The basic idea is again to break the problem up into the family of autonomous linear problems and "feedback" via the Euler algorithm, exactly as for solutions with values in (L^p({{mathbb R}^{N}})) (in Sect. 4, 5).
Fig. 5 Voltage traces for the autonomous linear system (29)–(30) for various representative values of α, ