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

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.

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.

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.

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.

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