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.

Positive stable realizations of discrete-time linear systems

2012 ◽  
Vol 60 (3) ◽  
pp. 605-616
Author(s):  
T. Kaczorek
Keyword(s):  
Transfer Function ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Numerical Examples ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

Abstract The problem of existence and determination of the set of positive asymptotically stable realizations of a proper transfer function of linear discrete-time systems is formulated and solved. Necessary and sufficient conditions for existence of the set of the realizations are established. A procedure for computation of the set of realizations are proposed and illustrated by numerical examples.

scholarly journals Impulsive controllability of linear dynamical systems with applications to maneuvers of spacecraft

1996 ◽  
Vol 2 (4) ◽  
pp. 277-299 ◽  
Author(s):  
Xinzhi Liu ◽  
Allan R. Willms
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Dynamical Systems ◽  
Novel Approach ◽  
Necessary And Sufficient

Necessary and sufficient conditions for impulsive controllability of linear dynamical systems are obtained, which provide a novel approach to problems that are basically defined by continuous dynamical systems, but on which only discrete-time actions are exercised. As an application, impulsive maneuvering of a spacecraft is discussed.

Switchability in Discontinuous, Discrete Dynamical Systems

2013 ◽  
Author(s):  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Discrete Dynamical Systems ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

Tin this paper, a theory for switchability and singularity of discontinuous, discrete dynamical systems. G-functions for the discrete dynamical systems are introduced through the boundary, and the necessary and sufficient conditions for the switchability of discrete mappings are presented.

ON FLOW BARRIERS AND SWITCHABILITY IN DISCONTINUOUS DYNAMICAL SYSTEMS

2011 ◽  
Vol 21 (01) ◽  
pp. 1-76 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Friction Model ◽  
Necessary And Sufficient Conditions ◽  
Separation Boundary ◽  
Boundary Flow ◽  
Pass Through ◽  
Necessary And Sufficient ◽  
And Control

In this paper, the theory of flow barriers in discontinuous dynamical systems is systematically presented as a new theory for the first time, which helps one rethink the existing theories of stability and control in dynamical systems. The concept of flow barriers in discontinuous dynamical systems is introduced, and the passability of a flow to the separation boundary with flow barriers is presented. Because the flow barriers exist on the separation boundary, the switchability of a flow to such a separation boundary is changed accordingly. The coming and leaving flow barriers in passable flows are discussed first, and the necessary and sufficient conditions for a flow to pass through the boundary with flow barrier are developed. Flow barriers for sink and source flows are also discussed. Once the sink flow is formed, the boundary flow will exist. When the boundary flow disappears from the boundary, the boundary flow barrier on the boundary may exist, which is independent of vector fields in the corresponding domains. Thus, the necessary and sufficient conditions for formations and vanishing of the boundary flow are developed. A periodically forced friction model is presented as an example for a better understanding of flow barrier existence in physical problems. The flow barrier theory presented in this paper may provide a theoretic base to further develop control theory and stability.

scholarly journals Asymptotic stability of a full term linear difference equation with two parameters

2015 ◽  
Vol 63 (1) ◽  
pp. 283-290
Author(s):  
Petr Tomášek
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Linear Difference Equation ◽  
Full Term ◽  
Asymptotically Stable ◽  
Term Linear ◽  
Asymptotic Stability Region ◽  
Two Parameters ◽  
Necessary And Sufficient

Abstract The paper introduces an efficient form of necessary and sufficient conditions for a special full term linear difference equation with two real parameters to be asymptotically stable. The result is obtained utilizing the Schur Cohn criterion. The asymptotic stability region in the parameters plane is also illustrated in the paper.

On Weak Asymptotic Periodicity

2020 ◽  
Vol 30 (02) ◽  
pp. 2050030
Author(s):  
Karol Gryszka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Property ◽  
Metric Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Topological Conjugacy ◽  
Asymptotic Periodicity ◽  
Necessary And Sufficient

We introduce the asymptotic property associated with recurrence-like behavior of orbits in dynamical systems in general metric spaces. We define a notion of weak asymptotic periodicity and determine its elementary properties and relations including the invariance by topological conjugacy. We use the equicontinuity and the topology of the space to describe necessary and sufficient conditions for the existence of such a behavior.

Controllability of nonlinear stochastic fractional higher order dynamical systems

2019 ◽  
Vol 22 (4) ◽  
pp. 1063-1085
Author(s):  
R. Mabel Lizzy ◽  
K. Balachandran ◽  
Yong-Ki Ma
Keyword(s):  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Banach Fixed Point Theorem ◽  
Necessary And Sufficient Condition ◽  
Bounded Operators ◽  
Semilinear Systems ◽  
Operator Functions ◽  
Fractional System ◽  
Necessary And Sufficient ◽  
The Given

Abstract This paper deals with the study of controllability of stochastic fractional dynamical systems with 1 < α ≤ 2. Necessary and sufficient condition for controllability of linear stochastic fractional system is obtained. Sufficient conditions for controllability of stochastic fractional semilinear systems, integrodifferential systems, systems with neutral term, systems with delays in control and systems with Lévy noise is formulated and established. The solution is obtained in terms of Mittag-Leffler operator functions by considering bounded operators. The Banach fixed point theorem is used to obtain the desired results from an equivalent nonlinear integral equation of the given system.

DISSIPATIVE DYNAMICAL SYSTEMS IN A BEHAVIORAL CONTEXT

1991 ◽  
Vol 01 (01) ◽  
pp. 1-25 ◽  
Author(s):  
SIEP WEILAND ◽  
JAN C. WILLEMS
Keyword(s):  
Dynamical Systems ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Dissipative System ◽  
Linear Time Invariant ◽  
Linear Time Invariant Systems ◽  
Dissipative Dynamical Systems ◽  
Necessary And Sufficient ◽  
Lq Theory ◽  
Time Invariant

Various conceptual definitions of dissipativeness of time invariant dynamical systems are introduced. A formal distinction is made between external and internal dissipativeness and it is shown that, under certain conditions, these notions are equivalent. A characterization of the class of internal storage functions associated with a dissipative system is given. The results are applied to the class of finite-dimensional linear time invariant systems. Necessary and sufficient conditions for dissipativeness of systems in this class are derived and the relation to LQ-theory is discussed.

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