case of linear dynamics away from switching boundaries, we obtain an explicit expression for the iPRC. We present examples from cell biology (Glass networks) and neuroscience (central pattern generator models). We apply the iPRCs obtained to study synchronization and phase-locking in piecewise smooth LC systems in which synchronization arises solely due to the crossing of switching manifolds.

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22

Mandal, Dhrubajyoti. "Dynamics of Two Dimensional Piecewise Smooth Maps with Stochastically Varying Border." Advances in Dynamical Systems and Applications 14, no. 2 (December 8, 2019): 245. http://dx.doi.org/10.37622/adsa/14.2.2019.245-255.

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23

Biswas, Dhrubajyoti, Soumyajit Seth, and Mita Bor. "A Study of the Dynamics of a New Piecewise Smooth Map." International Journal of Bifurcation and Chaos 30, no. 01 (January 2020): 2050018. http://dx.doi.org/10.1142/s0218127420500182.

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In this article, we have studied a [Formula: see text]D map, which is formed by combining the two well-known maps, i.e. the tent and the logistic maps in the unit interval, i.e. [Formula: see text]. The point of discontinuity of the map (known as border) denotes the transition from tent map to logistic map. The proposed map can behave as the piecewise smooth or nonsmooth map or both (depending on the behavior of the map just before and after the border) and the dynamics of the map has been studied using analytical tools and numerical simulations. Characterization has been done by primarily studying the Lyapunov exponents and the corresponding bifurcation diagrams. Some peculiar dynamics of this map have been shown numerically. Finally, a Simulink implementation of the proposed map has been demonstrated.

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24

Jeffrey, Mike R. "Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 25, no. 10 (October 2015): 103125. http://dx.doi.org/10.1063/1.4934204.

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25

Cánovas, Jose S., Anastasiia Panchuk, and Tönu Puu. "Asymptotic dynamics of a piecewise smooth map modelling a competitive market." Mathematics and Computers in Simulation 117 (November 2015): 20–38. http://dx.doi.org/10.1016/j.matcom.2015.05.004.

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26

Françoise, J. P., Hongjun Ji, Dongmei Xiao, and Jiang Yu. "Global Dynamics of a Piecewise Smooth System for Brain Lactate Metabolism." Qualitative Theory of Dynamical Systems 18, no. 1 (September 15, 2018): 315–32. http://dx.doi.org/10.1007/s12346-018-0286-z.

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27

Avrutin, Viktor, and Zhanybai T. Zhusubaliyev. "Piecewise-Linear Map for Studying Border Collision Phenomena in DC/AC Converters." International Journal of Bifurcation and Chaos 30, no. 07 (June 15, 2020): 2030015. http://dx.doi.org/10.1142/s0218127420300153.

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Recently, while studying the dynamics of power electronic DC/AC converters we have demonstrated that the behavior of these systems can be modeled by piecewise-smooth maps which belong to a specific class of models not investigated before. The characteristic feature of these maps is the presence of a very high number of switching manifolds (border points in 1D). Obviously, the multitude of control strategies applied in the modern power electronics leads to different maps belonging to this class of models. However, in this paper we show that several models can be studied using the same piecewise-linear approximation, so that the bifurcation phenomena which can be observed in this approximation are generic for many models. Based on the results obtained before for piecewise-smooth models with different kinds of nonlinearities resulting from the corresponding control strategies, in the present paper we discuss the generic bifurcation patterns in the underlying piecewise-linear map.

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28

Cao, Qingjie, Marian Wiercigroch, Ekaterina E. Pavlovskaia, J. Michael T. Thompson, and Celso Grebogi. "Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, no. 1865 (August 13, 2007): 635–52. http://dx.doi.org/10.1098/rsta.2007.2115.

