ANALYTICAL EXPRESSIONS FOR STABLE AND UNSTABLE MANIFOLDS IN HIGHER DEGREE OF FREEDOM HAMILTONIAN SYSTEMS

1996 ◽  
Vol 06 (11) ◽  
pp. 1997-2013 ◽  
Author(s):  
HARRY DANKOWICZ
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Initial Conditions ◽  
Saddle Points ◽  
Melnikov Method ◽  
Type Systems ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Analytical Expressions

Perturbations of completely integrable Hamiltonian systems with three or more degrees of freedom are studied. In particular, the unperturbed systems are assumed to be separable into a product of simple oscillator-type systems and a system containing homo- or heteroclinic connections consisting of stable and unstable manifolds of saddle points. Under a perturbation, the manifolds persist but separate and may no longer intersect. In this paper we show how, with proper choices for initial conditions, one may solve the variational equations to obtain analytical expressions for orbits on the perturbed manifolds in the form of expansions in the small parameter characterizing the perturbation. The derivation also shows how the distance between the manifolds can be uniquely defined, and thus provides an alternative to the traditional higher dimensional Melnikov method. It is finally argued that the approximate knowledge of the shape and position of the perturbed manifolds could be utilized for the study of large-scale phase-space motions, such as those associated with Arnold diffusion. The theory is further illuminated in two example problems.

A class of C∞-integrable Hamiltonian systems

1989 ◽  
Vol 113 (3-4) ◽  
pp. 293-314 ◽  
Author(s):  
W. M. Oliva ◽  
M. S. A. C. Castilla
Keyword(s):  
Hamiltonian Systems ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
First Integrals ◽  
Integrable Hamiltonian Systems ◽  
Integral Of Motion ◽  
Toda System ◽  
Nonnegative Coefficients ◽  
Special Case ◽  
Three Degrees Of Freedom

SynopsisWe discuss the C∞ complete integrability of Hamiltonian systems of type q = —grad V(q) = F(q), in which the closure of the cone generated (with nonnegative coefficients) by the vectors F(q), q ϵ ℝn, does not contain a line. The components of the asymptotic velocities are first integrals and the main aim is to prove their smoothness as functions of the initial conditions. The Toda-like system with potential V(q)=ΣNi=1 exp(fi∣ q) is a special case of the considered systems ifthe cone C(f1,…,fN)={ΣNi=1cifi,ci≧0} does notcontain a line. In any number of degrees of freedom, if C(f1,…,fN) has amplitude not too large (ang (fi, fj ≦π/2i,j=1,2,…, N), the first integrals are C∞ functions. In two degrees of freedom, without restriction on the amplitude of the cone, C∞-integrability is proved even in a case in which it is known that there is no other meromorphic integral of motion independent of energy. In three degrees of freedom the C∞-integrability of a deformation of the classic nonperiodic Toda system is proved. Some other examples are also discussed.

Melnikov Method for a Class of Planar Hybrid Piecewise-Smooth Systems

2016 ◽  
Vol 26 (02) ◽  
pp. 1650030 ◽  
Author(s):  
Shuangbao Li ◽  
Wensai Ma ◽  
Wei Zhang ◽  
Yuxin Hao
Keyword(s):  
Melnikov Method ◽  
Homoclinic Solution ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Unperturbed System ◽  
Piecewise Smooth Systems ◽  
Stable And Unstable Manifolds ◽  
Straight Line ◽  
Unstable Manifolds ◽  
Smooth System

In this paper, we extend the well-known Melnikov method for smooth systems to a class of periodic perturbed planar hybrid piecewise-smooth systems. In this class, the switching manifold is a straight line which divides the plane into two zones, and the dynamics in each zone is governed by a smooth system. When a trajectory reaches the separation line, then a reset map is applied instantaneously before entering the trajectory in the other zone. We assume that the unperturbed system is a piecewise Hamiltonian system which possesses a piecewise-smooth homoclinic solution transversally crossing the switching manifold. Then, we study the persistence of the homoclinic orbit under a nonautonomous periodic perturbation and the reset map. To achieve this objective, we obtain the Melnikov function to measure the distance of the perturbed stable and unstable manifolds and present the theorem for homoclinic bifurcations for the class of planar hybrid piecewise-smooth systems. Furthermore, we employ the obtained Melnikov function to detect the chaotic boundaries for a concrete planar hybrid piecewise-smooth system.

