Lyapunov exponents for higher dimensional random maps

A random map is a discrete time dynamical system in which one of a number of transformations is selected randomly and implemented. Random maps have been used recently to model interference effects in quantum physics. The main results of this paper deal with the Lyapunov exponents for higher dimensional random maps, where the individual maps are Jabloński maps on the n-dimensional cube.

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Invariant densities of random maps have lower bounds on their supports

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/jamsa/2006/79175 ◽

2006 ◽

Vol 2006 ◽

pp. 1-13

Author(s):

Paweł Góra ◽

Abraham Boyarsky ◽

Md Shafiqul Islam

Keyword(s):

Dynamical System ◽

Lower Bounds ◽

Discrete Time ◽

Asymptotic Properties ◽

Invariant Density ◽

Random Maps ◽

Discrete Time Dynamical System ◽

Invariant Densities

A random map is a discrete-time dynamical system in which one of a number of transformations is randomly selected and applied at each iteration of the process. The asymptotic properties of a random map are described by its invariant densities. If Pelikan's average expanding condition is satisfied, then the random map has invariant densities. For individual maps, piecewise expanding is sufficient to establish many important properties of the invariant densities, in particular, the fact that the densities are bounded away from 0 on their supports. It is of interest to see if this property is transferred to random maps satisfying Pelikan's condition. We show that if all the maps constituting the random map are piecewise expanding, then the same result is true. However, if one or more of the maps are not expanding, this may not be true: we present an example where Pelikan's condition is satisfied, but not all the maps are piecewise expanding, and show that the invariant density is not separated from 0.

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Ocean ecosystem discrete time dynamics generated by ℓ-Volterra operators

International Journal of Biomathematics ◽

10.1142/s1793524519500153 ◽

2019 ◽

Vol 12 (02) ◽

pp. 1950015 ◽

Cited By ~ 1

Author(s):

U. A. Rozikov ◽

S. K. Shoyimardonov

Keyword(s):

Dynamical System ◽

Fixed Points ◽

Discrete Time ◽

Volterra Operator ◽

Saddle Points ◽

Time Dynamics ◽

Discrete Time Dynamical System ◽

Starting Point ◽

Ocean Ecosystem ◽

Initial Starting

We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters [Formula: see text]) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a [Formula: see text]-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if [Formula: see text], then (under some conditions on [Formula: see text]) this [Formula: see text]-Volterra operator may have up to three or a countable set of fixed points; if [Formula: see text], then the operator has up to three fixed points. Depending on the parameters, the fixed points may be attracting, repelling or saddle points. The limit behaviors of trajectories of the dynamical system are studied. It is shown that independently on values of parameters and on initial (starting) point, all trajectories converge. Thus, the operator (dynamical system) is regular. We give some biological interpretations of our results.

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Feedback Anticontrol of Discrete Chaos

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127498001236 ◽

1998 ◽

Vol 08 (07) ◽

pp. 1585-1590 ◽

Cited By ~ 130

Author(s):

Guanrong Chen ◽

Dejian Lai

Keyword(s):

Dynamical System ◽

Discrete Time ◽

State Feedback ◽

Design Method ◽

Control Design ◽

Rigorous Approach ◽

Discrete Time Dynamical System ◽

Discrete Chaos ◽

Simple Feedback ◽

Discrete Time Dynamical Systems

In this paper, a simple feedback control design method earlier proposed by us for discrete-time dynamical systems is proved to be a mathematically rigorous approach for anticontrol of chaos, in the sense that any given discrete-time dynamical system can be made chaotic by the designed state-feedback controller along with the mod-operations.

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