Search results for "Hamiltonian system"
showing 10 items of 58 documents
Critical Attractor and Universality in a Renormalization Scheme for Three Frequency Hamiltonian Systems
1998
We study an approximate renormalization-group transformation to analyze the breakup of invariant tori for three degrees of freedom Hamiltonian systems. The scheme is implemented for the spiral mean torus. We find numerically that the critical surface is the stable manifold of a critical nonperiodic attractor. We compute scaling exponents associated with this fixed set, and find that they can be expected to be universal.
Nondegeneracy in the Perturbation Theory of Integrable Dynamical Systems
1990
The most general nondegeneracy condition for the existence of invariant tori in nearly integrable and analytic Hamiltonian systems is formulated.
Stationary problems for equation of the KdV type and dynamical r-matrices
1995
We study a quite general family of dynamical $r$-matrices for an auxiliary loop algebra ${\cal L}({su(2)})$ related to restricted flows for equations of the KdV type. This underlying $r$-matrix structure allows to reconstruct Lax representations and to find variables of separation for a wide set of the integrable natural Hamiltonian systems. As an example, we discuss the Henon-Heiles system and a quartic system of two degrees of freedom in detail.
Bifurcations of phase portraits of a Singular Nonlinear Equation of the Second Class
2014
Abstract The soliton dynamics is studied using the Frenkel Kontorova (FK) model with non-convex interparticle interactions immersed in a parameterized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Non-convex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. In the continuum limit for such a model, the particles are governed by a Singular Nonlinear Equation of the Second Class. The dynamical behavior of traveling wave solutions is studied by using the theory of bifurcations of dynamical systems. Under different para…
Classical Chern–Simons Mechanics
2001
We are interested in a completely integrable Hamiltonian system \((\mathscr{M}_{2N},\omega,H).\) Local coordinates on the 2N-dimensional phase space \(\mathscr{M}_{2N}\) are denoted by η a = (p, q), a = 1, 2, … 2N and the symplectic 2-form ω is given by
Correct thermodynamic forces in Tsallis Thermodynamics: connection with Hill Nanothermodynamics
2005
The equivalence between Tsallis Thermodynamics and Hill Nanothermodynamics is established. The correct thermodynamic forces in Tsallis thermodynamics are pointed out. Through this connection we also find a general expression for the entropic index $q$ which we illustrate with two physical examples, allowing in both cases to relate $q$ to the underlying dynamics of the Hamiltonian systems.
Group Foundations of Quantum and Classical Dynamics : Towards a Globalization and Classification of Some of Their Structures
1987
This paper is devoted to a constructiveand critical analysis of the structure of certain dynamical systems from a group manifold point of view recently developed. This approach is especially suitable for discussing the structure of the quantum theory, the classical limit, the Hamilton-Jacobi theory and other problems such as the definition and globalization of the Poincare-Cartan form which appears in the variational approach to higher order dynamical systems. At the same time, i t opens a way for the classification of all hamiltonian and lagrangian systems associated with suitably defined dynamical groups. Both classical and quantum dynamics are discussed, and examples of all the different…
Instabilities of optical solitons and Hamiltonian singular solutions in a medium of finite extension
2011
International audience; We analyze the role of soliton solutions and Hamiltonian singularities in the dynamics of counterpropagating waves in a medium of finite spatial extension. The soliton solution can become unstable due to the finite extension of the system. We show that the spatiotemporal dynamics then relaxes toward a Hamiltonian singular state of a nature different than that of the soliton state. This phenomenon can be explained through a geometrical analysis of the singularities of the stationary Hamiltonian system.
Poincaré Surface of Sections, Mappings
2001
We consider a system with two degrees of freedom, which we describe in four-dimensional phase space. In this (finite) space we define an (oriented) two-dimensional surface. If we then consider the trajectory in phase space, we are interested primarily in its piercing points through this surface. This piercing can occur repeatedly in the same direction. If the motion of the trajectory is determined by the Hamiltonian equations, then the n + 1-th piercing point depends only on the nth. The Hamiltonian thus induces a mapping n → n + 1 in the “Poincare surface of section” (PSS). The mapping transforms points of the PSS into other (or the same) points of the PSS. In the following we shall limit …
Infinite orbit depth and length of Melnikov functions
2019
Abstract In this paper we study polynomial Hamiltonian systems d F = 0 in the plane and their small perturbations: d F + ϵ ω = 0 . The first nonzero Melnikov function M μ = M μ ( F , γ , ω ) of the Poincare map along a loop γ of d F = 0 is given by an iterated integral [3] . In [7] , we bounded the length of the iterated integral M μ by a geometric number k = k ( F , γ ) which we call orbit depth. We conjectured that the bound is optimal. Here, we give a simple example of a Hamiltonian system F and its orbit γ having infinite orbit depth. If our conjecture is true, for this example there should exist deformations d F + ϵ ω with arbitrary high length first nonzero Melnikov function M μ along…