Optimal Control of Homoclinic Bifurcation: Theoretical Treatment and Practical Reduction of Safe Basin Erosion in the Helmholtz Oscillator

2003 ◽  
Vol 9 (3-4) ◽  
pp. 281-315 ◽  
Author(s):  
Stefano Lenci ◽  
Giuseppe Rega
Keyword(s):  
Finite Number ◽  
Control Method ◽  
Mathematical Problem ◽  
Melnikov Method ◽  
Excitation Amplitude ◽  
Homoclinic Bifurcation ◽  
Theoretical Treatment ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
Basin Erosion

A control method of the homoclinic bifurcation is developed and applied to the nonlinear dynamics of the Helmholtz oscillator. The method consists of choosing the shape of external and/or parametric periodic excitations, which permits us to avoid, in an optimal manner, the transverse intersection of the stable and unstable manifolds of the hilltop saddle. The homoclinic bifurcation is detected by the Melnikov method, and its dependence on the shape of the excitation is shown. We successively investigate the mathematical problem of optimization, which consists of determining the theoretical optimal excitation that maximizes the distance between stable and unstable manifolds for fixed excitation amplitude or, equivalently, the critical amplitude for homoclinic bifurcation. The optimal excitations in the reduced case with a finite number of superharmonic corrections are first determined, and then the optimization problem with infinite superharmonics is investigated and solved under a constraint on the relevant amplitudes, which is necessary to guarantee the physical admissibility of the mathematical solution. The mixed case of a finite number of constrained superharmonics is also considered. Some numerical simulations are then performed aimed at verifying the Melnikov's theoretical predictions of the homoclinic bifurcations and showing how the optimal excitations are indeed able to separate stable and unstable manifolds. Finally, we numerically investigate in detail the effectiveness of the control method with respect to the basin erosion and escape phenomena, which are the most important and dangerous practical aspects of the Helmholtz oscillator.

Heteroclinic Bifurcations and Invariant Manifolds in Rocking Block Dynamics

1991 ◽  
Vol 46 (6) ◽  
pp. 481-490 ◽  
Author(s):  
B. Bruhn ◽  
B. P. Koch
Keyword(s):  
Invariant Manifolds ◽  
Melnikov Method ◽  
Heteroclinic Bifurcation ◽  
Stable And Unstable Manifolds ◽  
Smale Horseshoe ◽  
Driven Systems ◽  
Rocking Block ◽  
Block Motion ◽  
Analytical Formulas ◽  
Unstable Manifolds

Abstract A simple model of rigid block motion under the influence of external perturbations is discussed. For periodic forcings we prove the existence of Smale horseshoe chaos in the dynamics. For slender blocks a heteroclinic bifurcation condition is calculated exactly, i.e. without using perturbation methods. That means that our results are valid for arbitrary excitation amplitudes. Furthermore, analytical formulas for the first pieces of the stable and unstable manifolds are derived not only for periodically but also for transiently driven systems. In the case of small excitation and damping the Melnikov method is used to treat the full nonlinear problem

scholarly journals A numerical method for the approximation of stable and unstable manifolds of microscopic simulators

2021 ◽  
Author(s):  
Constantinos Siettos ◽  
Lucia Russo
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Stochastic Systems ◽  
Black Box ◽  
Coarse Grained ◽  
Stationary States ◽  
Macroscopic Description ◽  
Stable And Unstable Manifolds ◽  
Agent Based ◽  
Unstable Manifolds

AbstractWe address a numerical methodology for the approximation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a macroscopic description does not exist analytically in a closed form. Thus, the underlying hypothesis is that we have a detailed microscopic simulator (Monte Carlo, molecular dynamics, agent-based model etc.) that describes the dynamics of the subunits of a complex system (or a black-box large-scale simulator) but we do not have explicitly available a dynamical model in a closed form that describes the emergent coarse-grained/macroscopic dynamics. Our numerical scheme is based on the equation-free multiscale framework, and it is a three-tier procedure including (a) the convergence on the coarse-grained saddle equilibrium, (b) its coarse-grained stability analysis, and (c) the approximation of the local invariant stable and unstable manifolds; the later task is achieved by the numerical solution of a set of homological/functional equations for the coefficients of a polynomial approximation of the manifolds.

scholarly journals Topological and geometric hyperbolicity criteria for polynomial automorphisms of

