Autonomous Hamiltonian Systems Research Articles

Estimation of the number of ultrasubharmonics for a two-dimensional almost autonomous Hamiltonian system periodic in time

UDC 517.9 The Arnold method for the detection of fixed points of symplectic diffeomorphisms is used to establish lower estimates for the number of ultrasubharmonics in a Hamiltonian system on a two-dimensional symplectic manifold with an almost autonomous Hamiltonian periodic in time. It is shown that the asymptotic behavior of these estimates (as the small parameter of perturbation tends to zero) depends on the zone (from the set four zones of an annular domain foliated by the closed level curves of the unperturbed Hamiltonian) containing the generating unperturbed ultrasubharmonics.

Autonomous three-dimensional Newtonian systems which admit Lie and Noether point symmetries

We determine the autonomous three-dimensional Newtonian systems which admit Lie point symmetries and the three-dimensional autonomous Newtonian Hamiltonian systems which admit Noether point symmetries. We apply the results in order to determine the two-dimensional Hamiltonian dynamical systems which move in a space of constant non-vanishing curvature and are integrable via Noether point symmetries. The derivation of the results is geometric and can be extended naturally to higher dimensions.

Modeling by autonomous Hamiltonian system: fixing the sign of a parameter

We illustrate how the application of autonomous Hamiltonian dynamical system can definitively fix a sign for an otherwise indefinite sign of a certain astrophysical parameter. To illustrate it, we shall consider the Mannheim–Kazanas-de Sitter solution of Weyl conformal gravity containing the parameter γ, which is believed to be significant in the galactic halo gravity. However, the sign of γ has so far been ambiguous. The strategy we adopt to fix it is to look for a finite maximum radius up to which the halo supports stable material circular orbits inside the de Sitter radius. The maximum radius for several observed lenses are calculated for both signs of γ, and with the observed value of cosmological constant Λ. These lenses (all having approximately the Einstein radius R E ≈ 1023 cm) consistently yield a maximum radius ( $$ R_{ \max }^{\text{stable}} \cong 4.25 \times 10^{27} {\text{cm}} $$ ) inside the de Sitter radius of the universe only when γ is negative, while a positive γ yields $$ R_{ \max }^{\text{stable}} $$ always exceeding the de Sitter radius.

Explicit construction of an autonomous Hamiltonian system exhibiting continual directed flow

We construct a prototypical example of a spatially-open autonomous Hamiltonian system in which localised, but otherwise unbiased, ensembles of initial conditions break spatio-temporal symmetries in the subsequent ensemble dynamics, despite time reversal symmetry of the equations of motion. Together with transient chaos, this provides the mechanism for the occurrence of a current. Transporting trajectories pass through transient chaos and subsequently cross surfaces of no-return, after which they perform solely regular motion so that the current is of continual ballistic nature.

On the entropy of relaxing deterministic systems

In this paper, we re-visit Gibbs' second (unresolved) paradox, namely the constancy of the fine-grained Gibbs entropy for autonomous Hamiltonian systems. We compare and contrast the different roles played by dissipation and entropy both at equilibrium where dissipation is identically zero and away from equilibrium where entropy cannot be defined and seems unnecessary in any case. Away from equilibrium dissipation is a powerful quantity that can always be defined and that appears as the central argument of numerous exact theorems: the fluctuation, relaxation, and dissipation theorems and the newly derived Clausius inequality.

Estimate for the number of perturbed ultrasubharmonics of a system with one and a half degrees of freedom close to a Hamiltonian system

UDC 517.9 We consider a two-dimensional autonomous Hamiltonian system with heteroclinic contour under the action of a time-periodic perturbation. It is shown that the number of ultrasubharmonics in the perturbed system is estimated from below by a function proportional to the square of the logarithm of the perturbation parameter when this parameter tends to zero.

Heteroclinic orbits for a discrete pendulum equation

About 20 years ago, Rabinowitz showed that there exist heteroclinic orbits of autonomous Hamiltonian systems joining two equilibria. A special case of autonomous Hamiltonian systems is the classical pendulum equation. The phase plane analysis of pendulum equation shows the existence of heteroclinic orbits joining two equilibria, which coincide with the result of Rabinowitz. However, the phase plane of discrete pendulum equation is similar to that of the classical pendulum equation, which suggests the existence of heteroclinic orbits for discrete pendulum equation also. By using variational method and delicate analysis technique, we show that there indeed exist heteroclinic orbits of discrete pendulum equation joining every two adjacent points of .

