Lecture 16. Geodesic Flows and Their Generalizations. Discontinuous Dynamical Systems. Stable and Unstable Manifolds

2017 ◽  
pp. 175-193
Keyword(s):  
Dynamical Systems ◽  
Geodesic Flows ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds

scholarly journals Finite-time Lagrangian transport analysis: stable and unstable manifolds of hyperbolic trajectories and finite-time Lyapunov exponents

2010 ◽  
Vol 17 (1) ◽  
pp. 1-36 ◽  
Author(s):  
M. Branicki ◽  
S. Wiggins
Keyword(s):  
Dynamical Systems ◽  
Finite Time ◽  
Vector Fields ◽  
Transient Flow ◽  
Time Interval ◽  
Stable And Unstable Manifolds ◽  
Lagrangian Transport ◽  
Transport Barriers ◽  
Unstable Manifolds ◽  
Hyperbolic Trajectories

Abstract. We consider issues associated with the Lagrangian characterisation of flow structures arising in aperiodically time-dependent vector fields that are only known on a finite time interval. A major motivation for the consideration of this problem arises from the desire to study transport and mixing problems in geophysical flows where the flow is obtained from a numerical solution, on a finite space-time grid, of an appropriate partial differential equation model for the velocity field. Of particular interest is the characterisation, location, and evolution of transport barriers in the flow, i.e. material curves and surfaces. We argue that a general theory of Lagrangian transport has to account for the effects of transient flow phenomena which are not captured by the infinite-time notions of hyperbolicity even for flows defined for all time. Notions of finite-time hyperbolic trajectories, their finite time stable and unstable manifolds, as well as finite-time Lyapunov exponent (FTLE) fields and associated Lagrangian coherent structures have been the main tools for characterising transport barriers in the time-aperiodic situation. In this paper we consider a variety of examples, some with explicit solutions, that illustrate in a concrete manner the issues and phenomena that arise in the setting of finite-time dynamical systems. Of particular significance for geophysical applications is the notion of flow transition which occurs when finite-time hyperbolicity is lost or gained. The phenomena discovered and analysed in our examples point the way to a variety of directions for rigorous mathematical research in this rapidly developing and important area of dynamical systems theory.

FROM SINGLE WELL CHAOS TO CROSS WELL CHAOS: A DETAILED EXPLANATION IN TERMS OF MANIFOLD INTERSECTIONS

1994 ◽  
Vol 04 (04) ◽  
pp. 933-941 ◽  
Author(s):  
ANDREW L. KATZ ◽  
EARL H. DOWELL
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Stable And Unstable Manifolds ◽  
Detailed Explanation ◽  
Single Well ◽  
Unstable Manifolds ◽  
Parameter Values

The study of stable and unstable manifolds, and their intersections with each other, is a powerful technique for interpreting complex bifurcations of nonlinear systems. The escape phenomenon in the twin-well Duffing oscillator is one such bifurcation that is elucidated through the analysis of manifold intersections. In this paper, two escape scenarios in the twin-well Duffing oscillator are presented. In each scenario, the relevant manifold structures are examined for parameter values on either side of the escape bifurcation. Included is a description of the role of the hilltop saddle stable manifolds, which are known to separate the single well basins (should single well attractors exist). In each of the two bifurcation scenarios, it is shown through a detailed analysis of Poincaré maps that a homoclinic intersection of the manifolds of a specific period-3 saddle implies the destruction of the single well chaotic attractor. Although the Duffing oscillator is used to illustrate the ideas advanced here, it is thought that the approach will be useful for a variety of dynamical systems.

scholarly journals Shooting methods and topological transversality

1999 ◽  
Vol 129 (6) ◽  
pp. 1137-1155 ◽  
Author(s):  
B. Buffoni
Keyword(s):  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Shooting Method ◽  
Heteroclinic Orbits ◽  
Kolmogorov Equation ◽  
Shooting Methods ◽  
Stable And Unstable Manifolds ◽  
Heteroclinic Solutions ◽  
Unstable Manifolds

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

Study on the Geometrical Properties and Existence of Orbit Homoclinic to a Saddle Point in n-Dimensional Autonomous Vector Field with Polynomials

2018 ◽  
Vol 28 (14) ◽  
pp. 1850169
Author(s):  
Lingli Xie
Keyword(s):  
Vector Field ◽  
Saddle Point ◽  
Equilibrium Point ◽  
Necessary Conditions ◽  
Continuous System ◽  
Homoclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Geometrical Properties ◽  
Unstable Manifolds

According to the theory of stable and unstable manifolds of an equilibrium point, we firstly find out some geometrical properties of orbits on the stable and unstable manifolds of a saddle point under some brief conditions of nonlinear terms composed of polynomials for [Formula: see text]-dimensional time continuous system. These properties show that the orbits on stable and unstable manifolds of the saddle point will stay on the corresponding stable and unstable subspaces in the [Formula: see text]-neighborhood of the saddle point. Furthermore, the necessary conditions of existence for orbit homoclinic to a saddle point are exposed. Some examples including homoclinic bifurcation are given to indicate the application of the results. Finally, the conclusions are presented.

scholarly journals Hadamard–Perron theorems and effective hyperbolicity

2014 ◽  
Vol 36 (1) ◽  
pp. 23-63 ◽  
Author(s):  
VAUGHN CLIMENHAGA ◽  
YAKOV PESIN
Keyword(s):  
Closing Lemma ◽  
Stable And Unstable Manifolds ◽  
Local Dynamics ◽  
Local Diffeomorphisms ◽  
Finite Amount ◽  
Amount Of Information ◽  
Unstable Manifolds

We prove several new versions of the Hadamard–Perron theorem, which relates infinitesimal dynamics to local dynamics for a sequence of local diffeomorphisms, and in particular establishes the existence of local stable and unstable manifolds. Our results imply the classical Hadamard–Perron theorem in both its uniform and non-uniform versions, but also apply much more generally. We introduce a notion of ‘effective hyperbolicity’ and show that if the rate of effective hyperbolicity is asymptotically positive, then the local manifolds are well behaved with positive asymptotic frequency. By applying effective hyperbolicity to finite-orbit segments, we prove a closing lemma whose conditions can be verified with a finite amount of information.

ON NONLINEAR WAVE EQUATIONS WITH BREAKING LOOP-SOLUTIONS

2010 ◽  
Vol 20 (02) ◽  
pp. 519-537 ◽  
Author(s):  
JIBIN LI ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Nonlinear Wave ◽  
Traveling Wave ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Level Curves ◽  
Straight Line ◽  
Fixed Parameter ◽  
Unstable Manifolds

Using analytic methods from the dynamical systems theory, some new nonlinear wave equations are investigated, which have exact explicit parametric representations of breaking loop-solutions under some fixed parameter conditions. It is shown that these parametric representations are associated with some families of open level-curves of traveling wave systems corresponding to such nonlinear wave equations, each of which lies in an area bounded by a singular straight line and the stable and the unstable manifolds of a saddle point of such a system.

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