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.

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.

On the existence of homoclinic orbits in some class of three-dimensional piecewise affine systems

2018 ◽  
Vol 37 (5) ◽  
pp. 6022-6033 ◽  
Author(s):  
Yanli Chen ◽  
Tiantian Wu ◽  
Xiaosong Yang
Keyword(s):  
Three Dimensional ◽  
Homoclinic Orbits ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

Chaotic Dynamics in Four-Dimensional Piecewise Affine Systems with Bifocal Heteroclinic Cycles

2018 ◽  
Vol 28 (11) ◽  
pp. 1850141 ◽  
Author(s):  
Tiantian Wu ◽  
Lei Wang ◽  
Xiao-Song Yang
Keyword(s):  
Chaotic Dynamics ◽  
Homoclinic Orbits ◽  
Invariant Sets ◽  
Heteroclinic Cycle ◽  
Piecewise Affine Systems ◽  
Heteroclinic Cycles ◽  
Affine Systems ◽  
Piecewise Affine ◽  
Nonsmooth Systems ◽  
Full Shift

The well-known Shil’nikov type theory provides an approach to proving the existence of chaotic invariant sets for some classes of smooth dynamical systems with homoclinic orbits or heteroclinic cycles. However, it cannot be applied to nonsmooth systems directly. Based on the similar ideas, this paper studies the existence of chaotic invariant sets for a class of two-zone four-dimensional piecewise affine systems with bifocal heteroclinic cycles that cross the switching manifold transversally at two points. It turns out that there exist countable infinite chaotic invariant sets in a neighborhood of the bifocal heteroclinic cycle under some eigenvalue conditions. Moreover, the horseshoes of the corresponding Poincaré map are topologically semi-conjugated to a full shift on four symbols.

scholarly journals A new class of 3-dimensional piecewise affine systems with homoclinic orbits

2016 ◽  
Vol 36 (9) ◽  
pp. 5119-5129 ◽  
Author(s):  
Tiantian Wu ◽  
Xiao-Song Yang
Keyword(s):  
Homoclinic Orbits ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
3 Dimensional ◽  
Piecewise Affine ◽  
New Class

On the existence of homoclinic orbits in n-dimensional piecewise affine systems

2018 ◽  
Vol 27 ◽  
pp. 366-389 ◽  
Author(s):  
Tiantian Wu ◽  
Xiao-Song Yang
Keyword(s):  
Homoclinic Orbits ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

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.

Homoclinic orbits and an invariant chaotic set in a new 4D piecewise affine systems

2018 ◽  
Vol 93 (4) ◽  
pp. 2445-2459 ◽  
Author(s):  
Qigui Yang ◽  
Kai Lu
Keyword(s):  
Homoclinic Orbits ◽  
Piecewise Affine Systems ◽  
Affine Systems ◽  
Piecewise Affine

Construction of a Class of Four-Dimensional Piecewise Affine Systems with Homoclinic Orbits

2016 ◽  
Vol 26 (06) ◽  
pp. 1650099 ◽  
Author(s):  
Tiantian Wu ◽  
Xiao-Song Yang
Keyword(s):  
Mathematical Analysis ◽  
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 four-dimensional piecewise affine systems. In addition, an example is provided to illustrate the effectiveness of the method.

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.

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