Stability of Nonsmooth Dynamical Systems

2016 ◽  
pp. 417-476
Author(s):  
Bernard Brogliato
Keyword(s):  
Dynamical Systems ◽  
Nonsmooth Dynamical Systems

Control of a class of nonsmooth dynamical systems

2013 ◽  
Vol 21 (11) ◽  
pp. 2212-2222 ◽  
Author(s):  
Hamid Reza Erfanian ◽  
Mohammad Hadi Noori Skandari ◽  
Ali Vahidian Kamyad
Keyword(s):  
Dynamical Systems ◽  
Nonsmooth Dynamical Systems

New Discontinuity-Induced Bifurcations in Chua's Circuit

2015 ◽  
Vol 25 (06) ◽  
pp. 1550090 ◽  
Author(s):  
Shihui Fu ◽  
Qishao Lu ◽  
Xiangying Meng
Keyword(s):  
Dynamical Systems ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Chua’S Circuit ◽  
Jacobian Matrices ◽  
Chua's Circuit ◽  
Before And After ◽  
Boundary Equilibrium ◽  
Discontinuous Bifurcations ◽  
Nonsmooth Dynamical Systems

Chua's circuit, an archetypal example of nonsmooth dynamical systems, exhibits mostly discontinuous bifurcations. More complex dynamical phenomena of Chua's circuit are presented here due to discontinuity-induced bifurcations. Some new kinds of classical bifurcations are revealed and analyzed, including the coexistence of two classical bifurcations and bifurcations of equilibrium manifolds. The local dynamical behavior of the boundary equilibrium points located on switch boundaries is found to be determined jointly by the Jacobian matrices evaluated before and after switching. Some new discontinuous bifurcations are also observed, such as the coexistence of two discontinuous and one classical bifurcation.

scholarly journals Attractor and Vector Structure Analyses of Bursting Oscillation with Sliding Bifurcation in Filippov Systems

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Rui Qu ◽  
Shaolong Li
Keyword(s):  
Dynamical Systems ◽  
Simple Model ◽  
Geometric Structure ◽  
Constant Voltage ◽  
Periodic Excitation ◽  
Practical Application ◽  
Sliding Bifurcation ◽  
Vector Structure ◽  
Nonsmooth Dynamical Systems ◽  
Smooth System

The main purpose of this paper is to investigate the mechanism of sliding phenomenon in Filippov (nonsmooth) dynamical systems by attractor analysis and vector analysis. A corresponding simple model based on Chua’s circuit with periodic excitation was introduced as an example. The attractor analysis proposed in our previous work is used to discuss the complicated oscillations of the Filippov system. However, it failed to perfectly explain the sliding phenomena and establish an analytical method of constant voltage control. Therefore, the geometric structure and analytic conditions of sliding bifurcations in the general n-dimensional piecewise smooth system are discussed in detail by vector structure analysis. The prospects of practical application of this method are also discussed in the end.

scholarly journals An Iteration Scheme Suitable for Solving Limit Cycles of Nonsmooth Dynamical Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Q. X. Liu ◽  
Y. M. Chen ◽  
J. K. Liu
Keyword(s):  
Dynamical Systems ◽  
Limit Cycles ◽  
Nonlinear Damping ◽  
Iteration Scheme ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Nonsmooth Systems ◽  
Nonlinear Algebraic Equations ◽  
Nonsmooth Dynamical Systems ◽  
Previous Iteration

The Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities and/or nonlinear damping terms. If the MIM is implemented routinely, the unknown frequency and amplitude of limit cycle (LC) would couple to each other in complicated nonlinear algebraic equations at each iteration. It is cumbersome to solve these algebraic equations, especially for nonsmooth systems. In the modified procedures, the unknown frequency is substituted by the determined value obtained at the previous iteration. By this means, the frequency is decoupled from the nonlinear terms. Numerical examples show that the LCs obtained by the modified MIM agree well with numerical results. The presented method is very suitable for solving self-excited systems, especially those with nonlinear damping and nonsmooth nonlinearities.

scholarly journals Transition to high-dimensional chaos in nonsmooth dynamical systems

2018 ◽  
Vol 98 (5) ◽  
Author(s):  
Ru-Hai Du ◽  
Shi-Xian Qu ◽  
Ying-Cheng Lai
Keyword(s):  
Dynamical Systems ◽  
High Dimensional ◽  
Nonsmooth Dynamical Systems

Grazing and Chaos in a Periodically Forced, Piecewise Linear System

2005 ◽  
Vol 128 (1) ◽  
pp. 28-34 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Linear System ◽  
Strange Attractor ◽  
Piecewise Linear ◽  
Mathematical Structure ◽  
Strange Attractors ◽  
Grazing Bifurcation ◽  
Piecewise Linear System ◽  
Nonsmooth Dynamical Systems ◽  
Analytic Prediction

The criteria for the grazing bifurcation of a periodically forced, piecewise linear system are developed and the initial grazing manifolds are obtained. The initial grazing manifold is invariant. The grazing flows are illustrated to verify the analytic prediction of grazing. The mechanism of the strange attractors fragmentation caused by the grazing is discussed, and an illustration of the fragmentized strange attractor is given through the Poincaré mapping. This fragmentation phenomenon exists extensively in nonsmooth dynamical systems. The mathematical structure of the fragmentized strange attractors should be further developed.

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