Exponentially asymptotically stable dynamical systems

1987 ◽  
Vol 25 (4) ◽  
pp. 243-252 ◽  
Author(s):  
Saber Elaydi ◽  
Hani R. Farran
Keyword(s):  
Dynamical Systems ◽  
Asymptotically Stable

scholarly journals ON DIFFUSION DRIVEN OSCILLATIONS IN COUPLED DYNAMICAL SYSTEMS

1999 ◽  
Vol 09 (04) ◽  
pp. 629-644 ◽  
Author(s):  
ALEXANDER POGROMSKY ◽  
TORKEL GLAD ◽  
HENK NIJMEIJER
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Systems ◽  
Threshold Value ◽  
Oscillatory Behavior ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Nonminimum Phase ◽  
Nonminimum Phase Systems ◽  
Diffusive Medium ◽  
Phase Systems

The paper deals with the problem of destabilization of diffusively coupled identical systems. Following a question of Smale [1976], it is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillatory behavior. It is shown that if the diffusive medium consists of hyperbolically nonminimum phase systems and the diffusive factors exceed some threshold value, the origin of the overall system undergoes a Poincaré–Andronov–Hopf bifurcation resulting in oscillatory behavior.

A New Result on Fractional Differential Inequality and Applications to Control of Dynamical Systems

2019 ◽  
Vol 141 (9) ◽  
Author(s):  
Ngo Van Hoa ◽  
Tran Minh Duc ◽  
Ho Vu
Keyword(s):  
Dynamical Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Dynamic Systems ◽  
Differential Inequality ◽  
Sliding Mode ◽  
Control Law ◽  
Asymptotically Stable ◽  
Mode Control ◽  
Fractional Differential

In this work, we establish a new estimate result for fractional differential inequality, and this inequality is used to derive a robust sliding mode control law for the fractional-order (FO) dynamic systems. The sliding mode control law is provided to make the states of the system asymptotically stable. Some examples are given to illustrate the results.

EMERGENT CHAOS SYNCHRONIZATION IN NONCHAOTIC CNNS

2008 ◽  
Vol 18 (05) ◽  
pp. 1337-1342 ◽  
Author(s):  
XIAO-SONG YANG ◽  
QUAN YUAN
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Chaos Synchronization ◽  
Chaotic Dynamics ◽  
Cellular Neural Networks ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable

It is shown that emergent chaos synchronization can take place in coupled nonchaotic unit dynamical systems. This is demonstrated by coupling two nonchaotic cellular neural networks, in which the couplings give rise to a synchronous chaotic dynamics and in the meanwhile the synchronous dynamics is globally asymptotically stable, thus chaos synchronization takes place under the suitable couplings.

LYAPUNOV EXPONENTS ON METRIC SPACES

2017 ◽  
Vol 97 (1) ◽  
pp. 153-162 ◽  
Author(s):  
C. A. MORALES ◽  
P. THIEULLEN ◽  
H. VILLAVICENCIO
Keyword(s):  
Lyapunov Exponent ◽  
Dynamical Systems ◽  
Ergodic Theory ◽  
Riemannian Manifolds ◽  
Metric Spaces ◽  
Lipschitz Constant ◽  
Topological Conjugacy ◽  
Asymptotically Stable ◽  
Characteristic Exponents ◽  
Stable Points

We use the pointwise Lipschitz constant to define an upper Lyapunov exponent for maps on metric spaces different to that given by Kifer [‘Characteristic exponents of dynamical systems in metric spaces’, Ergodic Theory Dynam. Systems3(1) (1983), 119–127]. We prove that this exponent reduces to that of Bessa and Silva on Riemannian manifolds and is not larger than that of Kifer at stable points. We also prove that it is invariant along orbits in the case of (topological) diffeomorphisms and under topological conjugacy. Moreover, the periodic orbits where this exponent is negative are asymptotically stable. Finally, we estimate this exponent for certain hyperbolic homeomorphisms.

Necessary and sufficient conditions for stable synchronization in random dynamical systems

2017 ◽  
Vol 38 (5) ◽  
pp. 1857-1875 ◽  
Author(s):  
JULIAN NEWMAN
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Compact Space ◽  
Sufficient Conditions ◽  
Invariant Set ◽  
Stochastic Flow ◽  
Asymptotically Stable ◽  
Random Maps ◽  
Necessary And Sufficient ◽  
Being Moved

For a composition of independent and identically distributed random maps or a memoryless stochastic flow on a compact space$X$, we find conditions under which the presence of locally asymptotically stable trajectories (e.g. as given by negative Lyapunov exponents) implies almost-sure mutual convergence of any given pair of trajectories (‘synchronization’). Namely, we find that synchronization occurs and is ‘stable’ if and only if the system exhibits the following properties: (i) there is asmallestnon-empty invariant set$K\subset X$; (ii) any two points in$K$are capable of being moved closer together; and (iii) $K$admits asymptotically stable trajectories.

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