Hartman-Grobman theorems along hyperbolic stationary trajectories

Abstract

Abstract. We extend the Hartman-Grobman theorems for discrete random dynamical systems (RDS), proved in [7], in two directions: for continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between travelling neighborhoods of trajectories and neighborhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.1. Introduction. The celebrated Hartman-Grobman theorem (HGT, for short) plays a fundamental rule in the theory of dynamical systems. Essentially, among other features, it allows one to make topological classification of the dynamics in a neighborhood of hyperbolic fixed points. This classification is based on the existence of a conjugacy of the local dynamics with the linearized system at a hyperbolic fixed point. For the original papers we mention Hartman [10] and [11], and Grobman [9]. One of the first results concerning HGT in random dynamical systems (RDS, for short) is due to Wanner [19] for discrete systems. His argument was based on random difference equation, such that the construction was made ω-wise. His proof is completed by showing that the choice of random homeomorphisms is, in fact, measurable. In our intrinsic approach, Coayla-Teran and Ruffino [7], we have looked for the conjugacy in an appropriate Banach space of random homeomorphisms. The arguments in [7] correspond to an appropriate extension of the deterministic arguments, with the state space enlarged by the probability space. Hence the norms and other constants were composed with L 1 (Ω) norm. In this article, we extend

On the Critical Coupling for Kuramoto Oscillators

“…This implies that all vector fields H λ , λ ∈ [0, 1], are locally topologically conjugate [39] near a hyperbolic equilibrium point x * ∈ E. In particular, near x * ∈ E, trajectories of the gradient vector field (5.2) can be continuously deformed to match trajectories of the Hamiltonian vector field (5.3) while preserving parameterization of time. This topological conjugacy holds also for hyperbolic equilibrium trajectories [17,Theorem 6] considered in synchronization. The similarity between second-order Hamiltonian systems and the corresponding first-order gradient flows is well-known in mechanical control systems [31,32], in dynamic optimization [4,5,25], and in transient stability studies for power networks [12,11,15], but we are not aware of any result as general as Theorem 5.1.…”

“…We now prove the final conjugacy statement. By the generalized Hartman-Grobman theorem [17,Theorem 6], the trajectories of the three vector fields (5.11), (5.6)-(5.7) (formulated in a rotating frame), and (5.8)-(5.9) are locally topologically conjugate to the flow generated by their respective linearized vector fields (locally near (Φ γ,0 (t), 0 m×1 ). Since the three vector fields (5.11), (5.6)-(5.7), and (5.8)-(5.9) are hyperbolic with respect to (Φ γ,0 (t), 0 m×1 ) and their respective Jacobians have the same hyperbolic inertia (besides the common one-dimensional center eigenspace corresponding to (Φ γ,0 (t), 0 m×1 ), the corresponding three linearized dynamics are topologically conjugate [39,Theorem 7.1].…”

Climate dynamics and fluid mechanics: Natural variability and related uncertainties

“…Moreover, by reinterpreting other results of Kaijser [102] in our RDS framework, we can show that the random attractor is in fact a random fixed point; this, in turn, allows us to conclude that the corresponding linearized cocycle at this random fixed point is hyperbolic. Next, by using the Hartman-Grobman theorem for RDSs [118,119,120], we can conjugate the nonlinear cocycle with its linearization; in fact, Theorem 3.1 of [119] says that this conjugacy is global.…”

Linearization and local stability of random dynamical systems

“…Surprisingly, the literature does not contain results which would be quite suitable for applications in this field. The main tools offered for local analysis in the context of stochastic dynamics (see the classical monograph by L. Arnold [1]) are stochastic analogues of the Hartman-Grobman theorem [17] - [19] developed in [38] and [5] - [7], and closely related results on stable manifold theorems [4,27,29,30,34,35]. For the most part, these are delicate results, the use of which requires the verification of complex conditions.…”