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Random attractors and almost-sure stability under discretization of a stochastic autoparametric system

Chuchu Chen, Jialin Hong, Yibo Wang

arXiv:2608.29149v1math.DSmath.NA

TL;DR

The paper asks whether stochastic autoparametric systems’ global asymptotic dynamics and single-mode almost-sure stability classification are preserved by discretization. It proves existence of a continuous random attractor, constructs a discrete random dynamical system, and establishes attractor convergence and Lyapunov-exponent sign preservation for sufficiently small steps.

  • Problem

    Existing results had not established random attractors for these stochastic autoparametric systems, while simultaneous attractor approximation and stability-classification preservation under discretization remained less understood.

  • Method

    The paper formulates the stochastic model as a multiplicative-noise SDE, constructs its continuous random dynamical system, and introduces a discretization with an auxiliary transformation restoring discrete dissipativity.

  • Results

    The continuous system admits a random attractor; the discrete attractor converges to it as the step size vanishes, and the numerical Lyapunov exponent preserves the continuous exponent’s sign for sufficiently small steps.

  • Takeaways & Limitations

    The proposed discretization captures both the continuous system’s global asymptotic dynamics and its almost-sure stability classification.

  • Takeaways & Limitations

    The discrete one-step map may not be invertible for all noise realizations, preventing extension of the cocycle to negative times in general.

Abstract

from arXiv · show

For a stochastic autoparametric block-and-pendulum system, the long-time dynamics exhibit two fundamental features: the almost-sure stability of the single mode solution, characterized by its Lyapunov exponent, and the global asymptotic dynamics when this single mode solution loses stability. This naturally raises the question of whether these dynamical features are preserved under discretization, since such preservation is essential for the resulting discrete system to faithfully capture the qualitative behavior of the continuous system. To address this question, we first establish the existence of a random attractor for the continuous system subject to multiplicative stochastic excitation, providing a rigorous characterization of the global asymptotic dynamics. We then propose a numerical discretization that induces a discrete random dynamical system and prove the convergence of its random attractor to the continuous one as the step size tends to zero. In addition, we show that the numerical Lyapunov exponent of the single mode solution has the same sign as its continuous counterpart for sufficiently small step sizes, thus preserving the corresponding almost-sure stability or instability classification. These results demonstrate that the proposed discretization captures both the global asymptotic dynamics and the stability characteristics of the underlying stochastic autoparametric system.

1. Introduction

The paper studies stochastic autoparametric block-and-pendulum dynamics, focusing on single-mode stability, global asymptotic behavior, and whether both features survive time discretization. It establishes random-attractor and Lyapunov-exponent results for continuous and discrete systems.

  • Motivation: Stochastic excitation makes stability thresholds and long-time behavior path-dependent, motivating almost-sure stability and global asymptotic analysis.
  • Single mode solution: The system contains a single mode in which the block oscillates while the pendulum remains vertical, with transverse stability determined by the top Lyapunov exponent.
  • Open problems: Prior work did not establish random attractors for stochastic autoparametric systems, leaving their post-activation global dynamics unresolved.
  • Research questions: The paper asks whether the continuous system generates a random attractor and whether its attractor and stability classification can be approximated under discretization.
  • Contributions: The proposed discretization induces a discrete random dynamical system whose random attractor converges to the continuous attractor as the step size tends to zero.
  • Continuous formulation: Resolving the second-order equations produces state-dependent diffusion coefficients, so the first-order formulation has multiplicative noise that complicates dissipative estimates.
  • Numerical approach: An auxiliary transformation restores a suitable discrete dissipative structure, enabling a random absorbing set and attractor convergence despite discretization difficulties.

2. Preliminary

This section formulates the stochastic autoparametric system as an RDS, identifies its invariant single-mode dynamics, and develops Lyapunov-exponent tools for transverse stability analysis.

  • Random dynamical systems: The system is modeled as a continuous RDS over an ergodic metric dynamical system, with a cocycle describing noise-perturbed state evolution.The discrete-time counterpart uses T = Z, while the continuous system uses T = R.
  • Random attractors: A random attractor is an invariant random compact set that pullback-attracts all compact tempered sets and describes the asymptotic regime.Attraction is measured using the Hausdorff semi-distance.
  • Model formulation: The stochastic equations form a first-order SDE on N = R2 × R/(2πZ) × R with multiplicative noise determined by the pendulum angle.The drift and diffusion maps are specified through f, f1, f2, and g.
  • Invariant single-mode dynamics: The single mode solution lies on the invariant manifold N0, where the pendulum remains vertical, while Nπ represents the inverted-pendulum configuration.The associated stationary measures are ν0 and νπ, and trajectories initialized on these subsets converge to them in distribution.
  • Lyapunov exponents: Transverse linearization reduces stability of the single mode solution to the Lyapunov exponent λ0 of a parametrically excited pendulum perturbation.The block perturbation is damped and unforced, so the long-time stability question concerns the angular perturbation driven by the single mode vibration.
  • Lyapunov exponents: The exponent λ0 exists almost surely and admits a Furstenberg–Khasminskii representation as a stationary average of the logarithmic growth rate Q.A unique stationary measure for the reduced process supports this representation.
  • Lyapunov-exponent estimate: For sufficiently small forcing and damping, a suitable damping choice can yield ˆλ0(ε) > 0, implying instability of the single mode solution.The existence of an additional stationary measure away from N0 and Nπ remains conjectural.

