Bridging the Gap between Reactivity, Contraction, and Finite-Time Lyapunov Exponents

TL;DR

Bridging the Gap between Reactivity, Contraction, and Finite-Time Lyapunov Exponents develops a unified framework to analyze transient stability in discrete-time dynamical systems by introducing sup-mean contractivity and the concept of $p$-iteration. It demonstrates that average contraction over $N$ steps suffices to guarantee attractor stability and that network synchronization can be assessed via average transverse reactivity, linking to the master stability function approach. The paper connects reactivity with finite-time Lyapunov exponents, showing how the $p$-iteration continuum spans from local reactivity to asymptotic Lyapunov behavior as $p\to\infty$, and it provides concrete results for time-varying linear and nonlinear maps, including logistic and Henon systems. The framework offers actionable tools for predicting stability regions in networks, at the cost of increased computation for larger $p$, and sets the stage for future work on stopping criteria and scalable synchronization analysis.

Abstract

Reactivity, contractivity, and Lyapunov exponents are powerful tools for studying the stability properties of dynamical systems and have been extensively investigated in the literature for decades. In this paper, we review and extend the concepts of reactivity, contractivity, and finite-time Lyapunov exponents for discrete-time dynamical systems and establish connections among them. We focus on time-invariant maps, time-varying linear maps, and certain classes of time-varying nonlinear maps. In particular, we show that if the corresponding $p$-iteration systems (with p > 1) are contractive, then the original systems admit stable attractors such as fixed points or limit cycles. We demonstrate the application of these results to the analysis of synchronization stability in coupled networks and discuss how p-iteration systems can serve as a useful framework for studying network synchronization.

Figures (4)

• Figure 1: The reactivity of the $p$-iteration system. For $p= 1$, $r_2[\mathcal{A}_k^{(p)}] > 0$ for $k = 0,1, 2$. Here, $\lambda = 0.9$ in Example 1.
• Figure 2: (a) The maximum value $w^*$ from \ref{['eq:optperiodic']}; (b,c) the optimal values of $a, b, c, d$ from \ref{['eq:optperiodic']} as functions of the parameter $\alpha$ of the time-invariant Logistic map in Example 2, for iteration numbers $p = 2$ and $p = 4$, respectively. The gaps within the curves correspond to the infeasibility of the optimization for $\alpha$ close to $3.4$.
• Figure 3: (a) The maximum value $w^*$ from \ref{['eq:optperiodic2']}; (b,c) the optimal values of $a, b, c, d$ from \ref{['eq:optperiodic2']} as functions of the parameter $e$ of the time-varying Logistic map \ref{['eq:logistictv']}, for iteration numbers $p = 2$ and $p = 4$, respectively.
• Figure 4: (a) Synchronizability criteria based on \ref{['eq:synchronizabilitymean']}, computed using either $R_2^{(\infty)} (p)$$\left(\beta_\text{mean}^{(p)}\right)$ or $\bar{R}_2 (p)$$\left(\beta_{\max}^{(p)}\right)$. (b) The unweighted and undirected network used in this example. (c) Bounds of the synchronous coupling strength interval $\kappa$, as defined in \ref{['eq:kappa']}. The green region corresponds to the interval based on $R^{(\infty)}_2(p)$, while the pink region is based on $\bar{R}_2(p)$. Dashed black lines indicate the interval of $\kappa$ in which the synchronization of the full system \ref{['eq:synch']} is stable. Synchronization is considered achieved when after simulating \ref{['eq:synch']}, the error $E = \frac{1}{k_f - k_0} \sum_{k=k_0}^{k_f} \frac{1}{n} \sum_{i=1}^{n} \| \pmb{x}^i_k - \frac{1}{n} \sum_{j=1}^{n} \pmb{x}^j_k \|_2$ falls below $10^{-10}$. In this example, we set $k_f = 10000$ and $k_0 = 9000$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Proposition 1
• Remark 1
• Definition 5
• Remark 2
• Proposition 2
• Proposition 3