Computing Invariant Tori and Circles in Dynamical Systems

2000 ◽  
pp. 407-437 ◽  
Author(s):  
Volker Reichelt
Keyword(s):  
Dynamical Systems ◽  
Invariant Tori

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.

scholarly journals Barriers to transport in aperiodically time-dependent two-dimensional velocity fields: Nekhoroshev's theorem and "Nearly Invariant" tori

2014 ◽  
Vol 21 (1) ◽  
pp. 165-185 ◽  
Author(s):  
S. Wiggins ◽  
A. M. Mancho
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Fluid Transport ◽  
Invariant Tori ◽  
Point Of View ◽  
Time Dependent ◽  
Time Interval ◽  
Two Dimensional ◽  
Kam Theorem ◽  
Nekhoroshev Theorem

Abstract. In this paper we consider fluid transport in two-dimensional flows from the dynamical systems point of view, with the focus on elliptic behaviour and aperiodic and finite time dependence. We give an overview of previous work on general nonautonomous and finite time vector fields with the purpose of bringing to the attention of those working on fluid transport from the dynamical systems point of view a body of work that is extremely relevant, but appears not to be so well known. We then focus on the Kolmogorov–Arnold–Moser (KAM) theorem and the Nekhoroshev theorem. While there is no finite time or aperiodically time-dependent version of the KAM theorem, the Nekhoroshev theorem, by its very nature, is a finite time result, but for a "very long" (i.e. exponentially long with respect to the size of the perturbation) time interval and provides a rigorous quantification of "nearly invariant tori" over this very long timescale. We discuss an aperiodically time-dependent version of the Nekhoroshev theorem due to Giorgilli and Zehnder (1992) (recently refined by Bounemoura, 2013 and Fortunati and Wiggins, 2013) which is directly relevant to fluid transport problems. We give a detailed discussion of issues associated with the applicability of the KAM and Nekhoroshev theorems in specific flows. Finally, we consider a specific example of an aperiodically time-dependent flow where we show that the results of the Nekhoroshev theorem hold.

scholarly journals Dynamics of Symmetric Dynamical Systems with Delayed Switching

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1111-1140 ◽  
Author(s):  
J. Sieber ◽  
P. Kowalczyk ◽  
S.J. Hogan ◽  
M. Di Bernardo
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Poincaré Map ◽  
Invariant Tori ◽  
Poincare Map ◽  
Single Degree Of Freedom ◽  
Finite Dimensional ◽  
Symmetric Periodic Orbits ◽  
Full Reflection

We study dynamical systems that switch between two different vector fields depending on a discrete variable and with a delay. When the delay reaches a problem-dependent critical value, so-called event collisions occur. This paper classifies and analyzes event collisions, a special type of discontinuity-induced bifurcations, for periodic orbits. Our focus is on event collisions of symmetric periodic orbits in systems with full reflection symmetry, a symmetry that is prevalent in applications. We derive an implicit expression for the Poincaré map near the colliding periodic orbit. The Poincaré map is piecewise smooth, finite-dimensional, and changes the dimension of its image at the collision. In the second part of the paper we apply this general result to the class of unstable linear single-degree-of-freedom oscillators where we detect and continue numerically collisions of invariant tori. Moreover, we observe that attracting closed invariant polygons emerge at the torus collision.

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

A NEWTON METHOD FOR LOCATING INVARIANT TORI OF MAPS

2006 ◽  
Vol 16 (05) ◽  
pp. 1491-1503 ◽  
Author(s):  
HARRY DANKOWICZ ◽  
GUNJAN THAKUR
Keyword(s):  
Dynamical Systems ◽  
Invariant Torus ◽  
Local Stability ◽  
Discrete System ◽  
Invariant Tori ◽  
A Priori ◽  
Spatial Location ◽  
Fixed Time ◽  
Periodic Trajectory ◽  
Shift Function

This paper presents an iterative method for locating invariant tori of maps. Specifically, through the introduction of a shift function on the torus corresponding to the projection of the dynamics on the torus onto toral parameters, a discrete system of equations may be formulated whose solution approximates the spatial location of the torus. As invariant tori of continuous flows may be considered invariant under the application of the flow for a fixed time, or may correspond to invariant tori (of one less dimension) of suitably introduced Poincaré maps, the methodology applies to discrete and continuous dynamical systems alike. Moreover, the insensitivity of the method to the local stability characteristics of the invariant torus as well as to the precise nature of the flow on the torus imply, for example, that the method can be employed for the continuation of unstable invariant tori on which the dynamics are attracted to a periodic trajectory as long as the torus is sufficiently smooth. The proposed methodology as well as reduced formulations that result from a priori knowledge about the invariant torus are illustrated through some sample dynamical systems.

SUFFICIENT CONDITIONS FOR THE EXISTENCE OF INVARIANT TORI IN MULTIDIMENSIONAL DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 681-689
Author(s):  
V. S. AFRAIMOVICH ◽  
A. L. ZHELEZNYAK ◽  
I. L. ZHELEZNYAK
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Strange Attractors

A method for analyzing the existence of a multidimensional torus for certain systems of ordinary differential equations is proposed. Using this method, the existence of a multidimensional torus in one of such systems is analyzed. This system is a mathematical model of the dynamics of interacting structures within the drift hydrodynamical systems. The behavior of trajectories on the multidimensional torus is numerically investigated. The existence of two- and three-dimensional tori as well as strange attractors are considered.

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