3. Random attractor for continuous RDS

The continuous stochastic system is transformed into an equivalent additive-noise formulation, enabling construction of a continuous RDS and proof of a random attractor. The argument combines conjugacy, dissipative estimates, and pullback absorption.

  • Transformation and RDS construction: A stationary Ornstein–Uhlenbeck transformation converts the multiplicative-noise SDE into an equivalent random differential equation with additive noise.The transformation is built from dz = −αzdt + σdω(t) and preserves the dynamics through conjugacy.
  • Dissipative estimates: Tempered-noise and tempered-set estimates yield a pullback absorbing set for the transformed RDS.The construction uses sublinear Brownian growth, Gronwall’s inequality, and a finite random radius.
  • Random-attractor existence: For every ω and x, the SDE has a unique global solution whose solution mapping generates a continuous RDS possessing a random attractor.The attractor for the original system is obtained from the transformed attractor under the inverse conjugacy.
  • Transformation and RDS construction: The map T is a diffeomorphism, so the transformed and original solution mappings are conjugate and generate equivalent continuous RDSs.A random attractor for the transformed system maps back to one for the original system via T^−1.
  • Dissipative estimates: The transformed system admits a coercive energy estimate whose quadratic dissipation is strictly positive for sufficiently small δ2 and ϵ.The energy is bounded above and below by constants times the squared state norm, up to an additive constant.

4. Numerical method and random attractor for discrete RDS

The paper designs a discretization that preserves the continuous system’s key long-time features by inducing a discrete RDS with a convergent random attractor.

  • Random attractor: Uniform moment bounds and an auxiliary transformation recover a discrete dissipative structure and yield a random absorbing set.These estimates support construction of the discrete RDS and its random attractor.
  • Numerical method: A variable transformation addresses the state-dependent diffusion and recovers an additive-noise formulation suitable for analysis.The transformed equations retain the original dynamics while making dissipative estimates tractable.
  • Numerical method: The proposed implicit scheme is uniquely solvable and consistently approximates the transformed SDE with a higher-order remainder term.The remainder satisfies ∥Rk∥≤Cτ(τ+|∆Wk|)(∥Vk∥2+1).
  • Scope boundary: The discrete RDS is generally forward-time only because its one-step map may not be invertible, although forward asymptotics remain sufficient here.Backward iterates are unavailable in general, while pullback formulation still uses the two-sided noise.
  • Random attractor: For sufficiently small step size τ>0, the discrete RDS possesses a random attractor Aτ(ω).The discrete cocycle is constructed over the shifted two-sided Wiener space and is conjugate to the transformed system.
  • Random attractor: The numerical random attractor Aτ(ω) converges to the continuous random attractor A(ω) under the Hausdorff semi-distance.The convergence is stated as the step size tends to zero.

5. Numerical Lyapunov exponent for the single mode solution

The section analyzes the numerical Lyapunov exponent of the discretized single mode solution and shows that sufficiently small steps preserve the continuous stability classification.

  • Main result: The numerical method preserves the sign of λ0, thereby preserving almost-sure stability or instability of the single mode solution for sufficiently small τ.This is the section’s principal qualitative conclusion.
  • Stability mechanism: The transverse stability is governed by the parametric excitation of the pendulum perturbation variables induced by the single mode vibration.The primary perturbation follows an exponentially stable unforced damped oscillator and decays asymptotically.
  • Estimation strategy: The exponent λτ is evaluated through polar-coordinate estimates, stationary measures, ergodicity, and comparison with the continuous exponent λ0.Direct evaluation is nontrivial because the variational process is driven by Brownian increments and colored noise.
  • Estimation strategy: For sufficiently small τ, Theorems 5.3 and 5.4 provide matching bounds connecting λτ to λ0.The bounds are obtained by estimating the long-time growth of the transverse variational process.
  • Main result: There exist sufficiently large K and sufficiently small τ0 such that λτ is sign-consistent with λ0 whenever k>K and τ<τ0.The proof treats separately the cases λ0<0 and λ0>0, yielding λτ<0 and λτ>0 respectively.

6. Proofs of propositions in Section 5

The proofs establish stationary-measure properties and convergence estimates needed to connect the discrete Lyapunov exponent with its continuous counterpart.