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In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α , tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load–deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α =0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α =0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

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DE, SOMA, PARTHA SHARATHI DUTTA, SOUMITRO BANERJEE, and AKHIL RANJAN ROY. "LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS." International Journal of Bifurcation and Chaos 21, no. 06 (June 2011): 1617–36. http://dx.doi.org/10.1142/s0218127411029318.

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In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

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30

Li, Yuxi, Zhouchao Wei, Wei Zhang, and Ming Yi. "Melnikov-type method for a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation: Homoclinic orbits." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 7 (July 2022): 073119. http://dx.doi.org/10.1063/5.0096086.

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The Melnikov method is extended to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation when an unperturbed system is a piecewise Hamiltonian system with a homoclinic orbit. The homoclinic orbit continuously crosses the first switching manifold and transversally jumps across the second switching manifold by the impulsive effect. The trajectory of the corresponding perturbed system crosses the first switching manifold by applying the reset map describing the impact rule instantaneously. Then, the random Melnikov process of such systems is derived and the criteria for the onset of chaos with or without noise excitation are established. In addition, the complicated dynamics of concrete piecewise-smooth systems with or without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations are investigated by this extended method and numerical simulations.

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31

Mora, Karin, Chris Budd, Paul Glendinning, and Patrick Keogh. "Non-smooth Hopf-type bifurcations arising from impact–friction contact events in rotating machinery." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2171 (November 8, 2014): 20140490. http://dx.doi.org/10.1098/rspa.2014.0490.

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We analyse the novel dynamics arising in a nonlinear rotor dynamic system by investigating the discontinuity-induced bifurcations corresponding to collisions with the rotor housing (touchdown bearing surface interactions). The simplified Föppl/Jeffcott rotor with clearance and mass unbalance is modelled by a two degree of freedom impact–friction oscillator, as appropriate for a rigid rotor levitated by magnetic bearings. Two types of motion observed in experiments are of interest in this paper: no contact and repeated instantaneous contact. We study how these are affected by damping and stiffness present in the system using analytical and numerical piecewise-smooth dynamical systems methods. By studying the impact map, we show that these types of motion arise at a novel non-smooth Hopf-type bifurcation from a boundary equilibrium bifurcation point for certain parameter values. A local analysis of this bifurcation point allows us a complete understanding of this behaviour in a general setting. The analysis identifies criteria for the existence of such smooth and non-smooth bifurcations, which is an essential step towards achieving reliable and robust controllers that can take compensating action.

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Askar, S. S., and A. Al-khedhairi. "The dynamics of a business game: A 2D-piecewise smooth nonlinear map." Physica A: Statistical Mechanics and its Applications 537 (January 2020): 122766. http://dx.doi.org/10.1016/j.physa.2019.122766.

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Gardini, Laura, and Fabio Tramontana. "Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps." Chaos, Solitons & Fractals 45, no. 11 (November 2012): 1328–42. http://dx.doi.org/10.1016/j.chaos.2012.07.007.

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Baake, Michael, Uwe Grimm, and Harald Jockusch. "Freely forming groups: Trying to be rare." ANZIAM Journal 48, no. 1 (July 2006): 1–10. http://dx.doi.org/10.1017/s1446181100003370.

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AbstractA simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical system that reaches its globally stable equilibrium in finite time. This dynamical system is sufficiently simple to permit an explicit solution, built piecewise from solutions of the logistic equation in continuous time. It displays an interesting tree-like structure of coalescing components.

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35

TABORDA, J. A., F. ANGULO, and G. OLIVAR. "MANDELBROT-LIKE BIFURCATION STRUCTURES IN CHAOS BAND SCENARIO OF SWITCHED CONVERTER WITH DELAYED-PWM CONTROL." International Journal of Bifurcation and Chaos 20, no. 01 (January 2010): 99–119. http://dx.doi.org/10.1142/s0218127410025430.

