### Abstract

Grazing bifurcations can cause impact oscillators to exhibit chaotic motions. Such dynamical behaviour can be described by a normal form map (called the Nordmark map). A main feature of the Nordmark map is that it has a square-root term. The purpose of this paper is to study the structure of the chaotic attractor of the Nordmark map from the topological point of view. First, the trapping region of the asymptotic dynamics of the map is constructed. It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω-limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. Besides, the dynamics on the invariant set is topologically mixing. Accordingly the invariant set is a chaotic attractor of the map.

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

Pages (from-to) | 164-170 |

Number of pages | 7 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 398 |

Early online date | 29 Mar 2019 |

DOIs | |

Publication status | Published - Nov 2019 |

### Fingerprint

### Keywords

- Chaotic attractor
- Impact oscillator
- Nordmark map
- Trapping region
- DYNAMICS
- SYSTEMS
- STRANGE ATTRACTORS
- BORDER-COLLISION BIFURCATIONS

### ASJC Scopus subject areas

- Condensed Matter Physics
- Applied Mathematics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*398*, 164-170. https://doi.org/10.1016/j.physd.2019.03.007

**Chaotic attractor of the normal form map for grazing bifurcations of impact oscillators.** / Miao, Pengcheng; Li, Denghui (Corresponding Author); Yue, Yuan; Xie, Jianhua; Grebogi, Celso.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 398, pp. 164-170. https://doi.org/10.1016/j.physd.2019.03.007

}

TY - JOUR

T1 - Chaotic attractor of the normal form map for grazing bifurcations of impact oscillators

AU - Miao, Pengcheng

AU - Li, Denghui

AU - Yue, Yuan

AU - Xie, Jianhua

AU - Grebogi, Celso

N1 - This work is supported by the National Natural Science Foundation of China (11572263, 11672249 and 11732014).

PY - 2019/11

Y1 - 2019/11

N2 - Grazing bifurcations can cause impact oscillators to exhibit chaotic motions. Such dynamical behaviour can be described by a normal form map (called the Nordmark map). A main feature of the Nordmark map is that it has a square-root term. The purpose of this paper is to study the structure of the chaotic attractor of the Nordmark map from the topological point of view. First, the trapping region of the asymptotic dynamics of the map is constructed. It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω-limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. Besides, the dynamics on the invariant set is topologically mixing. Accordingly the invariant set is a chaotic attractor of the map.

AB - Grazing bifurcations can cause impact oscillators to exhibit chaotic motions. Such dynamical behaviour can be described by a normal form map (called the Nordmark map). A main feature of the Nordmark map is that it has a square-root term. The purpose of this paper is to study the structure of the chaotic attractor of the Nordmark map from the topological point of view. First, the trapping region of the asymptotic dynamics of the map is constructed. It is then proven that, for some set of parameter values having positive Lebesgue measure, the ω-limit set of each point of the trapping region is contained in a invariant set which is just the closure of the unstable manifold of the hyperbolic fixed point of the map. Besides, the dynamics on the invariant set is topologically mixing. Accordingly the invariant set is a chaotic attractor of the map.

KW - Chaotic attractor

KW - Impact oscillator

KW - Nordmark map

KW - Trapping region

KW - DYNAMICS

KW - SYSTEMS

KW - STRANGE ATTRACTORS

KW - BORDER-COLLISION BIFURCATIONS

UR - http://www.scopus.com/inward/record.url?scp=85064695956&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/chaotic-attractor-normal-form-map-grazing-bifurcations-impact-oscillators

U2 - 10.1016/j.physd.2019.03.007

DO - 10.1016/j.physd.2019.03.007

M3 - Article

VL - 398

SP - 164

EP - 170

JO - Physica. D, Nonlinear Phenomena

JF - Physica. D, Nonlinear Phenomena

SN - 0167-2789

ER -