Moduli of continuity of the derivatives of invariant tori for linear extensions of dynamical systems

2000 ◽  
Vol 36 (1) ◽  
pp. 120-131 ◽  
Author(s):  
A. M. Samoilenko ◽  
A. A. Burilko ◽  
I. N. Grod
Keyword(s):  
Dynamical Systems ◽  
Invariant Tori ◽  
Moduli Of Continuity ◽  
Linear Extensions ◽  
Derivatives Of

Existence of Green-Samoilenko Functions of Some Linear Extensions of Dynamical Systems

2004 ◽  
Vol 7 (4) ◽  
pp. 454-460
Author(s):  
H. M. Kulyk ◽  
V. L. Kulyk
Keyword(s):  
Dynamical Systems ◽  
Linear Extensions

ROTATION VECTORS FOR TORAL MAPS AND FLOWS: A TUTORIAL

1995 ◽  
Vol 05 (02) ◽  
pp. 321-348 ◽  
Author(s):  
JAMES A. WALSH
Keyword(s):  
Dynamical Systems ◽  
Rotation Number ◽  
Chaotic Behavior ◽  
Invariant Tori ◽  
Mode Locking ◽  
Rotation Vector ◽  
Higher Genus ◽  
Rotation Vectors ◽  
Dissipative Dynamical Systems ◽  
Rotation Set

This paper is an introduction to the concept of rotation vector defined for maps and flows on the m-torus. The rotation vector plays an important role in understanding mode locking and chaos in dissipative dynamical systems, and in understanding the transition from quasiperiodic motion on attracting invariant tori in phase space to chaotic behavior on strange attractors. Throughout this article the connection between the rotation vector and the dynamics of the map or flow is emphasized. We begin with a brief introduction to the dimension one setting, in which case the rotation vector reduces to the well known rotation number of H. Poincaré. A survey of the main results concerning the rotation number and bifurcations of circle maps is presented. The various definitions of rotation vector in the higher dimensional setting are then introduced with emphasis again placed on how certain properties of the rotation set relate to the dynamics of the map or flow. The dramatic differences between results in dimension two and results in higher dimensions are also presented. The tutorial concludes with a brief introduction to extensions of the concept of rotation vector to the setting of dynamical systems defined on surfaces of higher genus.

Simulation and Modelling Knowledge-Mining Architectures Using Recurrent Hybrid Nets

2008 ◽  
pp. 266-306
Author(s):  
David Al-Dabass
Keyword(s):  
Signal Processing ◽  
Dynamical Systems ◽  
Intelligent Systems ◽  
Behaviour Pattern ◽  
Complex Behaviour ◽  
Knowledge Mining ◽  
Two Stages ◽  
Mining Model ◽  
Derivatives Of ◽  
Theoretical Foundations

Hybrid recurrent nets combine arithmetic and integrator elements to form nodes for modeling the complex behaviour of intelligent systems with dynamics. Given the behaviour pattern of such nodes it is required to determine the values of their causal parameters. The architecture of this knowledge mining process consists of two stages: time derivatives of the trajectory are determined first, followed by the parameters. Hybrid recurrent nets of first order are employed to compute derivatives continuously as the behaviour is monitored. A further layer of arithmetic and hybrid nets is then used to track the values of the causal parameters of the knowledge mining model. Applications to signal processing are used to illustrate the techniques. The theoretical foundations of this knowledge mining process is presented in the first part of the chapter, where the application of dynamical systems theory is extended to abstract systems to illustrate its broad relevance to any system including biological and non physical processes. It models the complexity of systems in terms of observability and controllability.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

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