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In this paper, we report Mandelbrot-like bifurcation structures in a one-dimensional parameter space of real numbers corresponding to a dc-dc power converter modeled as a piecewise-smooth system with three zones. These fractal patterns have been studied in two-dimensional parameter space for smooth systems, but for nonsmooth systems has not been reported yet. The Mandelbrot-like sets we found are created in transition from the torus band to chaos band scenarios exhibited by a dc-dc buck power converter controlled by Delayed Pulse-Width Modulator (PWM) based on Zero Average Dynamics (or ZAD strategy), which corresponds to a piecewise-smooth system (PWS). The real parameter is provided by the PWM control strategy, namely ZAD strategy, and it can be varied in a large range, ideally (-∞, +∞). At -∞ and +∞ the dynamical behavior is the same, and thus we will describe the synamics in an ring-like parameter space. Mandelbrot-like borders are built by four chaotic bands, therefore these structures can be thought as instability islands where the state variables cannot be located. Using the Poincaré map approach we characterize the bifurcation structures and we describe recurrent patterns in different scales.

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36

Tang, Yilei. "Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems." Discrete & Continuous Dynamical Systems - A 38, no. 4 (2018): 2029–46. http://dx.doi.org/10.3934/dcds.2018082.

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37

Gao, Xue, Qian Chen, and Xianbin Liu. "Nonlinear dynamics and design for a class of piecewise smooth vibration isolation system." Nonlinear Dynamics 84, no. 3 (February 25, 2016): 1715–26. http://dx.doi.org/10.1007/s11071-016-2599-2.

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SHPILEVAYA, OLGA. "STABILITY OF SISO NONLINEAR SYSTEMS WITH PARAMETERS DISTURBANCES." International Journal of Modern Physics B 26, no. 25 (September 10, 2012): 1246008. http://dx.doi.org/10.1142/s0217979212460083.

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We study single-input single-output (SISO) control systems with the rapid piecewise-smooth parameters disturbances. The system dynamics are described by switched system models. The system output is regulated with the help of the nonlinear astatic controller with parameters which depend on some disturbance properties. The system stability is studied by second Lyapunov method.

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39

Wei, Zhouchao, Yuxi Li, Irene Moroz, and Wei Zhang. "Melnikov-type method for a class of planar hybrid piecewise-smooth systems with impulsive effect and noise excitation: Heteroclinic orbits." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 10 (October 2022): 103127. http://dx.doi.org/10.1063/5.0106073.

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The classical Melnikov method for heteroclinic orbits is extended theoretically to a class of hybrid piecewise-smooth systems with impulsive effect and noise excitation. We assume that the unperturbed system is a piecewise Hamiltonian system with a pair of heteroclinic orbits. The heteroclinic orbit transversally jumps across the first switching manifold by an impulsive effect and crosses the second switching manifold continuously. In effect, the trajectory of the corresponding perturbed system crosses the second switching manifold by applying the reset map describing the impact rule instantaneously. The random Melnikov process of such systems is then derived by measuring the distance of perturbed stable and unstable manifolds, and the criteria for the onset of chaos with or without noise excitation is established. In this derivation process, we overcome the difficulty that the derivation method of the corresponding homoclinic case cannot be directly used due to the difference between the symmetry of the homoclinic orbit and the asymmetry of the heteroclinic orbit. Finally, we investigate the complicated dynamics of a particular piecewise-smooth system with and without noise excitation under the reset maps, impulsive effect, and non-autonomous periodic and damping perturbations by this new extended method and numerical simulations. The numerical results verify the correctness of the theoretical results and demonstrate that this extended method is simple and effective for studying the dynamics of such systems.

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40

Wang, Fanrui, Zhouchao Wei, Wei Zhang, and Irene Moroz. "Coexistence of three heteroclinic cycles and chaos analyses for a class of 3D piecewise affine systems." Chaos: An Interdisciplinary Journal of Nonlinear Science 33, no. 2 (February 2023): 023108. http://dx.doi.org/10.1063/5.0132018.