  • Stationary measures: The proof of Proposition 5.1 verifies Lyapunov and minorization conditions for the Markov chain associated with the variational dynamics.These conditions imply existence and uniqueness of a stationary measure.
  • Stationary measures: The Lyapunov function Γ2(v,x,ψ):=κ1v2+x2+δ4vx+M controls the discrete state and supports dissipative estimates.The parameters satisfy δ4∈(0,√κ1) and M≥1.
  • Stationary measures: The variational Markov chain has a unique stationary measure µτ, while the primary numerical process has a unique stationary measure ντ.Both measures are accompanied by moment bounds needed in the exponent estimates.
  • Convergence estimates: Finite-time strong convergence between the numerical process Ck and the continuous process c(tk) supplies the key discretization-error estimate.The comparison is established on every finite interval [0,T] for sufficiently small τ.
  • Convergence estimates: Exponential integrability and moment boundedness control the stochastic remainder terms in the long-time comparison of λτ and λ0.The proof combines Taylor expansions, Hölder and Cauchy–Schwarz inequalities, martingale limits, and ergodic theorems.

7. Numerical experiments

Numerical experiments examine Lyapunov-exponent stability, attractor robustness under timestep refinement, and noise-dependent transitions in long-time dynamics.

  • Lyapunov-exponent stability: The numerical Lyapunov exponent increases with noise intensity, and positive regions indicate loss of almost-sure stability of the single mode solution.The left panel varies σ and ζ2; the red line marks λτ = 0.
  • Lyapunov-exponent stability: With ζ2 = 0.1, the single-mode and four-dimensional-system exponents both increase with σ and become positive beyond a threshold.The comparison is qualitative rather than quantitatively equivalent.
  • Random attractors: At σ = 2 with λτ > 0, an ensemble of 10^5 initial conditions exhibits complex dynamics in projections and physical block-and-pendulum configurations.The simulation uses T = 50 and τ = 5 × 10^-4 under a fixed noise realization.
  • Discretization robustness: Projected attractors are nearly indistinguishable across τ = k × 10^-3 for k = 0.5, 1, 2, 4, indicating convergence as the timestep is refined.The comparison uses the same noise realization and projections onto U_k-U_k and V_k-V_k planes.
  • Noise-dependent dynamics: For small σ with λτ < 0, attractor projections collapse to a stable point; for larger σ with λτ > 0, they broaden into bands indicating a transition to chaotic dynamics.The bifurcation diagrams project computed random attractors onto the U_k- and V_k-axes.

Appendix A. Estimate of Ek in the proof of Theorem 4.4

The appendix estimates the discrete error term E_k using algebraic rewrites, inequalities, Taylor expansion, and timestep-dependent bounds.

  • Error estimate: The estimate begins by rewriting E_k using an earlier relation and then applying the triangle inequality.
  • Error estimate: The calculation collects timestep-, state-, and noise-dependent terms to establish the required discrete dissipative estimate.
  • Error estimate: Taylor expansion and Young’s inequality provide additional bounds when δ3 is sufficiently small.
  • Error estimate: Relations involving Ū_k, Z_k, and V̄_k are combined with Young’s inequality to control the discrete terms.The argument uses Ū_{k+1} = Ū_k + τ S̄_k Ū_{k+1} and |Ū_k| ≤ 2π.
  • Error estimate: The resulting estimates introduce positive constants c5 and c̄5 after choosing ε sufficiently small.

Appendix B. Proof of Lemma 5.2

The lemma proof derives componentwise relations for B_k and bounds the transformed variable B̃_{k+1} using timestep and noise-increment terms.

  • Component estimates: Writing B_k in polar form introduces c_k = cos ψ_k and s_k = sin ψ_k for the subsequent component estimates.
  • Component estimates: The discrete relations for β_k and β̄_k are obtained directly from the scheme and are used to verify the stated lemma identities.
  • Component estimates: The transformed quantity B̃_{k+1} is bounded by a noise-increment factor multiplying B_{k+1}, ΔW_k, Z_{k+1}, and timestep terms, plus |B̄_{k+1}|.
  • Component estimates: The proof further controls B_{k+1} through timestep and noise-dependent factors before concluding the lemma.

Appendix C. Calculation of ck and sk

The appendix derives the discrete evolution of angular components c_k and s_k from the scheme, using polar coordinates and auxiliary estimates.

  • Derivation of c_k and s_k: The evolution of B_{k+1} is substituted into the transformed variables to obtain bounds needed for the angular-component calculation.
  • Derivation of c_k and s_k: Moment boundedness of Z_k is used to control the resulting expressions for any p ≥ 1.
  • Derivation of c_k and s_k: The vector C_k = (η_k, η̄_k, c_k, s_k)^⊤ is shown to satisfy a discrete stochastic equation with drift, noise, and correction terms.
  • Derivation of c_k and s_k: The first two components follow from the original discrete equations, while the c_k and s_k components are derived from β_{k+1} = β_k + τβ̄_{k+1}.
  • Derivation of c_k and s_k: The equations for c_k and s_k are handled similarly, with the function Φ3 collecting the resulting nonlinear terms.
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