American Institute of Mathematical Sciences

Advanced
September  2006, 5(3): 493-503. doi: 10.3934/cpaa.2006.5.493

Dangerous Border-Collision bifurcations of a piecewise-smooth map

Hun Ki Baek 1, and  Younghae Do 1,

1. 

Department of Mathematics, Kyungpook National University, Daegu, 702-701, South Korea, South Korea

Received  June 2005 Revised  February 2006 Published  June 2006

In this paper we study the dangerous border-collision bifurcations [8] which recently have been numerically found on piecewise smooth maps characterized by non-differentiability on some surface in the phase space. The striking feature of such bifurcations is 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. We consider a specific variable space in order to do an analytical investigation of such bifurcations and prove the stability of fixed points. We also extend these bifurcation phenomena for the fixed points to the multiple coexisting attractors.
Keywords: chaos., border-collision bifurcation, Fixed points, nonsmooth.
Mathematics Subject Classification: 37C05, 37C37, 37G35, 37H2.
Citation: Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493
[1]

Iryna Sushko, Anna Agliari, Laura Gardini. Bistability and border-collision bifurcations for a family of unimodal piecewise smooth maps. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 881-897. doi: 10.3934/dcdsb.2005.5.881
[2]

Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547
[3]

Paula Kemp. Fixed points and complete lattices. Conference Publications, 2007, 2007 (Special) : 568-572. doi: 10.3934/proc.2007.2007.568
[4]

John Franks, Michael Handel, Kamlesh Parwani. Fixed points of Abelian actions. Journal of Modern Dynamics, 2007, 1 (3) : 443-464. doi: 10.3934/jmd.2007.1.443
[5]

Laura Gardini, Roya Makrooni, Iryna Sushko. Cascades of alternating smooth bifurcations and border collision bifurcations with singularity in a family of discontinuous linear-power maps. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 701-729. doi: 10.3934/dcdsb.2018039
[6]

Alexey A. Petrov, Sergei Yu. Pilyugin. Shadowing near nonhyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3761-3772. doi: 10.3934/dcds.2014.34.3761
[7]

Juan Campos, Rafael Ortega. Location of fixed points and periodic solutions in the plane. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 517-523. doi: 10.3934/dcdsb.2008.9.517
[8]

Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, 2021, 29 (4) : 2553-2560. doi: 10.3934/era.2021001
[9]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
[10]

Younghae Do, Juan M. Lopez. Slow passage through multiple bifurcation points. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 95-107. doi: 10.3934/dcdsb.2013.18.95
[11]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[12]

Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[13]

Victoria Martín-Márquez, Simeon Reich, Shoham Sabach. Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1043-1063. doi: 10.3934/dcdss.2013.6.1043
[14]

Adrian Petruşel, Radu Precup, Marcel-Adrian Şerban. On the approximation of fixed points for non-self mappings on metric spaces. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 733-747. doi: 10.3934/dcdsb.2019264
[15]

Marian Gidea, Yitzchak Shmalo. Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6123-6148. doi: 10.3934/dcds.2018264
[16]

Anna Cima, Armengol Gasull, Víctor Mañosa. Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 889-904. doi: 10.3934/dcds.2018038
[17]

Lei Guo, Gui-Hua Lin. Globally convergent algorithm for solving stationary points for mathematical programs with complementarity constraints via nonsmooth reformulations. Journal of Industrial & Management Optimization, 2013, 9 (2) : 305-322. doi: 10.3934/jimo.2013.9.305
[18]

Inmaculada Baldomá, Ernest Fontich, Pau Martín. Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4159-4190. doi: 10.3934/dcds.2017177
[19]

Jifa Jiang, Lei Niu. On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 217-244. doi: 10.3934/dcds.2016.36.217
[20]

A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121.

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (92)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences