Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.     

Invariant normalization of non-autonomous Hamiltonian systems

A new algorithm is proposed for reducing non-autonomous Hamiltonian systems to normal Birkhoff form. The criterion for the normal form is the condition that the vector fields of the perturbed and unperturbed parts of the system should commute. The invariant character of the criterion enables the system to be normalized in a unified way, without first simplifying the unperturbed part and without distinguishing between resonance and non-resonance, or autonomous and non-autonomous, cases. The whole algorithm reduces to a one-dimensional recurrence formula. The result is obtained by using the Campbell-Hausdorff formula for the ring of asymptotic forms, as well as the solution of a homologicla equation in the form of a quadrature. Three examples are considered to illustrate the various special features of the new algorithm. One of the examples is of interest for nuclear magnetic resonance theory.     

On Hopf bifurcations of piecewise planar Hamiltonian systems

In this paper we study the number of limit cycles appearing in Hopf bifurcations of piecewise planar Hamiltonian systems. For the case that the Hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively. For some systems of special form we obtain the Hopf cyclicity.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

Limit cycle bifurcations by perturbing piecewise smooth Hamiltonian systems with multiple parameters

This paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near-Hamiltonian systems with multiple parameters. By the first Melnikov function, some novel criteria have been established for the existence of multiple limit cycles. Furthermore, an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in Llibre and Mereu (2014) [12]. Compared with the result in the above reference, one more limit cycle is found by our method.     

An explicit formula for the discussion of stability in autonomous Hamiltonian systems

The well known theorem of Arnold and Moser gives (in most cases) an answer to the question of the stability of the equilibria of autonomous Hamiltonian systems which can be reduced to two degrees of freedom. To apply their theorem, the Hamiltonian has to be transformed into its Birkhoff normal form of order two. In this paper this transformation is reduced to the multiplication of polynomials, which can easily be performed algebraically on a computer. The conical precessions of a dynamically symmetric satellite in a circular orbit serve as an example.

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Zusammenfassung Der bekannte Satz von Arnold und Moser liefert (auer in wenigen Ausnahmefällen) eine Antwort auf die Frage nach der Stabilität von Gleichgewichtslagen autonomer Hamiltonscher Systeme, die sich auf zwei Freiheitsgrade reduzieren lassen. Damit dieser Satz angewendet werden kann, mu die Hamiltonfunktion auf ihre Birkhoff-Normalform der Ordnung zwei transformiert werden. Diese Transformation wird hier auf Multiplikationen von Polynomen reduziert, die einfach, in symbolischer Weise, auf dem Computer ausgeführt werden können. Als Beispiel dienen die konischen Präzessionen eines dynamisch symmetrischen Satelliten auf einer Kreisbahn.

     

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.     

Hamiltonian systems with a stochastic force:nonlinear versus linear,and a girsanov formula

We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process     

The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems

In this paper, the iteration formula of the Maslov-type index theory for linear Hamiltonian systems with continuous, periodic, and symmetric coefficients is established. This formula yields a new method to determine the minimality of the period for solutions of nonlinear autonomous Hamiltonian systems via their Maslov-type indices. Applications of this formula give new results on the existence of periodic solutions with prescribed minimal period for such systems, and unify known results under various convexity conditions.

     

Nondegenerate systems and generic properties of the integrable Hamiltonian systems

The goal of the paper is to clarify whether the nondegenerate Hamiltinian systems are “typical” among all integrable systems. The importance of this problem is emphasized by a theorem of Fomenko-Zieschang, in which an isoenergy invariant determining the nondegenerate systems up to topological equivalence is constructed. Bibliography: 12 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 184–192.     

Extensions,dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.