Linear stability of high-dimensional symplectic systems

1998

We analytically compute asymptotic expansions of a 1-dimensional submanifold of stable and unstable manifolds in a 4-dimensional symplectic mapping by using the method called asymptotic expansions beyond all orders. This method enables us to capture exponentially small splitting of separatrices and also to obtain explicit functional approximations of the sub-manifolds. In addition, we show the condition with which homoclinic structure caused by crossing between the stable and unstable sub-manifolds is regarded as a direct product of 2-dimensional mappings.

Progress of Theoretical Physics, 1999

The homoclinic bifurcation of 4-dimensional symplectic mappings is studied asymptotically. We construct 2-dimensional stable and unstable manifolds near sub-manifolds that experience exponentially small splitting, and successfully obtain exponentially small oscillating terms in these 2

2010

We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way.

International Journal of Robust and Nonlinear Control, 1991

In this paper, we introduce and analyse robustness measures for the stability of discrete-time system x(t + 1) = A x (t) under parameter perturbations of the form A + A + BDC where B , C are given matrices. In particular we characterize the stability radius of the uncertain system x(t + 1) = (A + BDC)x(t), D an unknown complex perturbation matrix, via an associated symplectic pencil and present an algorithm for the computation of that radius. 1. INTRODUCTION Consider a dynamical system described by the following linear difference equation x(t + 1) = Ax(t), t E N (1) where A is an n x n matrix and x (t) denotes the n-dimensional state vector. Suppose that the system is CI-stable, i.e. the spectrum a(A) of A lies in the open unit disk C1 = [se C; I sl < 1). It follows from the continuity of the spectrum that this property is preserved under suficiently small perturbations of the entries of A. An important problem of robustness analysis is to determine to what extent stability is preserved when the entries of the nominal system matrix A are subjected to large parameter perturbations. In this paper we assume that the perturbed system matrix has the form A + BDC where BE IK""", CE IKPX" are fixed matrices defining the structure of the perturbations and D E IKmxp is an unknown disturbance matrix, IK = R or C. The perturbed system may be formally interpreted as a closed loop system with unknown static linear output feedback (Figure 1). Figure 1. Feedback interpretation of the perturbed system This paper was recommended for publication by editor M. Green

2010

We present a multiscale integrator for Hamiltonian systems with slowly varying quadratic stiff potentials that uses coarse timesteps (analogous to what the impulse method uses for constant quadratic stiff potentials). This method is based on the highly-non-trivial introduction of two efficient symplectic schemes for exponentiations of matrices that only require O(n) matrix multiplications operations at each coarse time step for a preset small number n. The proposed integrator is shown to be (i) uniformly convergent on positions; (ii) symplectic in both slow and fast variables; (iii) well adapted to high dimensional systems. Our framework also provides a general method for iteratively exponentiating a slowly varying sequence of (possibly high dimensional) matrices in an efficient way.

2006

Abstract We give a constructive argument to establish existence of asmooth singular value decomposition (SVD) for a generic C k symplectic function X. We rely on the explicit structure of the polar factorization of X in order to justify the form of the SVD. Our construction gives a new algorithm to find the SVD of X, which we have used to approximate the Lyapunov exponents of a Hamiltonian differential system. Algorithmic details and an example are given. Keywords: symplectic group, singular values, Lyapunov exponents

Proceedings of the 28th IEEE Conference on Decision and Control

In this paper, we introduce and analyze robustness measures for the stability of discrete-time systems x(t+1) = Ax(t) under parameter perturbations of the form A ! A + BDC where B; C are given matrices. In particular we characterize the stability radius of the uncertain system x(t+1) = (A+BDC)x(t), D an unknown complex perturbation matrix, via an associated symplectic pencil and present an algorithm for the computation of that radius.

African Diaspora Journal of Mathematics , 2017

This paper continues to carry out a foundational study of Banyaga's topologies of a closed symplectic manifold (M,ω) [4]. Our intention in writing this paper is to work out several “symplectic analogues” of some results found in the study of Hamiltonian dynamics. By symplectic analogue, we mean if the first de Rham's group (with real coefficients) of the manifold is trivial, then the results of this paper reduce to some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out an impact of the L∞−version of Hofer-like length in the investigation of the symplectic nature of the C0−limit of a sequence of symplectic maps. This yields a symplectic analogue of a result that was proved by Hofer-Zehnder [10] (for compactly supported Hamiltonian diffeomorphisms on R2n); then reformulated by Oh-Müller [14] for Hamiltonian diffeomorphisms in general. Furthermore, we show that Polterovich's regularization process for Hamiltonian paths extends over the whole group of symplectic isotopies, and then use it to prove the equality between the two versions of Hofer-like norms. This yields the symplectic analogue of the uniqueness result of Hofer's geometry proved by Polterovich [13]. Our results also include the symplectic analogues of some approximation lemmas found by Oh-Müller [14]. As a consequence of a result of this paper, we prove by other method a result found by McDuff-Salamon.

The European Physical Journal B, 2006

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d = 1 lattice array. The global coupling is modulated through a factor r −α , being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0 ≤ α ≤ 1, and short-range (integrable) when α > 1. We verify that the largest Lyapunov exponent λM scales as λM ∝ N −κ(α) , where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N → ∞ (hence λM → 0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration tc scales as tc ∝ N β(α) , where β(α) appears to be numerically consistent with the following behavior: β > 0 for 0 ≤ α < 1, and zero for α ≥ 1. All these results exhibit major conjectures formulated within nonextensive statistical mechanics (NSM). Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model, also studied in the frame of NSM.

2011

This paper develops a symplectic bifurcation theory for integrable systems in dimension four. We prove that if an integrable system has no hyperbolic singularities and its bifurcation diagram has no vertical tangencies, then the fibers of the induced singular Lagrangian fibration are connected. The image of this singular Lagrangian fibration is, up to smooth deformations, a planar region bounded by the graphs of two continuous functions. The bifurcation diagram consists of the boundary points in this image plus a countable collection of rank zero singularities, which are contained in the interior of the image. Because it recently has become clear to the mathematics and mathematical physics communities that the bifurcation diagram of an integrable system provides the best framework to study symplectic invariants, this paper provides a setting for studying quantization questions, and spectral theory of quantum integrable systems.