Stability of equilibrium solutions of Hamiltonian systems under the presence of a single resonance in the non-diagonalizable case

The problem of knowing the stability of one equilibrium solution of an analytic autonomous Hamiltonian system in a neighborhood of the equilibrium point in the case where all eigenvalues are pure imaginary and the matrix of the linearized system is non-diagonalizable is considered. We give information about the stability of the equilibrium solution of Hamiltonian systems with two degrees of freedom in the critical case. We make a partial generalization of the results to Hamiltonian systems with n degrees of freedom, in particular, this generalization includes those in [1].      

On the Constructive Algorithm for Stability Analysis of an Equilibrium Point of a Periodic Hamiltonian System with Two Degrees of Freedom in the Second-order Resonance Case

This paper is concerned with a nonautonomous Hamiltonian system with two degrees of freedom whose Hamiltonian is a 2π-periodic function of time and analytic in a neighborhood of an equilibrium point. It is assumed that the system exhibits a secondorder resonance, i. e., the system linearized in a neighborhood of the equilibrium point has a double multiplier equal to ?1. The case of general position is considered when the monodromy matrix is not reduced to diagonal form and the equilibrium point is linearly unstable. In this case, a nonlinear analysis is required to draw conclusions on the stability (or instability) of the equilibrium point in the complete system.In this paper, a constructive algorithm for a rigorous stability analysis of the equilibrium point of the above-mentioned system is presented. This algorithm has been developed on the basis of a method proposed in [1]. The main idea of this method is to construct and normalize a symplectic map generated by the phase flow of a Hamiltonian system.It is shown that the normal form of the Hamiltonian function and the generating function of the corresponding symplectic map contain no third-degree terms. Explicit formulae are obtained which allow one to calculate the coefficients of the normal form of the Hamiltonian in terms of the coefficients of the generating function of a symplectic map.The developed algorithm is applied to solve the problem of stability of resonant rotations of a symmetric satellite.     

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.     

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

On stability in Hamiltonian systems with two degrees of freedom

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.     

Stability of a Hamiltonian system in a limiting case

We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the 1: −1 resonance case where the linearized system has double pure imaginary eigenvalues ±iω, ω ≠ 0 and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable.     

The critical case of a pair of zero roots in a two-degree-of-freedom hamiltonian system

The problem of the motion of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its equilibrium position is considered. It is assumed that the characteristic equation of the linearized system has a pair of pure imaginary roots. The roots of the other pair are assumed to be close to or equal to zero, and in the latter case non-simple elementary dividers correspond to these roots. The problem of the existence, bifurcations and orbital stability of families of periodic motions, generated from the equilibrium position, is solved. Conditionally periodic motions are analysed. The problem of the boundedness of the trajectories of the system in the neighbourhood of the equilibrium position in the case when it is Lyapunov unstable, is considered. Non-linear oscillations of an artificial satellite in the region of its steady rotation around the normal to the orbit plane are investigated as an application.     

The stability of the equilibrium position of Hamiltonian systems with two degrees of freedom

The stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom with a Hamiltonian, the unperturbed part of which generates oscillators with a cubic restoring force, is considered. It is proved that the equilibrium position is Lyapunov conditionally stable for initial values which do not belong to a certain surface of the Hamiltonian level. A reduction of the system onto this surface shows that, in the generic case, unconditional Lyapunov stability also occurs.     

Non-linear oscillations of a hamiltonian system with one degree of freedom and fourth-order resonance

Non-linear oscillations of a 2π-periodic Hamiltonian system with one degree of freedom are considered . It is assumed that the origin of coordinates is an equilibrium position, the linearized system is assumed to be stable, its characteristic exponents ±iv are pure imaginary, and the value of 4v is close to an integer. When the methods of classical perturbation theory are used, the investigation reduces to an analysis of a model system which can be described by the typical Hamiltonian of problems on the motion of Hamiltonian systems with one degree of freedom in the case of fourth-order resonance. The system is analysed in detail. The results for the model system are applied to the total system using Poincaré's theory of periodic motion and the KAM-theory. The existence, number and stability of 8π-periodic motions of the initial system are investigated. Trajectories of motion which start in a fairly small neighbourhood of the origin of coordinates are bounded. An estimate of the size of that neighbourhood is given. The examples considered are of a point mass above a curve in the shape of an ellipse which collides with the curve, and plane non-linear oscillations of a satellite in an elliptical orbit in the case of fourth-order resonance.     

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

Non-linear oscillations of an autonomous Hamiltonian system with two degrees of freedom in the neighbourhood of a stable equilibrium are considered. It is assumed that the frequency ratio of the linear oscillations is close to or equal to two, and that the Hamiltonian is sign-definite in the neighbourhood of the equilibrium. A solution is presented to the problem of the orbital stability of periodic motions emanating from the equilibrium position. Conditionally periodic motions of an approximate system are analysed taking into account terms of order up to and including three in the normalized Hamiltonian. The KAM theory is used to consider the problem of maintaining these motions taking into account fourth- and higher-order terms in the series expansion of the Hamiltonian in a sufficiently small neighbourhood of the equilibrium. The results are used to investigate non-linear oscillations of an elastic pendulum.     

Stability of equilibrium solutions of Hamiltonian systems with n-degrees of freedom and single resonance in the critical case

In this paper we give new results for the stability of one equilibrium solution of an autonomous analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom. Our Main Theorem generalizes several results existing in the literature and mainly we give information in the critical cases (i.e., the condition of stability and instability is not fulfilled). In particular, our Main Theorem provides necessary and sufficient conditions for stability of the equilibrium solutions under the existence of a single resonance. Using analogous tools used in the Main Theorem for the critical case, we study the stability or instability of degenerate equilibrium points in Hamiltonian systems with one degree of freedom. We apply our results to the stability of Hamiltonians of the type of cosmological models as in planar as in the spatial case.     

