Existence of Homoclinic Cycles and Periodic Orbits in a Class of Three-Dimensional Piecewise Affine Systems

2018 ◽  
Vol 28 (02) ◽  
pp. 1850024 ◽  
Author(s):  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Spatial Location ◽  
Homoclinic Bifurcation ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

For a class of three-dimensional piecewise affine systems, this paper focuses on the existence of homoclinic cycles and the phenomena of homoclinic bifurcation leading to periodic orbits. Based on the spatial location relation between the invariant manifolds of subsystems and the switching manifold, a concise necessary and sufficient condition for the existence of homoclinic cycles is obtained. Then the homoclinic bifurcation is studied and the sufficient conditions for the birth of a periodic orbit are obtained. Furthermore, the sufficient conditions are obtained for the periodic orbit to be a sink, a source or a saddle. As illustrations, several concrete examples are presented.

Discontinuous Lyapunov functions for a class of piecewise affine systems

2018 ◽  
Vol 41 (3) ◽  
pp. 729-736 ◽  
Author(s):  
Farideh Cheraghi-Shami ◽  
Ali-Akbar Gharaveisi ◽  
Malihe M Farsangi ◽  
Mohsen Mohammadian
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Convex Polytopes ◽  
Sufficient Conditions ◽  
Monotonicity Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Local Asymptotic Stability

In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

Designing Chaotic Systems by Piecewise Affine Systems

2016 ◽  
Vol 26 (09) ◽  
pp. 1650154 ◽  
Author(s):  
Tiantian Wu ◽  
Qingdu Li ◽  
Xiao-Song Yang
Keyword(s):  
Mathematical Analysis ◽  
Chaotic Systems ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

Based on mathematical analysis, this paper provides a methodology to ensure the existence of homoclinic orbits in a class of three-dimensional piecewise affine systems. In addition, two chaotic generators are provided to illustrate the effectiveness of the method.

scholarly journals Homoclinic Orbits in Several Classes of Three-Dimensional Piecewise Affine Systems with Two Switching Planes

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3285
Author(s):  
Yanli Chen ◽  
Lei Wang ◽  
Xiaosong Yang
Keyword(s):  
Analytic Solution ◽  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Heteroclinic Cycle ◽  
Sufficient Condition ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Analytic Proof ◽  
Piecewise Affine ◽  
Cycle Plays

The existence of homoclinic orbits or heteroclinic cycle plays a crucial role in chaos research. This paper investigates the existence of the homoclinic orbits to a saddle-focus equilibrium point in several classes of three-dimensional piecewise affine systems with two switching planes regardless of the symmetry. An analytic proof is provided using the concrete expression forms of the analytic solution, stable manifold, and unstable manifold. Meanwhile, a sufficient condition for the existence of two homoclinic orbits is also obtained. Furthermore, two concrete piecewise affine asymmetric systems with two homoclinic orbits have been constructed successfully, demonstrating the method’s effectiveness.

Chaos Generated by a Class of 3D Three-Zone Piecewise Affine Systems with Coexisting Singular Cycles

2020 ◽  
Vol 30 (14) ◽  
pp. 2050209
Author(s):  
Kai Lu ◽  
Wenjing Xu ◽  
Qigui Yang
Keyword(s):  
Mathematical Proof ◽  
Three Dimensional ◽  
Piecewise Affine Systems ◽  
Heteroclinic Cycles ◽  
Affine Systems ◽  
Numerical Examples ◽  
Piecewise Affine ◽  
Dynamical Behaviors ◽  
Nonsmooth Systems ◽  
Existence Conditions

It is a significant and challenging task to detect both the coexistence of singular cycles, mainly homoclinic and heteroclinic cycles, and chaos induced by the coexistence in nonsmooth systems. By analyzing the dynamical behaviors on manifolds, this paper proposes some criteria to accurately locate the coexistence of homoclinic cycles and of heteroclinic cycles in a class of three-dimensional (3D) piecewise affine systems (PASs), respectively. It further establishes the existence conditions of chaos arising from such coexistence, and presents a mathematical proof by analyzing the constructed Poincaré map. Finally, the simulations for two numerical examples are provided to validate the established results.

scholarly journals A reference governor approach for constrained piecewise affine systems

2009 ◽  
Author(s):  
Paolo Falcone ◽  
Francesco Borrelli ◽  
Jaroslav Pekar ◽  
Greg Stewart
Keyword(s):  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Reference Governor
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