### Abstract

We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the map T(x, z)=(1-a|x|+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative case b=1 separatrices and certain invariant tori are calculated.

Original language | English |
---|---|

Pages (from-to) | 195-221 |

Number of pages | 27 |

Journal | Journal of Statistical Physics |

Volume | 33 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 1983 |

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### Keywords

- evolution of strange attractors
- homoclinic and heteroclinic points
- invariant curves
- invariant tori
- phase diagram
- Two-dimensional map

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

**Invariant curves, attractors, and phase diagram of a piecewise linear map with chaos.** / Tél, T.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 33, no. 1, pp. 195-221. https://doi.org/10.1007/BF01009756

}

TY - JOUR

T1 - Invariant curves, attractors, and phase diagram of a piecewise linear map with chaos

AU - Tél, T.

PY - 1983/10

Y1 - 1983/10

N2 - We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the map T(x, z)=(1-a|x|+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative case b=1 separatrices and certain invariant tori are calculated.

AB - We introduce equations describing the invariant curves associated with periodic points in a wide class of two-dimensional invertible maps, which in the special case of the map T(x, z)=(1-a|x|+bz,x) can be solved by analytical methods. In the dissipative case several branches of the separatrices of the fixed points, as well as, of the period-2 and -4 points, are constructed. The regions of the parameter space where a given type of strange attractor exists are located. We point out that the disappearance of homoclinic intersections between the separatrices of the fixed point and that of heteroclinic intersections between the unstable manifolds of the period-2 points and the stable manifold of the fixed point may occur separately, and the latter leads already to the appearance of a two-piece strange attractor. This phenomenon may happen at weak dissipation in other maps, too. In the conservative case b=1 separatrices and certain invariant tori are calculated.

KW - evolution of strange attractors

KW - homoclinic and heteroclinic points

KW - invariant curves

KW - invariant tori

KW - phase diagram

KW - Two-dimensional map

UR - http://www.scopus.com/inward/record.url?scp=0013335183&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0013335183&partnerID=8YFLogxK

U2 - 10.1007/BF01009756

DO - 10.1007/BF01009756

M3 - Article

AN - SCOPUS:0013335183

VL - 33

SP - 195

EP - 221

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 1

ER -