2021 ◽  
pp. 1-21
Author(s):  
ERIC BEDFORD ◽  
ROMAIN DUJARDIN
Keyword(s):  
Saddle Points ◽  
Topological Conjugacy ◽  
Sufficient Condition ◽  
Stable And Unstable Manifolds ◽  
Polynomial Automorphisms ◽  
Uniform Hyperbolicity ◽  
Unstable Manifolds ◽  
The Way

Abstract We prove that uniform hyperbolicity is invariant under topological conjugacy for dissipative polynomial automorphisms of $\mathbb {C}^2$ . Along the way we also show that a sufficient condition for hyperbolicity is that local stable and unstable manifolds of saddle points have uniform geometry.

Recent developments in classical mechanics

1990 ◽  
Vol 332 (1625) ◽  
pp. 221-221
Keyword(s):  
Statistical Mechanics ◽  
Hamiltonian Systems ◽  
Chaotic Motion ◽  
Degrees Of Freedom ◽  
Variational Principles ◽  
Initial Conditions ◽  
Classical Mechanics ◽  
Fractal Boundary ◽  
Completely Regular ◽  
Regular And Chaotic Motion

Hamiltonian systems with a finite number of degrees of freedom have traditionally been divided into two types: those with few degrees of freedom, which were supposed to exhibit some kind of regular ordered motion, approximately soluble by hamiltonian perturbation theory, and those with large numbers of degrees of freedom for which the methods of statistical mechanics should be used. The past few decades have seen a complete change of view, affecting almost all practical applications of classical mechanics. The motion of a hamiltonian system is usually neither completely regular nor properly described by statistical mechanics. It exhibits both regular and chaotic motion for different initial conditions, and the transition between the two types of motion as the initial conditions are varied is subtle and complicated. Variational principles, cantori, and their role in determining the transport properties of chaotic motion in hamiltonian systems and modular smoothing, a method for the rapid calculation of critical functions, which form the fractal boundary between regular and chaotic motion, have appeared in Percival (1987, 1990).

Introduction

2020 ◽  
pp. 3-16
Author(s):  
Kaloshin Vadim ◽  
Zhang Ke
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Energy Surface ◽  
Initial Conditions ◽  
Invariant Tori ◽  
Kam Theory ◽  
Ergodic Hypothesis ◽  
Open Set ◽  
Nearly Integrable Systems

This introductory chapter provides an overview of Arnold diffusion. The famous question called the ergodic hypothesis, formulated by Maxwell and Boltzmann, suggests that for a typical Hamiltonian on a typical energy surface, all but a set of initial conditions of zero measure have trajectories dense in this energy surface. However, Kolmogorov-Arnold-Moser (KAM) theory showed that for an open set of (nearly integrable) Hamiltonian systems, there is a set of initial conditions of positive measure with almost periodic trajectories. This disproved the ergodic hypothesis and forced reconsideration of the problem. For autonomous nearly integrable systems of two degrees or time-periodic systems of one and a half degrees of freedom, the KAM invariant tori divide the phase space. These invariant tori forbid large scale instability. When the degrees of freedoms are larger than two, large scale instability is indeed possible, as evidenced by the examples given by Vladimir Arnold. The chapter explains that the book answers the question of the typicality of these instabilities in the two and a half degrees of freedom case.

HOMOCLINIC AND HETEROCLINIC INTERSECTIONS IN THE PERIODICALLY FORCED BRUSSELATOR

1989 ◽  
Vol 03 (04) ◽  
pp. 643-663 ◽  
Author(s):  
WAN-SUN NI ◽  
PEI-QING TONG ◽  
BAI-LIN HAO
Keyword(s):  
Invariant Manifolds ◽  
Saddle Points ◽  
Periodic Points ◽  
Extended Phase Space ◽  
Direct Integration ◽  
Time Axis ◽  
Extended Phase ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
The One

The homoclinic and heteroclinic intersections of the stable and unstable manifolds of the fixed and periodic points in the Poincaré maps of the periodically forced Brusselator have been studied by direct integration of the system using periodic-orbit following technique. Since the free limit cycle oscillator does not possess any saddle points where one may start the construction of invariant manifolds, one has to look into the Poincaré sections in the extended phase space with the time axis included. We have followed a series of homoclinic and heteroclinic crossings and the one-piece chaotic attractor appears to be the envelope of unstable manifolds of all orders.

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