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Register a new userNormal stability of slow manifolds in nearly-periodic Hamiltonian systems
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M. Kruskal showed that each nearly-periodic dynamical system admits a formal $U(1)$ symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly-invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly-periodic Hamiltonian systems on barely symplectic manifolds -- manifolds equipped with closed, non-degenerate $2$-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly-periodic system. We prove that one previous embedding, and two new embeddings enjoy long-term normal stability, and thereby strengthen the theoretical justification for these models.
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We consider the problem of the continuation with respect to a small parameter $epsilon$ of spatially localised and time periodic solutions in 1-dimensional dNLS lattices, where $epsilon$ represents the strength of the interaction among the sites on the lattice. Specifically, we consider different dNLS models and apply a recently developed normal form algorithm in order to investigate the continuation and the linear stability of degenerate localised periodic orbits on lower and full dimensional invariant resonant tori. We recover results already existing in the literature and provide new insightful ones, both for discrete solitons and for invariant subtori.
We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum methods. Using a topological generalisation of Lyapunovs result that an extremal critical point of a conserved quantity is stable, we show that a Poisson equilibrium is stable if it is an isolated point in the intersection of a level set of a conserved function with a subset of the phase space that is related to the non-Hausdorff nature of the symplectic leaf space at that point. This criterion is applied to generalise the Energy-Momentum method to Hamiltonian systems which are invariant under non-compact symmetry groups for which the coadjoint orbit space is not Hausdorff. We also show that a $G$-stable relative equilibrium satisfies the stronger condition of being $A$-stable, where $A$ is a specific group-theoretically defined subset of $G$ which contains the momentum isotropy subgroup of the relative equilibrium.
We consider the classical problem of the continuation of periodic orbits surviving to the breaking of invariant lower dimensional resonant tori in nearly integrable Hamiltonian systems. In particular we extend our previous results (presented in CNSNS, 61:198-224, 2018) for full dimensional resonant tori to lower dimensional ones. We develop a constructive normal form scheme that allows to identify and approximate the periodic orbits which continue to exist after the breaking of the resonant torus. A specific feature of our algorithm consists in the possibility of dealing with degenerate periodic orbits. Besides, under suitable hypothesis on the spectrum of the approximate periodic orbit, we obtain information on the linear stability of the periodic orbits feasible of continuation. A pedagogical example involving few degrees of freedom, but connected to the classical topic of discrete solitons in dNLS-lattices, is also provided.
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a suitable normal form construction that allows to identify and approximate the periodic orbits which survive to the breaking of the resonant torus. Our algorithm allows to treat the continuation of approximate orbits which are at leading order degenerate, hence not covered by classical averaging methods. We discuss possible future extensions and applications to localized periodic orbits in chains of weakly coupled oscillators.
In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: $$ H(x,y)=h(y)+eps P(x,y), qquad xinmathbb{T}^n, yinmathbb{R}^n,quad ngeq 3. $$ Under typical perturbation $eps P$, the system admits connecting orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>min h$.
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