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

2019 ◽  
Vol 398 ◽  
pp. 164-170 ◽  
Author(s):  
Pengcheng Miao ◽  
Denghui Li ◽  
Yuan Yue ◽  
Jianhua Xie ◽  
Celso Grebogi
Keyword(s):  
Normal Form ◽  
Chaotic Attractor ◽  
Impact Oscillators ◽  
Grazing Bifurcations

scholarly journals Grazing bifurcations in impact oscillators

1994 ◽  
Vol 50 (6) ◽  
pp. 4427-4444 ◽  
Author(s):  
Wai Chin ◽  
Edward Ott ◽  
Helena E. Nusse ◽  
Celso Grebogi
Keyword(s):  
Impact Oscillators ◽  
Grazing Bifurcations

Universal grazing bifurcations in impact oscillators

1996 ◽  
Vol 53 (1) ◽  
pp. 134-139 ◽  
Author(s):  
Fernando Casas ◽  
Wai Chin ◽  
Celso Grebogi ◽  
Edward Ott
Keyword(s):  
Impact Oscillators ◽  
Grazing Bifurcations

Symmetry-Breaking and Chaos in Oscillatory Neural Networks

2001 ◽  
Author(s):  
Yonghong Chen ◽  
Jianxue Xu ◽  
Tong Fang
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Symmetry Breaking ◽  
Chaotic Attractor ◽  
Phase Locking ◽  
Dynamical Behavior ◽  
Second Order ◽  
Poincaré Maps ◽  
Poincare Maps

Abstract Complex dynamical behavior of neural networks may lead to new methodology of information processing. In this paper the dynamics of a neural network designed by the normal form for Hopf bifurcation is studied. The secondary Hopf bifurcation of the network is discussed and a two-torus is observed. Examining the phase-locking motions on the two-torus, we present the conditions of symmetry-breaking occurring in the system. If the ratio of the two frequencies of the codimension two Hopf bifurcation is represented by an irreducible fraction, then the symmetry-breaking will occur when either the numerator or the denominator of the fraction is an even number. Chaotic attractors may be created with the sigmoid nonlinearities added to the right hand side of the normal form equations. The phase trajectory and the second order Poincaré maps of the chaotic attractor are given. The chaotic attractor looks like a butterfly on some of the second order Poincaré maps. This is a marvelous example for chaos to mimic nature.

Co-dimension-Two Grazing Bifurcations in Single-Degree-of-Freedom Impact Oscillators

2006 ◽  
Vol 1 (4) ◽  
pp. 328-335 ◽  
Author(s):  
Phanikrishna Thota ◽  
Xiaopeng Zhao ◽  
Harry Dankowicz
Keyword(s):  
Degree Of Freedom ◽  
Mapping Technique ◽  
Global Dynamics ◽  
Grazing Bifurcation ◽  
Single Degree Of Freedom ◽  
Bifurcation Points ◽  
Impact Oscillators ◽  
Single Degree ◽  
Dimension One ◽  
Grazing Bifurcations

Grazing bifurcations in impact oscillators characterize the transition in asymptotic dynamics between impacting and nonimpacting motions. Several different grazing bifurcation scenarios under variations of a single system parameter have been previously documented in the literature. In the present paper, the transition between two characteristically different co-dimension-one grazing bifurcation scenarios is found to be associated with the presence of certain co-dimension-two grazing bifurcation points and their unfolding in parameter space. The analysis investigates the distribution of such degenerate bifurcation points along the grazing bifurcation manifold in examples of single-degree-of-freedom oscillators. Unfoldings obtained with the discontinuity-mapping technique are used to explore the possible influence on the global dynamics of the smooth co-dimension-one bifurcations of the impacting dynamics that emanate from such co-dimension-two points. It is shown that attracting impacting motion may result from parameter variations through a co-dimension-two grazing bifurcation of an initially unstable limit cycle in a nonlinear micro-electro-mechanical systems (MEMS) oscillator.

Control of Chaotic Impacts

1997 ◽  
Vol 07 (04) ◽  
pp. 951-955 ◽  
Author(s):  
Fernando Casas ◽  
Celso Grebogi
Keyword(s):  
Periodic Orbit ◽  
Unstable Periodic Orbit ◽  
Controlling Chaos ◽  
Impact Oscillators ◽  
Universal Properties ◽  
Grazing Bifurcations

We apply controlling chaos techniques to select the desired sequence of impacts in a map that captures universal properties of impact oscillators near grazing. For instance, we can choose the period and then stabilize an unstable periodic orbit with, say, one impact per period involved in the grazing bifurcations that take place in the system.

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