Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase

2000 ◽  
pp. 99-114
Author(s):  
Ken Palmer
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Asymptotic Phase

Smooth conjugacy of centre manifolds

1992 ◽  
Vol 120 (1-2) ◽  
pp. 61-77 ◽  
Author(s):  
Almut Burchard ◽  
Bo Deng ◽  
Kening Lu
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Ordinary Differential Equations ◽  
Parabolic Equations ◽  
Geometric Theory ◽  
Unstable Manifolds ◽  
Centre Manifolds ◽  
Smooth Conjugacy

SynopsisIn this paper, we prove that for a system of ordinary differential equations of class Cr+1,1, r≧0 and two arbitrary Cr+1, 1 local centre manifolds of a given equilibrium point, the equations when restricted to the centre manifolds are Cr conjugate. The same result is proved for similinear parabolic equations. The method is based on the geometric theory of invariant foliations for centre-stable and centre-unstable manifolds.

scholarly journals Locating oscillatory orbits of the parametrically-excited pendulum

1996 ◽  
Vol 37 (3) ◽  
pp. 309-319 ◽  
Author(s):  
M. J. Clifford ◽  
S. R. Bishop
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Numerical Techniques ◽  
Stable And Unstable Manifolds ◽  
Parametrically Excited ◽  
Countable Infinity ◽  
Unstable Manifolds

AbstractA method is considered for locating oscillating, nonrotating solutions for the parametrically-excited pendulum by inferring that a particular horseshoe exists in the stable and unstable manifolds of the local saddles. In particular, odd-periodic solutions are determined which are difficult to locate by alternative numerical techniques. A pseudo-Anosov braid is also located which implies the existence of a countable infinity of periodic orbits without the horseshoe assumption being necessary.

ALFVÉN INTERIOR CRISIS

2004 ◽  
Vol 14 (07) ◽  
pp. 2375-2380 ◽  
Author(s):  
F. A. BOROTTO ◽  
A. C.-L. CHIAN ◽  
E. L. REMPEL
Keyword(s):  
Periodic Orbits ◽  
Large Amplitude ◽  
Numerical Study ◽  
Alfven Wave ◽  
Unstable Periodic Orbits ◽  
Stable And Unstable Manifolds ◽  
Laboratory Plasmas ◽  
Derivative Nonlinear Schrödinger Equation ◽  
Unstable Manifolds ◽  
Low Dimensional

A numerical study of an interior crisis of a large-amplitude Alfvén wave described by the driven-dissipative derivative nonlinear Schrödinger equation, in the low-dimensional limit, is reported. An example of Alfvén interior crisis is characterized using the unstable periodic orbits and their associated invariant stable and unstable manifolds in the Poincaré plane. We suggest that this type of chaotic transition can be observed in space and laboratory plasmas.

Smoothness of Asymptotic Phase Revisited

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Flaviano Battelli ◽  
Kenneth J. Palmer
Keyword(s):  
Periodic Solution ◽  
Vector Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stable Manifold ◽  
Autonomous System ◽  
Asymptotic Phase

AbstractIt is well-known that solutions on the stable manifold of a hyperbolic periodic solution of an autonomous system of ordinary differential equations have an asymptotic phase which has the same order of smoothness as the vector field. In this paper we show if the system depends on a parameter that, in general, the asymptotic phase loses one order of smoothness in the parameter.

scholarly journals On Using Curvature to Demonstrate Stability

2008 ◽  
Vol 2008 ◽  
pp. 1-7
Author(s):  
C. Connell McCluskey
Keyword(s):  
Differential Equations ◽  
Global Stability ◽  
Ordinary Differential Equations ◽  
Periodic Orbits ◽  
Large Amplitude ◽  
Minimum Distance ◽  
New Approach ◽  
Sudden Appearance ◽  
Globally Stable ◽  
Rule Out

A new approach for demonstrating the global stability of ordinary differential equations is given. It is shown that if the curvature of solutions is bounded on some set, then any nonconstant orbits that remain in the set, must contain points that lie some minimum distance apart from each other. This is used to establish a negative-criterion for periodic orbits. This is extended to give a method of proving an equilibrium to be globally stable. The approach can also be used to rule out the sudden appearance of large-amplitude periodic orbits.

Sign in / Sign up

Export Citation Format

Share Document