Related papers: Integrals and chaos in generalized H\'{e}non-Heile…

Homoclinic chaos is usually examined with the hypothesis of hyperbolicity of the critical point. We consider here, following a (suitably adjusted) classical analytic method, the case of non-hyperbolic points and show that, under a…

In general relativity, the motion of an extended body moving in a given spacetime can be described by a particle on a (generally non-geodesic) worldline. In first approximation, this worldline is a geodesic of the underlying spacetime, and…

The Hamiltonian Hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1: -1 resonance. At the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn…

Following the formalism of Gell-Mann and Hartle, phenomenological equations of motion are derived from the decoherence functional formalism of quantum mechanics, using a path-integral description. This is done explicitly for the case of a…

In recent paper Fakkousy et al. show that the 3D H\'{e}non-Heiles system with Hamiltonian $ H = \frac{1}{2} (p_1 ^2 + p_2 ^2 + p_3 ^2) +\frac{1}{2} (A q_1 ^2 + C q_2 ^2 + B q_3 ^2) + (\alpha q_1 ^2 + \gamma q_2 ^2)q_3 + \frac{\beta}{3}q_3…

In this paper we consider cases of existence of invariant measure, additional first integrals, and Poisson structure in a problem of rigid body's rolling without sliding on plane and sphere. The problem of rigid body's motion on plane was…

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra $\wt{\gr{gl}}^{+*}(2,{\bf R})$ are integrated by separation of variables in…

We consider three-dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We…

We investigate how the transition from integrability to nonintegrability occurs by changing the parameters of the Hamiltonian of a Heisenberg spin-1/2 chain with defects. Randomly distributed defects may lead to quantum chaos. A similar…

In systems where one coordinate undergoes periodic oscillation, the net displacement in any other coordinate over a single period is shown to be given by differentiation of the action integral associated with the oscillating coordinate.…

Homoclinic and unstable periodic orbits in chaotic systems play central roles in various semiclassical sum rules. The interferences between terms are governed by the action functions and Maslov indices. In this article, we identify…

An infinite family of quasi-maximally superintegrable Hamiltonians with a common set of (2N-3) integrals of the motion is introduced. The integrability properties of all these Hamiltonians are shown to be a consequence of a hidden…

A Hamiltonian formulation of generic many-particle systems with space-dependent balanced loss and gain coefficients is presented. It is shown that the balancing of loss and gain necessarily occurs in a pair-wise fashion. Further, using a…

Recently, the author and collaborators have developed a systematic program for proving the existence of homoclinic orbits in partial differential equations. Two typical forms of homoclinic orbits thus obtained are: (1). transversal…

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H…

We investigate a general system of two coupled harmonic oscillators with cubic nonlinearity. Without damping, the system is Hamiltonian, with the origin as an elliptic equilibrium characterized by two distinct linear frequencies. To…

Numerical integrations of the Solar System reveal a remarkable stability of the orbits of the inner planets over billions of years, in spite of their chaotic variations characterized by a Lyapunov time of only 5 million years and the lack…

In this article, we review the analytical and numerical approaches for computing the phase space structures in two degrees-of-freedom Hamiltonian systems that arise in chemical reactions. In particular, these phase space structures are the…

We consider two stable heteroclinic cycles rotating in opposite directions, coupled via diffusive terms. A complete synchronization in this system is impossible, and numerical exploration shows that chaos is abundant at low levels of…

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We…