On some problems in perturbation theory of smooth invariant tori of dynamical systems

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.

Download Full-text

Energy Surfaces of Ellipsoidal Billiards

Zeitschrift für Naturforschung A ◽

10.1515/zna-1996-0401 ◽

1996 ◽

Vol 51 (4) ◽

pp. 219-241 ◽

Cited By ~ 1

Author(s):

Jan Wiersig ◽

Peter H. Richter

Keyword(s):

Perturbation Theory ◽

Invariant Tori ◽

Actual Computation ◽

Different Types ◽

Energy Surfaces ◽

Action Variables

Abstract Energy surfaces in the space of action variables are calculated and graphically presented for general triaxial ellipsoidal billiards. As was demonstrated by Jacobi in 1838, the system may be integrated in terms of hyperelliptic functions. The actual computation, however, has never been done. It is found that generic energy surfaces consist of seven pieces, representing topologically different types of invariant tori. The character of the corresponding motion is discussed. Frequencies, winding numbers, and the location of resonances are also determined. The results may serve as a basis for perturbation theory of slightly modified systems, and for semi-classical quantization.

Download Full-text

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

Nonlinear Processes in Geophysics ◽

10.5194/npg-21-165-2014 ◽

2014 ◽

Vol 21 (1) ◽

pp. 165-185 ◽

Cited By ~ 16

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.

Download Full-text

Dynamics of Symmetric Dynamical Systems with Delayed Switching

Journal of Vibration and Control ◽

10.1177/1077546309341124 ◽

2010 ◽

Vol 16 (7-8) ◽

pp. 1111-1140 ◽

Cited By ~ 13

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.

Download Full-text