On the stability of a nonautonomous hamiltonian system under a parametric resonance of essential type

The problem of the stability of the equilibrium position of an nonautonomous Hamiltonian system with periodic coefficients, in which two multipliers of the linearized system are equal, is analyzed in a nonlinear setting. The stability in the finite approximation, and formal Liapunov stability or instability are proved, depending on the Hamiltonian's coefficients.     

Time Delay-Induced Instabilities and Hopf Bifurcations in General Reaction–Diffusion Systems

The distribution of the roots of a second order transcendental polynomial is analyzed, and it is used for solving the purely imaginary eigenvalue of a transcendental characteristic equation with two transcendental terms. The results are applied to the stability and associated Hopf bifurcation of a constant equilibrium of a general reaction–diffusion system or a system of ordinary differential equations with delay effects. Examples from biochemical reaction and predator–prey models are analyzed using the new techniques.     

A Study of the Motions of an Autonomous Hamiltonian System at a 1:1 Resonance

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.     

Resonant periodic motions of Hamiltonian systems with one degree of freedom when the Hamiltonian is degenerate

The motions of a non-autonomous Hamiltonian system with one degree of freedom which is periodic in time and where the Hamiltonian contains a small parameter is considered. The origin of coordinates of the phase space is the equilibrium position of the unperturbed or complete system, which is stable in the linear approximation. It is assumed that there is degeneracy in the unperturbed Hamiltonian when account is taken of terms no higher than the fourth degree (the frequency of the small linear oscillations depends on the amplitude) and, in this case, one of the resonances of up to the fourth order inclusive is realized in the system. Model Hamiltonians are constructed for each case of resonance and a qualitative investigation of the motions of the model system is carried out. Using Poincaré's theory of periodic motions and KAM-theory, a rigorous solution is given of the problem of the existence, bifurcations and stability of the periodic motions of the initial system, which are analytic with respect to fractional powers of the small parameter. The resonant periodic motions (in the case of the degeneracy being considered) of a spherical pendulum with an oscillating suspension point are investigated as an application.     

Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response

We consider a delayed predator-prey system with Beddington-DeAngelis functional response. The stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. An example is given and numerical simulations are performed to illustrate the obtained results.     

Non-linear oscillations of a Hamiltonian system in the case of 3:1 resonance

The motion of an autonomous Hamiltonian system with two degrees of freedom near its equilibrium position is considered. It is assumed that, in a certain region of the equilibrium position, the Hamiltonian is an analytic and sign-definite function, while the frequencies of linear oscillations satisfy a 3:1 ratio. A detailed analysis of the truncated system, corresponding to the normalized Hamiltonian is given, in which terms of higher than the fourth order are dropped. It is shown that the truncated system can be integrated in terms of Jacobi elliptic functions, and its solutions describe either periodic motions or motions that are asymptotic to periodic motions, or conventionally periodic motions. It is established, using the KAM-theory methods, that the majority of conventionally periodic motions are also preserved in the complete system. Moreover, in a fairly small neighbourhood of the equilibrium position, the trajectories of the complete system, which are not conventionally periodic, form a set of exponentially small measure. The results of the investigation are used in the problem of the motion of a dynamically symmetrical satellite in the region of its cylindrical precession.     

On stability at the Hamiltonian Hopf Bifurcation

For a 2 d.o.f. Hamiltonian system we prove the Lyapunov stability of its equilibrium with two double pure imaginary eigenvalues and non-semisimple Jordan form for the linearization matrix, when some coefficient in the 4th order normal form is positive (the equilibrium is known to be unstable, if this coefficient is negative). Such the degenerate equilibrium is met generically in one-parameter unfoldings, the related bifurcation is called to be the Hamiltonian Hopf Bifurcation. Though the stability is known since 1977, proofs that were published are either incorrect or not complete. Our proof is based on the KAM theory and a work with the Weierstrass elliptic functions, estimates of power series and scaling.      

The numerical approximation of center manifolds in Hamiltonian systems

In this paper we develop a numerical method for computing higher order local approximations of center manifolds near steady states in Hamiltonian systems. The underlying system is assumed to be large in the sense that a large sparse Jacobian at the equilibrium occurs, for which only a linear solver and a low-dimensional invariant subspace is available. Our method combines this restriction from linear algebra with the requirement that the center manifold is parametrized by a symplectic mapping and that the reduced equation preserves the Hamiltonian form. Our approach can be considered as a special adaptation of a general method from Numer. Math. 80 (1998) 1-38 to the Hamiltonian case such that approximations of the reduced Hamiltonian are obtained simultaneously. As an application we treat a finite difference system for an elliptic problem on an infinite strip.     

On nonlinear motions of Hamiltonian system in case of fourth order resonance

We deal with an autonomous Hamiltonian system with two degrees of freedom. We assume that the Hamiltonian function is analytic in a neighborhood of the phase space origin, which is an equilibrium point. We consider the case when two imaginary eigenvalues of the matrix of the linearized system are in the ratio 3: 1. We study nonlinear conditionally periodic motions of the system in the vicinity of the equilibrium point. Omitting the terms of order higher then five in the normalized Hamiltonian we analyze the so-called truncated system in detail. We show that its general solution can be given in terms of elliptic integrals and elliptic functions. The motions of truncated system are either periodic, or asymptotic to a periodic one, or conditionally periodic. By using the KAM theory methods we show that most of the conditionally periodic trajectories of the truncated systems persist also in the full system. Moreover, the trajectories that are not conditionally periodic in the full system belong to a subset of exponentially small measure. The results of the study are applied for the analysis of nonlinear motions of a symmetric satellite in a neighborhood of its cylindric precession.