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A mechanism for dangerous border collision bifurcations

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  • Do, Younghae

Abstract

In this paper we investigate a mechanism causing dangerous border collision bifurcations which is numerically characterized by exhibiting a stable fixed point before and after the critical bifurcation point, but the unbounded behavior of orbits at the critical bifurcation point. In particular, we provide that at the critical bifurcation value μ0, the qualitative type of the fixed point without having Jacobian information is saddle, which can be induced by invariant manifolds of the periodic saddle orbit on the boundary of the basin of attractor at infinity at the parameter μ≠μ0. In addition, we show that invariant sets of such fixed point are nonsmooth curve, and also point out a dangerous situation of numerical simulation.

Suggested Citation

  • Do, Younghae, 2007. "A mechanism for dangerous border collision bifurcations," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 352-362.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:2:p:352-362
    DOI: 10.1016/j.chaos.2006.07.018
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    Cited by:

    1. Baek, Hun Ki & Do, Younghae, 2009. "Existence of homoclinic orbits of an area-preserving map with a nonhyperbolic structure," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 2154-2162.
    2. Kang, Yoseb & Lee, Eun Hye & Kim, Sang-Hyun & Jang, Yong Hyun & Do, Younghae, 2021. "Complexity and multistability of a nonsmooth atopic dermatitis system," Chaos, Solitons & Fractals, Elsevier, vol. 153(P2).
    3. Do, Younghae & Kim, Sang Dong & Kim, Phil Su, 2008. "Stability of fixed points placed on the border in the piecewise linear systems," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 391-399.

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