Symplectic algorithms based on the principle of least action and generating functions

SUMMARYIn this paper, the principle of least action and generating functions are used to construct symplectic numerical algorithms for finite dimensional autonomous Hamiltonian systems. The approximate action is obtained by approximating the generalized coordinates and momentums by Lagrange polynomials and performing Gaussian quadrature. Based on the principle of least action and the requirements of a canonical transformation, different types of symplectic algorithms have been constructed by choosing different types of independent variables at two ends of the time step. The symmetric property of the four types of symplectic algorithms proposed in this paper is discussed, and the exact linear stability domain for small m, n and g is discussed. The linear stability and precision of different types of symplectic algorithms are tested using numerical examples. Copyright © 2011 John Wiley & Sons, Ltd.

CHAINS OF ROTATIONAL TORI AND FILAMENTARY STRUCTURES CLOSE TO HIGH MULTIPLICITY PERIODIC ORBITS IN A 3D GALACTIC POTENTIAL

This paper discusses phase space structures encountered in the neighborhood of periodic orbits with high order multiplicity in a 3D autonomous Hamiltonian system with a potential of galactic type. We consider 4D spaces of section and we use the method of color and rotation [Patsis & Zachilas, 1994] in order to visualize them. As examples, we use the case of two orbits, one 2-periodic and one 7-periodic. We investigate the structure of multiple tori around them in the 4D surface of section and in addition, we study the orbital behavior in the neighborhood of the corresponding simple unstable periodic orbits. By considering initially a few consequents in the neighborhood of the orbits in both cases we find a structure in the space of section, which is in direct correspondence with what is observed in a resonance zone of a 2D autonomous Hamiltonian system. However, in our 3D case we have instead of stability islands rotational tori, while the chaotic zone connecting the points of the unstable periodic orbit is replaced by filaments extending in 4D following a smooth color variation. For more intersections, the consequents of the orbit which started in the neighborhood of the unstable periodic orbit, diffuse in phase space and form a cloud that occupies a large volume surrounding the region containing the rotational tori. In this cloud the colors of the points are mixed. The same structures have been observed in the neighborhood of all m-periodic orbits we have examined in the system. This indicates a generic behavior.

THE STRUCTURE AND EVOLUTION OF CONFINED TORI NEAR A HAMILTONIAN HOPF BIFURCATION

We study the orbital behavior at the neighborhood of complex unstable periodic orbits in a 3D autonomous Hamiltonian system of galactic type. At a transition of a family of periodic orbits from stability to complex instability (also known as Hamiltonian Hopf Bifurcation) the four eigenvalues of the stable periodic orbits move out of the unit circle. Then the periodic orbits become complex unstable. In this paper, we first integrate initial conditions close to the ones of a complex unstable periodic orbit, which is close to the transition point. Then, we plot the consequents of the corresponding orbit in a 4D surface of section. To visualize this surface of section we use the method of color and rotation [Patsis & Zachilas, 1994]. We find that the consequents are contained in 2D "confined tori". Then, we investigate the structure of the phase space in the neighborhood of complex unstable periodic orbits, which are further away from the transition point. In these cases we observe clouds of points in the 4D surfaces of section. The transition between the two types of orbital behavior is abrupt.

Okamoto’s Space for the First Painlevé Equation in Boutroux Coordinates

We study the completeness and connectedness of asymptotic behaviours of solutions of the first Painlev\'e equation $\op{d}^2y/\op{d}x^2=6\, y^2+ x$, in the limit $x\to\infty$, $x\in\C$. This problem arises in various physical contexts including the critical behaviour near gradient catastrophe for the focusing nonlinear Schr\"odinger equation. We prove that the complex limit set of solutions is non-empty, compact and invariant under the flow of the limiting autonomous Hamiltonian system, that the infinity set of the vector field is a repellor for the dynamics and obtain new proofs for solutions near the equilibrium points of the autonomous flow. The results rely on a realization of Okamoto's space, i.e., the space of initial values compactified and regularized by embedding in $\C\Proj 2$ through an explicit construction of nine blow-ups.

On the Behavior of Periodic Solutions of Planar Autonomous Hamiltonian Systems with Multivalued Periodic Perturbations

Aim of the paper is to provide a method to analyze the behavior of T -periodic solutions x_\varepsilon , \varepsilon>0 , of a perturbed planar Hamiltonian system near a cycle x_0 , of smallest period T , of the unperturbed system. The perturbation is represented by a T -periodic multivalued map which vanishes as \varepsilon\to 0 . In several problems from nonsmooth mechanical systems this multivalued perturbation comes from the Filippov regularization of a nonlinear discontinuous T -periodic term. Through the paper, assuming the existence of a T -periodic solution x_\varepsilon for \varepsilon>0 small, under the condition that x_0 is a nondegenerate cycle of the linearized unperturbed Hamiltonian system we provide a formula for the distance between any point x_0(t) and the trajectories x_\varepsilon([0,T]) along a transversal direction to x_0(t) .

Numerical integration of variational equations

We present and compare different numerical schemes for the integration of the variational equations of autonomous Hamiltonian systems whose kinetic energy is quadratic in the generalized momenta and whose potential is a function of the generalized positions. We apply these techniques to Hamiltonian systems of various degrees of freedom and investigate their efficiency in accurately reproducing well-known properties of chaos indicators such as the Lyapunov characteristic exponents and the generalized alignment indices. We find that the best numerical performance is exhibited by the "tangent map method," a scheme based on symplectic integration techniques which proves to be optimal in speed and accuracy. According to this method, a symplectic integrator is used to approximate the solution of the Hamilton equations of motion by the repeated action of a symplectic map S , while the corresponding tangent map TS is used for the integration of the variational equations. A simple and systematic technique to construct TS is also presented.

Emergence of continual directed flow in Hamiltonian systems

We propose a minimal model for the emergence of a directed flow in autonomous hamiltonian systems. It is shown that internal breaking of the spatiotemporal symmetries, via localized initial conditions, which are unbiased with respect to the transporting degree of freedom, and transient chaos conspire to form the physical mechanism for the occurrence of a current. Most importantly, after passage through the transient chaos, trajectories perform solely regular transporting motion so that the resulting current is of continual ballistic nature. This has to be distinguished from the features of transport reported previously for driven hamiltonian systems with mixed phase space where transport is determined by intermittent behavior exhibiting power-law decay statistics of the duration of regular ballistic periods.

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with n-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.

On the structure of the set of bifurcation points of periodic solutions for multiparameter Hamiltonian systems

This paper deals with periodic solutions of the Hamilton equation , where and is a parameter. Theorems on global bifurcation of solutions with periods , , from a stationary point are proved. can be singular. However, it is assumed that the local topological degree of ∇xH(·, λ0) at x0 is nonzero. For systems satisfying ∇xH(x0, λ) = 0 for all it is shown that (global) bifurcation points of solutions with periods can be identified with zeros of appropriate continuous functions . If, for all , , where A(λ) and B(λ) are (n × n)-matrices, then Fj can be defined by Fj(λ) = det[A(λ)B(λ) − j2I]. Symmetry breaking results concerning bifurcation of solutions with different minimal periods are obtained. A geometric description of the set of bifurcation points is given. Examples of constructive application of the theorems proved to analytical and numerical investigation and visualization of the set of all bifurcation points in given domain are provided.This paper is based on a part of the author's thesis (Radzki 2005 Branching points of periodic solutions of autonomous Hamiltonian systems (Polish) PhD Thesis Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, Toruń).

Instantons re-examined: Dynamical tunneling and resonant tunneling

Starting from trace formulas for the tunneling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex trajectories that are selected with appropriate complex-time paths. We show how this instantonlike approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunneling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunneling or resonant tunneling. Even though only one-dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunneling.

10.1007/s11208-008-2007-7

We consider the planar circular restricted three-body problem. It is described by an autonomous Hamiltonian system with two degrees of freedom and one parameter μ ∈ [0, 1/2], which is the mass ratio of the two massive bodies. Periodic solutions of this problem form two-parameter families. We propose methods of computation of symmetric periodic solutions for all values of the parameter μ. Each solution has a period and two traces, namely, the plane and the vertical one. Two characteristics of a family, i.e., its intersection with the symmetry plane, are plotted in the four coordinate systems used in the investigations: two global and two local ones related to the two massive bodies. We also describe generating families, i.e., the limits of families as μ → 0, which are known almost explicitly. As examples, we consider the family c, which begins at the fixed collinear point L 1, and the family i, which begins with direct circular orbits of an infinitely small radius around the primary P 1 of a bigger mass. We give a complete description of the generating families c and i for μ = 0.

Non-linear oscillations of a Hamiltonian system with two degrees of freedom with 2:1 resonance

The motions of an autonomous Hamiltonian system with two degrees of freedom close to an equilibrium position, stable in the linear approximation, are considered. It is assumed that in this neighbourhood the quadratic part of the Hamiltonian of the system is sign-variable, and the ratio of the frequencies of the linear oscillations are close to or equal to two. It is also assumed that the corresponding resonance terms in the third-degree terms of the Hamiltonian are small. The problem of the existence, bifurcations and orbital stability of the periodic motions of the system near the equilibrium position is solved. Conditionally periodic motions of the system are investigated. An estimate is obtained of the region in which the motions of the system are bounded in the neighbourhood of an unstable equilibrium in the case of exact resonance. The motions of a heavy dynamically symmetrical rigid body with a fixed point in the neighbourhood of its permanent rotations around the vertical for 2:1 resonance are considered as an application.

Periodic solutions with prescribed minimal period for the second order Hamiltonian systems with even potentials

In this paper, under a similar but stronger condition than that of Ambrosetti and Rabinowitz we find a T-periodic solution of the autonomous superquadratic second order Hamiltonian system with even potential for any T > 0; moreover, such a solution has T as its minimal period.