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Our objective is to investigate the innovative dynamics of piecewise smooth systems with multiple discontinuous switching manifolds. This paper establishes the coexistence of heteroclinic cycles in a class of 3D piecewise affine systems with three switching manifolds through rigorous mathematical analysis. By constructing suitable Poincaré maps adjacent to heteroclinic cycles, we demonstrate the occurrence of two distinct types of horseshoes and show the conditions for the presence of chaotic invariant sets. A family of attractors that satisfy the criteria are presented using this technique. It is shown that the outcomes of numerical simulation accurately reflect those of our theoretical results.

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Dankowicz, Harry, and Jenny Jerrelind. "Control of near-grazing dynamics in impact oscillators." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2063 (September 2005): 3365–80. http://dx.doi.org/10.1098/rspa.2005.1516.

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A method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities. In particular, it is shown that a discrete, linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory, such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact. The implementation relies on a local analysis of the near-grazing dynamics using the concept of discontinuity mappings. Numerical results are presented for a linear and a nonlinear oscillator.

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42

Ma, Yinsong, and Guangcai Liu. "Automated Path Planning for unmanned aerial vehicle in Urban Dynamic Area." Journal of Physics: Conference Series 2252, no. 1 (April 1, 2022): 012045. http://dx.doi.org/10.1088/1742-6596/2252/1/012045.

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Abstract This paper addresses the path planning and autonomous obstacle avoidance problem of UAVs in urban dynamic area. A flight path planning strategy for UAVs in complex urban environments is proposed.First, the A* algorithm is used to construct a desired global path in a 3D static environment, which is used as the static reference path for dynamic obstacle avoidance below.The environmental and the key points of algorithm are also elaborated. In this paper, the dynamic obstacles are divided into three categories, then, in order to avoid the collision between dynamic obstacles and static optimal paths, two strategies to achieve local online path adjustment are proposed. Finally, the seven order minimum snap trajectories generation based on piecewise polynomials is utilized to smooth the flight path.We can obtain a smooth trajectory based on UAV dynamics and safety. The simulation results verify the effectiveness of the proposed UAV path planning strategy in the dynamic and complex urban area.

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43

Ma, Yinsong, and Guangcai Liu. "Automated Path Planning for unmanned aerial vehicle in Urban Dynamic Area." Journal of Physics: Conference Series 2252, no. 1 (April 1, 2022): 012045. http://dx.doi.org/10.1088/1742-6596/2252/1/012045.

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Abstract This paper addresses the path planning and autonomous obstacle avoidance problem of UAVs in urban dynamic area. A flight path planning strategy for UAVs in complex urban environments is proposed.First, the A* algorithm is used to construct a desired global path in a 3D static environment, which is used as the static reference path for dynamic obstacle avoidance below.The environmental and the key points of algorithm are also elaborated. In this paper, the dynamic obstacles are divided into three categories, then, in order to avoid the collision between dynamic obstacles and static optimal paths, two strategies to achieve local online path adjustment are proposed. Finally, the seven order minimum snap trajectories generation based on piecewise polynomials is utilized to smooth the flight path.We can obtain a smooth trajectory based on UAV dynamics and safety. The simulation results verify the effectiveness of the proposed UAV path planning strategy in the dynamic and complex urban area.

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JOCHMANN, F., and L. RECKE. "WELL-POSEDNESS OF AN INITIAL BOUNDARY VALUE PROBLEM FROM LASER DYNAMICS." Mathematical Models and Methods in Applied Sciences 12, no. 04 (April 2002): 593–606. http://dx.doi.org/10.1142/s0218202502001805.

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In this paper a mathematical model, consisting of nonlinear first-order ordinary and partial differential equations with initial and boundary conditions, for the dynamical behavior of multisection DFB (distributed feedback) semiconductor lasers is investigated. We introduce a suitable weak formulation and prove existence, uniqueness and regularity properties of the solutions. The assumptions on the data are quite general, in particular, the physically relevant case of piecewise smooth, but discontinuous with respect to space and time coefficients in the equations and in the boundary conditions is included.

