[Paper Review] Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

Marat Akhmet, Madina Tleubergenova+1 more|arXiv (Cornell University)|Nov 18, 2021

Advanced Differential Equations and Dynamical SystemsMathematics19 references21 citations

TL;DR

This paper introduces modulo periodic Poisson stable (MPPS) solutions for quasilinear differential equations by combining a periodic component with a Poisson stable component. Using a novel verification method for Poisson stability and assuming the Poisson number κω = 0, the authors prove the existence and asymptotic stability of unique MPPS solutions, with numerical simulations demonstrating their feasibility in dynamical systems, particularly relevant to neuroscience and engineering applications.

ABSTRACT

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.

Motivation & Objective

• To introduce a new class of recurrent solutions—modulo periodic Poisson stable (MPPS) functions—by combining periodic and Poisson stable components.
• To establish sufficient conditions for the existence and asymptotic stability of MPPS solutions in quasilinear systems with periodic and Poisson stable coefficients.
• To develop a verifiable method for Poisson stability based on the Poisson number κω = 0, enabling theoretical and numerical analysis.
• To provide numerical simulations of Poisson stable functions using hybrid dynamical systems, extending applicability to real-world systems like neural networks and mechanical oscillators.
• To explore the presence of periodic components in chaotic attractors (e.g., Lorenz, Rössler, Chua) and their implications for optimization and control.

Proposed method

• Define MPPS functions as the sum of a continuous ω-periodic function φ(t) and a Poisson stable function ψ(t), with the Poisson number κω = 0 ensuring recurrence.
• Use the fundamental matrix X(t,s) of the homogeneous system x′ = A(t)x to construct the solution x(t) = ∫_{-∞}^t X(t,s)[φ(s)+ψ(s)]ds.
• Apply the exponential decay estimate ∥X(t,s)∥ ≤ Ke^{-α(t−s)} under condition (C4) to bound solution differences.
• Leverage Lemma 1 to estimate the difference ∥X(t+τ,s+τ)−X(t,s)∥ using the norm of A(t+τ)−A(t), enabling stability analysis.
• Use the contraction argument with inequalities (6)–(8) to prove that xψ(t) is Poisson stable under κω = 0.
• Construct Poisson stable functions numerically via hybrid systems: discrete logistic map λ_{n+1} = μλ_n(1−λ_n) for μ ∈ [3.89, 4], and continuous solution Θ(t) = ∫_{-∞}^t e^{-2(t−s)}Ω(s)ds with piecewise-constant Ω(t).

Experimental results

Research questions

• RQ1 Can a new class of solutions combining periodicity and Poisson stability be rigorously defined and analyzed in quasilinear systems?
• RQ2 Under what conditions does the sum of a periodic function and a Poisson stable function yield a Poisson stable solution?
• RQ3 What role does the Poisson number κω play in determining the recurrence behavior of solutions?
• RQ4 How can Poisson stable functions be numerically simulated and visualized in dynamical systems?
• RQ5 Can MPPS solutions be extended to chaotic systems like Lorenz, Rössler, or Chua attractors, and what implications does this have for control and optimization?

Key findings

• The system x′(t) = A(t)x(t) + φ(t) + ψ(t) admits a unique asymptotically stable MPPS solution under conditions (C1)–(C4), including κω = 0.
• The solution xψ(t) = ∫_{-∞}^t X(t,s)ψ(s)ds is proven to be Poisson stable when the Poisson number κω = 0 and ∥A(t+tk)−A(t)∥ is small.
• The function Θ(t) = ∫_{-∞}^t e^{-2(t−s)}Ω(s)ds is bounded with sup_t |Θ(t)| ≤ 1/2 and is Poisson stable for Ω(t) derived from the logistic map with μ = 3.89.
• Numerical simulations confirm that Θ(t) exhibits Poisson stability: |Θ(t+ζn)−Θ(t)| → 0 uniformly on compact intervals as n → ∞.
• The function G(t, υ(t)) is Poisson stable if υ(t) is ω-periodic and G satisfies a Lipschitz condition, provided κω = 0.
• The results generalize prior work on unpredictable and Poisson stable solutions by incorporating periodic components, enabling broader applications in neuroscience and engineering.