Constructing Permutations that Approximate Lebesgue Measure Preserving Dynamical Systems Under Spatial Discretization

1997 ◽  
Vol 07 (02) ◽  
pp. 401-406 ◽  
Author(s):  
P. E. Kloeden ◽  
J. Mustard
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Discrete Time ◽  
Fast Algorithm ◽  
Dynamical Behavior ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Uniform Lattice ◽  
Discrete Time Dynamical System

A discrete-time dynamical system can sometimes display quite different dynamical behavior under spatial discretization. Systems generated by maps for which the Lebesgue measure is invariant are, however, robust in the sense that they can be approximated by permutations on a uniform lattice. A fast algorithm to construct such permutations is presented here and its implementation is illustrated with several examples of well–known one and two dimensional systems.

scholarly journals On Sufficient Conditions for Chaotic Behavior of Multidimensional Discrete Time Dynamical System

2021 ◽  
Author(s):  
Nor Syahmina Kamarudin ◽  
Syahida Che Dzul-Kifli
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Discrete Dynamical System ◽  
Sufficient Conditions ◽  
Discrete Dynamical Systems ◽  
Continuous Map ◽  
Discrete Time Dynamical System ◽  
Periodicity Property

AbstractIn this work, we look at the extension of classical discrete dynamical system to multidimensional discrete-time dynamical system by characterizing chaos notions on $${\mathbb {Z}}^d$$ Z d -action. The $${\mathbb {Z}}^d$$ Z d -action on a space X has been defined in a very general manner, and therefore we introduce a $${\mathbb {Z}}^d$$ Z d -action on X which is induced by a continuous map, $$f:{\mathbb {Z}}\times X \rightarrow X$$ f : Z × X → X and denotes it as $$T_f:{\mathbb {Z}}^d \times X \rightarrow X$$ T f : Z d × X → X . Basically, we wish to relate the behavior of origin discrete dynamical systems (X, f) and its induced multidimensional discrete-time $$(X,T_f)$$ ( X , T f ) . The chaotic behaviors that we emphasized are the transitivity and dense periodicity property. Analogues to these chaos notions, we consider k-type transitivity and k-type dense periodicity property in the multidimensional discrete-time dynamical system. In the process, we obtain some conditions on $$(X,T_f)$$ ( X , T f ) under which the chaotic behavior of $$(X,T_f)$$ ( X , T f ) is inherited from the original dynamical system (X, f). The conditions varies whenever f is open, totally transitive or mixing. Some examples are given to illustrate these conditions.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Ocean ecosystem discrete time dynamics generated by ℓ-Volterra operators

2019 ◽  
Vol 12 (02) ◽  
pp. 1950015 ◽  
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.

scholarly journals Lyapunov maps, simplicial complexes and the Stone functor

1992 ◽  
Vol 12 (1) ◽  
pp. 153-183 ◽  
Author(s):  
Joel W. Robbin ◽  
Dietmar A. Salamon
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Discrete Time ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Simplicial Complexes ◽  
Attractor Network ◽  
Partially Ordered ◽  
Connection Matrices ◽  
Discrete Time Dynamical Systems

AbstractLet be an attractor network for a dynamical system ft: M → M, indexed by the lower sets of a partially ordered set P. Our main theorem asserts the existence of a Lyapunov map ψ:M → K(P) which defines the attractor network. This result is used to prove the existence of connection matrices for discrete-time dynamical systems.

Feedback Anticontrol of Discrete Chaos

1998 ◽  
Vol 08 (07) ◽  
pp. 1585-1590 ◽  
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.

ON SUPERSYMMETRIC NONLINEAR σ MODELS AND CLASSICAL DYNAMICAL SYSTEMS

1996 ◽  
Vol 11 (06) ◽  
pp. 1101-1115
Author(s):  
ANTTI J. NIEMI ◽  
KAUPO PALO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Partition Function ◽  
Integrable Systems ◽  
Morse Theory ◽  
General Class ◽  
Loop Space ◽  
Two Dimensional ◽  
Hamiltonian Flow ◽  
Classical Dynamical System

We construct a two-dimensional nonlinear σ model that describes the Hamiltonian flow in the loop space of a classical dynamical system. This model is obtained by equivariantizing the standard N=1 supersymmetric nonlinear σ model by the Hamiltonian flow. We use localization methods to evaluate the corresponding partition function for a general class of integrable systems, and find relations that can be viewed as generalizations of standard relations in classical Morse theory.

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