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Matrix measures, stability and contraction theory for dynamical systems on time scales

  • * Corresponding author: Giovanni Russo

    * Corresponding author: Giovanni Russo 
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  • This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.

    Mathematics Subject Classification: Primary: 34N05, 93C10; Secondary: 93D99.


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  • Figure 1.  Time evolution of (26) with the representative set of parameters of Section 8. Top panel: dynamics evolving on the time scale $ \mathbb{P}_{a,b} $, with $ a = 1 $ and $ b = 0.24 $. Both conditions (C1) and (C2) are satisfied. Bottom panel: dynamics evolving, with the same parameters, on the discrete time scale of Section 8 with $ c = 0.24 $ so that both (C1) and (C2) are satisfied. Code at: https://github.com/GIOVRUSSO/Control-Group-Code

    Figure 2.  Time behavior of (26) when $ {\mathbb T}\equiv {\mathbb R} $. The parameters are taken from [49]. In the simulation, we used $ \beta = 0.0373/N $ (i.e. the population is in lock-down), $ k_d = 1 $, $ k_{\Lambda} = N $ and initial conditions $ [0.25 N, 0.25 N, 0.25N, 0.25N] $. The code is available at: https://github.com/GIOVRUSSO/Control-Group-Code

    Figure 3.  Top panel: graph of the small world network considered in Section 11. The number of nodes is 100 and the nodes pinned by the stubborn agent are highlighted in red in the figure (colors online). In total, 49 nodes were pinned. The network was built following the Watts-Strogatz model [62] and by setting the mean node degree to $ 2 $ and the rewiring probability to $ 0.7 $. Bottom panel: time evolution for the network (the time evolution for $ x_r(t) $ is highlighted with a dashed black line). Code for the simulations at: https://github.com/GIOVRUSSO/Control-Group-Code

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