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45

Li, Shuangbao, Wensai Ma, Wei Zhang, and Yuxin Hao. "Melnikov Method for a Three-Zonal Planar Hybrid Piecewise-Smooth System and Application." International Journal of Bifurcation and Chaos 26, no. 01 (January 2016): 1650014. http://dx.doi.org/10.1142/s0218127416500140.

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In this paper, we extend the well-known Melnikov method for smooth systems to a class of planar hybrid piecewise-smooth systems, defined in three domains separated by two switching manifolds [Formula: see text] and [Formula: see text]. The dynamics in each domain is governed by a smooth system. When an orbit reaches the separation lines, then a reset map describing an impacting rule applies instantaneously before the orbit enters into another domain. We assume that the unperturbed system has a continuum of periodic orbits transversally crossing the separation lines. Then, we wish to study the persistence of the periodic orbits under an autonomous perturbation and the reset map. To achieve this objective, we first choose four appropriate switching sections and build a Poincaré map, after that, we present a displacement function and carry on the Taylor expansion of the displacement function to the first-order in the perturbation parameter [Formula: see text] near [Formula: see text]. We denote the first coefficient in the expansion as the first-order Melnikov function whose zeros provide us the persistence of periodic orbits under perturbation. Finally, we study periodic orbits of a concrete planar hybrid piecewise-smooth system by the obtained Melnikov function.

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46

Misra, Sambit, and Harry Dankowicz. "Control of near-grazing dynamics and discontinuity-induced bifurcations in piecewise-smooth dynamical systems." International Journal of Robust and Nonlinear Control 20, no. 16 (December 21, 2009): 1836–51. http://dx.doi.org/10.1002/rnc.1551.

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47

Nemirovskii, Yu V., and T. P. Romanova. "Dynamics of doubly connected plates in the plastic state with piecewise-smooth bearing contours." International Applied Mechanics 28, no. 4 (April 1992): 223–29. http://dx.doi.org/10.1007/bf00847280.

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48

Hogan, S. J., L. Higham, and T. C. L. Griffin. "Dynamics of a piecewise linear map with a gap." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2077 (July 26, 2006): 49–65. http://dx.doi.org/10.1098/rspa.2006.1735.

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In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 Int. J. Bifurcat. Chaos 13 , 3341–3351) and Banerjee et al . (Banerjee et al . 2004 IEEE Trans. Circ. Syst. II 51 , 649–654) and explain their numerical simulations.

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49

Simpson, David J. W. "Grazing-Sliding Bifurcations Creating Infinitely Many Attractors." International Journal of Bifurcation and Chaos 27, no. 12 (November 2017): 1730042. http://dx.doi.org/10.1142/s0218127417300427.

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As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

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50

Askar, S. S., and A. Al-khedhairi. "Complexity Analysis of a 2D-Piecewise Smooth Duopoly Model: New Products versus Remanufactured Products." Complexity 2022 (April 30, 2022): 1–12. http://dx.doi.org/10.1155/2022/8856009.

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Recent studies on remanufacturing duopoly games have handled them as smooth maps and have observed that the bifurcation types that occurred in such maps belong to generic classes like period-doubling or Neimark-Sacker bifurcations. Since those games yield piecewise smooth maps, their bifurcations belong to the so-called border-collision bifurcations, which occur when the map’s fixed points cross the borderline between the smooth regions in the phase space. In the current paper, we present a proper systematic analysis of the local stability of the map’s fixed points both analytically and numerically. This includes studying the border-collision bifurcation depending on the map’s parameters. We present different multistability scenarios of the dynamics of the game’s map and show different types of periodic cycles and chaotic attractors that jump from one region to another or just cross the borderline in the phase space.

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