Dissipative and Hamiltonian Systems with Chaotic Behavior: An Analytic Approach

Some classes of dissipative and Hamiltonian distributed systems are described. The dynamics of these systems is effectively reduced to finite-dimensional dynamics which can be unboundedly complex in a sense. Yarying the parameters of these systems, we can obtain an arbitrary (to within the orbital topological equivalence) structurally stable attractor in the dissipative case and an arbitrary polynomial weakly integrable Hamiltonian in the conservative case. As examples, we consider Hopfield neural networks and some reaction–diffusion systems in the dissipative case and a nonlinear string in the Hamiltonian case.     

Hamilton action as a function of phase variables

The Hamilton action is obtained in explicit form as a function of the phase coordinates and time for certain classes of conservative systems. The application of the action function to problems of investigating the stability of conservative systems is considered. It is shown that from the representation of the Hamiltonian action function in explicit form one can draw useful conclusions regarding the qualitative nature of the behaviour of the solutions of the systems considered.     

Line integral methods which preserve all invariants of conservative problems

Recently, the class of Hamiltonian Boundary Value Methods (HBVMs) has been introduced with the aim of preserving the energy associated with polynomial Hamiltonian systems (and, more in general, with all suitably regular Hamiltonian systems). However, many interesting problems admit other invariants besides the Hamiltonian function. It would be therefore useful to have methods able to preserve any number of independent invariants. This goal is achieved by generalizing the line-integral approach which HBVMs rely on, thus obtaining a number of generalizations which we collectively name Line Integral Methods. In fact, it turns out that this approach is quite general, so that it can be applied to any numerical method whose discrete solution can be suitably associated with a polynomial, such as a collocation method, as well as to any conservative problem. In particular, a completely conservative variant of both HBVMs and Gauss collocation methods is presented. Numerical experiments confirm the effectiveness of the proposed methods.     

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.     

New approach to the analysis of Hamiltonian and conservative systems

We suggest a new approach to the analysis of solutions of complicated conservative (in particular, Hamiltonian) systems, which implies the construction of an approximating extended two-parameter dissipative system of equations whose stable solutions (attractors) are arbitrarily exact approximations to solutions of the original conservative system. On the basis of numerical experiments for several conservative and Hamiltonian systems with two degrees of freedom, we show that, in all these systems, transition to chaos takes place not through the destruction of two-dimensional tori of the unperturbed system but, conversely, through the generation of complicated two-dimensional tori around cycles of the extended dissipative system and through an infinite cascade of bifurcations of the generation of new cycles and singular trajectories in accordance with the Feigenbaum-Sharkovskii-Magnitskii theory.     

Chaotic dynamics of homogeneous Yang-Mills fields with two degrees of freedom

In the present paper, we consider a scenario of transition to chaotic dynamics in the Hamiltonian system of homogeneous Yang-Mills fields with two degrees of freedom in the case of the Higgs mechanism. We show that in such a system, as well as in other Hamiltonian and conservative systems of equations, the nonlocal effect of multiplication of hyperbolic and elliptic cycles and tori around elliptic cycles in neighborhoods of the separatrix surfaces of hyperbolic cycles plays a key role on the initial stage of transition from a regular motion to a chaotic one. We observe that the new elliptic and hyperbolic cycles of the Hamiltonian system are generated as stable and saddle cycles of the extended dissipative system of equations not only as a result of saddle-node bifurcations but also as a result of fork-type bifurcations.     

Poincaré and Poincaré-Cartan integral invariants of a generalized Hamiltonian system

IntroductionUsually, only conservative or self-adjoint systems admit Hamiltonian canonical structure,i. e., a symplectic form and a Hamiltonian for such systems, under the experimenters' coordinateand time variables. Thus Hamilton's canonicaJ eqllations do not admit direct universality. ASa generajization of Hamilton's equations, Birkhoff's equations that are self-adjoint realize thedirect universajity and preserve the symbiotic character among the canonical formulation ofvariational princi…     

Hamilton系统的连续有限元法

利用常微分方程的连续有限元法,对非线性Hamilton系统证明了连续一次、二次有限元法分别是2阶和3阶的拟辛格式,且保持能量守恒;连续有限元法是辛算法对线性Hamilton系统,且保持能量守恒．在数值计算上探讨了辛性质和能量守恒性,与已有的辛算法进行对比,结果与理论相吻合．     

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic,chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.     

Higher order approximate periodic solutions for nonlinear oscillators with the Hamiltonian approach

In this work, the Hamiltonian approach is applied to obtain the natural frequency of the Duffing oscillator, the nonlinear oscillator with discontinuity and the quintic nonlinear oscillator. The Hamiltonian approach is then extended to the second and third orders to find more precise results. The accuracy of the results obtained is examined through time histories and error analyses for different values for the initial conditions. Excellent agreement of the approximate frequencies and the exact solution is demonstrated. It is shown that this method is powerful and accurate for solving nonlinear conservative oscillatory systems.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov-type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift-Hohenberg equation and reaction-diffusion systems of activator-inhibitor type.     

Lyapunov inequalities and stability for discrete linear Hamiltonian systems

In this paper, we establish several new Lyapunov type inequalities for discrete linear Hamiltonian systems when the end-points are not necessarily usual zeros, but rather, generalized zeros, which generalize and improve almost all related existing ones. Applying these inequalities, an optimal stability criterion is obtained.     

Sturmian theory for linear Hamiltonian systems without controllability

In this paper we present Sturmian separation and comparison theorems for linear Hamiltonian systems when no controllability assumption is imposed. This generalizes the traditional results of W. T. Reid for controllable (or normal) linear Hamiltonian systems to the possibly abnormal case. Our new theory is based on several recent results on linear Hamiltonian systems without controllability by W. Kratz, M. Wahrheit, V. Zeidan, and the author regarding the piecewise constant kernel for conjoined bases, the oscillation and eigenvalue theorems, and the Rayleigh principle. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Lyapunov inequality for linear Hamiltonian systems on time scales

The principal aim of this paper is to state and prove some Lyapunov inequalities for linear Hamiltonian system on an arbitrary time scale , so that the well-known case of differential linear Hamiltonian systems and the recently developed case of discrete Hamiltonian systems are unified. Applying these inequalities, a disconjugacy criterion for Hamiltonian systems on time scales is obtained.     

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.     

Binary nonlinearization of spectral problems of the perturbation AKNS systems

The method of nonlinearization of spectral problems is extended to the perturbation AKNS systems, and a new kind of finite-dimensional Hamiltonian systems is obtained. It is shown that the obtained Hamiltonian systems are just the perturbation systems of the well-known constrained AKNS flows and thus their Liouville integrability is established by restoring from the Liouville integrability of the constrained AKNS flows. As a byproduct, the process of binary nonlinearization of spectral problems and the process of perturbation of soliton equations commute in the case of the AKNS hierarchy.     

Mel’nikov-Samoilenko adiabatic stability problem

We develop a symplectic method for the investigation of invariant submanifolds of nonautonomous Hamiltonian systems and ergodic measures on them. The so-called Mel’nikov-Samoilenko problem for the case of adiabatically perturbed completely integrable oscillator-type Hamiltonian systems is studied on the basis of a new construction of “ virtual” canonical transformations. Dedicated to the memory of Viktor Koz’mich Mel’nikov, colleague and teacher, a talented Moscow mathematician, without whom the theory of dynamical systems would not be so attractive. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 6, pp. 787–803, June, 2006.