Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4974352 | Journal of the Franklin Institute | 2017 | 31 Pages |
Abstract
Given any finite family of real d-by-d nonsingular matrices {S1,â¦,Sl}, by extending the well-known Li-Yorke chaos of a deterministic nonlinear dynamical system to a discrete-time linear inclusion or hybrid or switched system:
xnâ{Skxnâ1;1â¤kâ¤l},x0âRdandnâ¥1,we study the chaotic dynamics of the state trajectory (xn(x0, Ï))n ⥠1 with initial state x0âRd, governed by a switching law Ï:Nâ{1,â¦,l}. Two sufficient conditions are given so that for a “large” set of switching laws Ï, there exhibits the scrambled dynamics as follows: for all x0,y0âRd,x0â y0,lim infnâ+ââ¥xn(x0,Ï)âxn(y0,Ï)â¥=0andlim supnâ+ââ¥xn(x0,Ï)âxn(y0,Ï)â¥=â.This implies that there coexist positive, zero and negative Lyapunov exponents and that the trajectories (xn(x0, Ï))n ⥠1 are extremely sensitive to the initial states x0âRd. We also show that a periodically stable linear inclusion system, which may be product unbounded, does not exhibit any such chaotic behavior. An explicit simple example shows the discontinuity of Lyapunov exponents with respect to the switching laws.
Related Topics
Physical Sciences and Engineering
Computer Science
Signal Processing
Authors
Xiongping Dai, Tingwen Huang, Yu Huang, Yi Luo, Gang Wang, Mingqing Xiao,