American Institute of Mathematical Sciences

Advanced
November  2007, 8(4): 943-970. doi: 10.3934/dcdsb.2007.8.943

Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator

Alexei Pokrovskii 1, , Oleg Rasskazov 1, and  Daniela Visetti 1,

1. 

Department of Applied Mathematics, University College, Cork, Ireland

Received  September 2006 Revised  July 2007 Published  August 2007

In this paper we consider the equation $\ddot x+x=\sin(\sqrt{2}t)+s(x)\,$ where $s(x)$ is a piece-wise linear map given by min$\{5x,1\}$ if $x\ge0$ and by max$\{-1, 5x\}$ if $x<0$. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.
Keywords: Chaotic behaviour, topological hyperbolicity, transversal homoclinic orbit..
Mathematics Subject Classification: Primary: 34C28; Secondary: 37D4.
Citation: Alexei Pokrovskii, Oleg Rasskazov, Daniela Visetti. Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 943-970. doi: 10.3934/dcdsb.2007.8.943
[1]

Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143
[2]

Chen-Chang Peng, Kuan-Ju Chen. Existence of transversal homoclinic orbits in higher dimensional discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1181-1197. doi: 10.3934/dcdsb.2010.14.1181
[3]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367
[4]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[5]

Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133
[6]

Maria João Costa. Chaotic behaviour of one-dimensional horseshoes. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 505-548. doi: 10.3934/dcds.2003.9.505
[7]

Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179
[8]

Enrique R. Pujals. Density of hyperbolicity and homoclinic bifurcations for attracting topologically hyperbolic sets. Discrete & Continuous Dynamical Systems, 2008, 20 (2) : 335-405. doi: 10.3934/dcds.2008.20.335
[9]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[10]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[11]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[12]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[13]

Xinfu Chen. Lorenz equations part II: "randomly" rotated homoclinic orbits and chaotic trajectories. Discrete & Continuous Dynamical Systems, 1996, 2 (1) : 121-140. doi: 10.3934/dcds.1996.2.121
[14]

Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991
[15]

Riadh Chteoui, Abdulrahman F. Aljohani, Anouar Ben Mabrouk. Classification and simulation of chaotic behaviour of the solutions of a mixed nonlinear Schrödinger system. Electronic Research Archive, 2021, 29 (4) : 2561-2597. doi: 10.3934/era.2021002
[16]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765
[17]

Xianjun Wang, Huaguang Gu, Bo Lu. Big homoclinic orbit bifurcation underlying post-inhibitory rebound spike and a novel threshold curve of a neuron. Electronic Research Archive, , () : -. doi: 10.3934/era.2021023
[18]

Minju Lee, Hee Oh. Topological proof of Benoist-Quint's orbit closure theorem for $ \boldsymbol{ \operatorname{SO}(d, 1)} $. Journal of Modern Dynamics, 2019, 15: 263-276. doi: 10.3934/jmd.2019021
[19]

Kengo Matsumoto. Continuous orbit equivalence of topological Markov shifts and KMS states on Cuntz–Krieger algebras. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5897-5909. doi: 10.3934/dcds.2020251
[20]

Kengo Matsumoto. Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021139

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences