Conservative Dynamical Systems Research Articles

Are there chaotic maps in the sphere?

Two of the most popular notions of chaoticity are the one due to Robert Devaney and the one that assumes positive Lyapunov exponents. In this note we discuss the coexistence of both definitions for conservative discrete dynamical systems in the two-sphere and with respect to the C 1 -generic point of view.

Abundance of elliptic dynamics on conservative three-flows

We consider a compact three-dimensional boundaryless Riemannian manifold M and the set of divergence-free (or zero divergence) vector fields without singularities, then we prove that this set has a C 1-residual (dense G δ) such that any vector field inside it is Anosov or else its elliptical orbits are dense in the manifold M. This is the flow-setting counterpart of Newhouse's Theorem 1.3 (S. Newhouse, Quasi-elliptic periodic points in conservative dynamical systems, Am. J. Math. 99 (1977), pp. 1061–1087). Our result follows from two theorems, the first one says that if Λ is a hyperbolic invariant set for some class C 1 zero divergence vector field X on M, then either X is Anosov, or else Λ has empty interior. The second one says that, if X is not Anosov, then for any open set U ⊆ M there exists Y arbitrarily close to X such that Y t has an elliptical closed orbit through U.

Conservation Laws for Partially Conservative Variable Mass Systems via d'Alembert's Principle

Conservation laws for partially conservative variable mass dynamical systems under symmetric infinitesimal transformations are determined. A generalization of Lagrange-d'Alembert's principle for a variable mass system in terms of asynchronous virtual variation is presented. The generalized Killing equations are obtained such that their solution yields the transformations and the associated conservation laws. An example illustrative of the theory is furnished at the end as well.

EVOLUTIONARY METHODS FOR THE APPROXIMATION OF THE STABILITY DOMAIN AND FREQUENCY OPTIMIZATION OF CONSERVATIVE MAPS

Two methodologies are presented for the numerical approximation of the "domain of stability" of nonlinear conservative maps: (a) the Evolutionary Estimation of the Domain of Stability (EEDS) and (b) the Evolutionary Frequency Optimization (EFO), optimizing certain frequency parameters of these maps so that the domain of stability encompasses the maximum possible "volume" of bounded motion, known in the accelerator literature as the dynamic aperture. The central components of the proposed approaches are: The Differential Evolution algorithm (DE) based on concepts of Computational Intelligence and the method of the Smaller ALignment Index (SALI) used for the determination of chaotic dynamics. Initially, we give a brief description of the two methodologies and then demonstrate their usefulness by applying them to some well-known examples of 2D and 4D Hénon maps. The proposed methodologies can be easily applied to "volume" preserving maps which are not necessarily symplectic as well as to continuous dynamical systems (flows) and can also be generalized to treat conservative dynamical systems of any dimension.

The discrete null space method for constrained mechanical systems in nonlinear structural and multibody dynamics

AbstractCorresponding to d'Alemberts principle in classical mechanics, the discrete null space method developed in [1] provides an energy‐momentum conserving time stepping scheme for conservative finite‐dimensional dynamical systems subject to constraints. The elimination of the Lagrange multipliers from the temporal discrete system leads to a reduced number of unknowns and to an improved condition number during the iterative solution of the nonlinear system. External and internal constraints are fulfilled likewise at the time nodes, wherefore the discrete null space method is particularly suited for the treatment of elastic multibody systems in structural dynamics. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Approximate Conserved Quantities of Conservative Dynamical Systems in R 3

Approximate generalized symmetries associated with the resonances of conservative dynamical systems in R 3 have been discussed. It has been shown that an approximate first integral (conserved quantities) can be found from the knowledge of two conservative approximate symmetries. Furthermore, the result has been applied to a physical system. Invariant curves have been obtained analytically and they have been compared with the numerical ones on the Poincaré surface of section. Approximate conserved quantity pinpoints the type of stability.

Systems with Escapes

There are two types of escapes in conservative dynamical systems with two degrees of freedom: escapes to infinity and escapes to certain singular points at a finite distance. In both cases the areas on a surface of section are not preserved. We consider the basins of escape to infinity in simple Hamiltonian systems. The initial conditions of orbits escaping after 1, 2, ... intersections with a surface of section form in general spiral fractal sets. Then we consider the sets of escapes into two fixed black holes for various values of the energy. The forms of these sets depend on the unstable periodic orbits and their asymptotic curves. We find the characteristics of the simple periodic orbits and their changes for various values of the energy.

On the solitary wave solutions of the CQNLS

Abstract In this paper, coexistence and simplified formulations of the solitary waves of the cubic–quintic non-linear Schrodinger equation (CQNLS) are investigated by analyzing the steady bifurcation and the energy integral of the conservative dynamical system satisfied by the wave packet. It is found that the bright solitary waves can coexist with kinks and anti-kinks in a range of the bifurcation control parameter. There exists a critical parameter value at which the dark solitary waves are distinguished from the bright solitary waves, kinks and anti-kinks. All of the simplified solitary wave solutions, kinks and anti-kinks are obtained by using our previously developed approximate method.

Regularization of the collision in the electromagnetic two-body problem

We derive a differential equation that is regular at the collision of two equal-mass bodies with attractive interaction in the relativistic action-at-a-distance electrodynamics. We use the energy constant related to the Poincare invariance of the theory to define finite variables with finite derivatives at the collision. The collision orbits are calculated numerically using the regular equation adapted in a self-consistent minimization method (a stable numerical method that chooses only nonrunaway solutions). This dynamical system appeared 100 years ago as an example of covariant time-symmetric two-body dynamics and acquired the status of electrodynamics in the 1940s by the works of Dirac, Wheeler, and Feynman. We outline the method with an emphasis on the physics of this complex conservative dynamical system.

Capture into resonance: a method for efficient control.

To achieve large changes in adiabatic invariants using small control input, a conservative dynamical system must possess an internal resonance. Capture into resonance is an inherently probabilistic process. We propose a control method to make it more structured. We study the motion of charged particles in an electromagnetic field as an example of such a system. When the nominal dynamics brings particles close to a resonance surface, a short control pulse forces the capture of a particle into the resonance with the wave. A captured particle is transported by the wave across the energy levels. The second pulse releases a particle from the resonance when the desired energy level is achieved. We discuss the distribution of energy achieved by the method.

Ultra-low Emittance Beam Generator Using Coulomb Crystals

Ultra-low Emittance Beam Generator Using Coulomb Crystals

KAM theory: The legacy of Kolmogorov's 1954 paper

Kolmogorov-Arnold-Moser (or kam) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and kam theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov’s pioneering work.

Radiation Induced Instability

In this paper we discuss the stability and instability properties of two classes of conservative dynamical systems which are comprised of a finite dimensional and an infinite dimensional subsystem. The finite dimensional component is a linear mechanical system with gyroscopic terms. The mechanical system is coupled to a wave equation defined on an infinite spatial domain via two different types of coupling which we denote by Lamb coupling and wave field coupling. We also investigate the effect of dispersion on the coupled system. In particular, we analyze the conditions under which coupling to a wave system induces instability in the finite dimensional system. Analytic results are compared to computer simulations.

Two-degree-of-freedom Hamiltonian for the time-symmetric two-body problem of the relativistic action-at-a-distance electrodynamics.

We find a two-degree-of-freedom Hamiltonian for the time-symmetric problem of straight line motion of two electrons in direct relativistic interaction. This time-symmetric dynamical system appeared 100 years ago and it was popularized in the 1940s by the work of Wheeler and Feynman in electrodynamics, which was left incomplete due to the lack of a Hamiltonian description. The form of our Hamiltonian is such that the action of a Lorentz transformation is explicitly described by a canonical transformation (with rescaling of the evolution parameter). The method is closed and defines the Hamitonian in implicit form without power expansions. We outline the method with an emphasis on the physics of this complex conservative dynamical system. The Hamiltonian orbits are calculated numerically at low energies using a self-consistent steepest-descent method (a stable numerical method that chooses only the nonrunaway solution). The two-degree-of-freedom Hamiltonian suggests a simple prescription for the canonical quantization of the relativistic two-body problem.

Stability of the HD 12661 Planetary System

In this paper we study the stability of the HD 12661 planetary system in the framework of the N- body problem. Using the initial conditions found and announced by the California & Carnegie Planet Search Team, (http://exoplanets.org/almanacframe.html), we estimate the dynamical limits on orbital parameters that provide sta- ble (quasi-periodic) motions of the system. We investigate the orbital stability by combining the MEGNO indicator analysis with short-term integrations of the orbital dynamics. The MEGNO technique, invented by Cincotta & Simo (2000), makes it possible to distinguish efficiently between chaotic and regular dynamics of a conservative dynamical system. The orbital evo- lution, derived simultaneously with MEGNO, helps to identify sources of instability. The nominal initial condition leads to a chaotic solution, with a Lyapunov time � 1300 yr. In spite of this, the system motion seems to be bounded. This was examined directly, by long-term, 1 Gyr integrations. During this time, the eccentricities vary in the range (0.1, 0.4). The system is locked in apsidal resonance with the critical argument librating about 180 ◦ , with a full amplitude varying typically between 40 ◦ and 180 ◦ . Using MEGNO, we found that the HD 12661 system evolves on a border of the 11:2 mean motion resonance. This reso- nance is stable and results in a quasi-periodic evolution of the system. From the viewpoint of global dynamics, the crucial factor for system stability is the presence of the apsidal resonance. We detected this resonance in a wide neighborhood of the initial condition in the space of orbital parameters of the system, and in wide ranges of relative inclination and masses of the planets. The center of libration can be 180 ◦ (as in the nominal system) or 0 ◦ . The regime depends on the initial values of the apsidal longitudes. Statistically, the system prefers almost exclusively one of these two resonance regimes. The HD 12661 system gives a very evident example of the dynamical role of secular resonances, and their influence on the stability of exosystems containing Jupiter-like planets. Data derived by numerical experiments are compared with the results of Laplace-Lagrange secular theory. The analytical theory gives a crude approximation of the secular dynamics, because the eccentricities and masses of the planets are large, and the nominal system is near the 11:2 mean motion resonance.

Nonstandard entropy production in the standard map

We investigate the time evolution of the entropy for a paradigmatic conservative dynamical system, the standard map, for different values of its controlling parameter a. When the phase space is sufficiently ‘chaotic’ (i.e., for large | a|), we reproduce previous results. For small values of | a|, when the phase space becomes an intricate structure with the coexistence of chaotic and regular regions, an anomalous regime emerges. We characterize this anomalous regime with the generalized nonextensive entropy, and we observe that for values of a approaching zero, it lasts for an increasingly large time. This scenario displays a striking analogy with recent observations made in isolated classical long-range N-body Hamiltonians, where, for a large class of initial conditions, a metastable state (whose duration diverges with 1/ N→0) is observed before it crosses over to the usual, Boltzmann–Gibbs regime.

On the numerical continuation of periodic orbits

This paper deals with the analytic continuation of periodic orbits of conservative dynamical systems with three degrees of freedom. For variations of any parameter (or integral), it relies on numerical analysis in order to implement a predictor-corrector algorithm to compute the initial conditions of the periodic orbits pertaining to the family. The method proposed here is not restricted to symmetric problems and, since the procedure involves the computation of the variational equations, a side effect is the trivial computation of the linear stability of the periodic orbits. As an illustration of the robustness of the method, several families of periodic orbits of the Restricted Three-Body Problem are computed.

Computing with certainty individual members of families of periodic orbits of a given period

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincare map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrodinger equation, and a hydrodynamic model.

Kolmogorov-Sinai Entropy Rate versus Physical Entropy

We elucidate the connection between the Kolmogorov-Sinai entropy rate $\ensuremath{\kappa}$ and the time evolution of the physical or statistical entropy $S$. For a large family of chaotic conservative dynamical systems including the simplest ones, the evolution of $S(t)$ for far-from-equilibrium processes includes a stage during which $S$ is a simple linear function of time whose slope is $\ensuremath{\kappa}$. We present numerical confirmation of this connection for a number of chaotic symplectic maps, ranging from the simplest two-dimensional ones to a four-dimensional and strongly nonlinear map.