\`x^2+y_1+z_12^34\`
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Coexisting singular cycles in a class of three-dimensional three-zone piecewise affine systems

  • * Corresponding author: Wenjing Xu

    * Corresponding author: Wenjing Xu

This work was supported by the National Natural Science Foundation of China (NSFC) grant No. 12101078

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  • Detecting an isolated homoclinic or heteroclinic cycle is a great challenge in a concrete system, letting alone the case of coexisting scenarios and more complicated chaotic behaviors. This paper systematically investigates the dynamics for a class of three-dimensional (3D) three-zone piecewise affine systems (PWASs) consisting of three sub-systems. Interestingly, under different conditions the considered system can display three types of coexisting singular cycles including: homoclinic and homoclinic cycles, heteroclinic and heteroclinic cycles, homoclinic and heteroclinic cycles. Furthermore, it establishes sufficient conditions for the presence of chaotic invariant sets emerged from such coexisting cycles. Finally, three numerical examples are provided to verify the proposed theoretical results.

    Mathematics Subject Classification: Primary: 34C37, 34C28; Secondary: 34H10.

    Citation:

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  • Figure 1.  Geometric structure of $ R_{k} $ and $ P_1(R_{k}) $

    Figure 2.  Geometric structure of $ P(R_{k}) $ and $ R_{ki} $ for $ i = 1, 2 $

    Figure 3.  Geometric structure of $ R_k $ and $ P_1(R_k) $

    Figure 4.  Geometric structure of Geometry of $ R_{ni} $ and $ R_{kij} $ for $ i = 1, 2 $, $ j = 1, 2 $

    Figure 5.  Geometric structure of Geometry of $ R_{ni} $ and $ R_{kij} $ for $ i = 1, 2 $, $ j = 1, 2 $

    Figure 6.  Geometric structure of $ R_{lk} $ and $ R_{rk} $ under Poincaré return map $ P $

    Figure 7.  System (35) in Example 1 has (a) a homoclinic $ \Upsilon_1 $, (b) a homoclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $

    Figure 8.  (a) The phase portrait of a chaotic set of system (35). The projection of the chaotic set in (b) x-y plane, (c)y-z plane

    Figure 9.  System (36) in Example 2 has (a) a heteroclinic cycle $ \Upsilon_1 $, (b) a heteroclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $

    Figure 10.  (a) The phase portrait of a chaotic set of system (36). The projection of the chaotic set in (b) x-z plane, (c)y-z plane

    Figure 11.  System (37) in Example 3 has (a) a heteroclinic cycle $ \Upsilon_1 $, (b) a homoclinic cycle $ \Upsilon_2 $, (c) coexisting $ \Upsilon_1 $ and $ \Upsilon_2 $

    Figure 12.  (a) The phase portrait of a chaotic set of system (37). The projection of the chaotic set in (b) x-y plane, (c)x-z plane

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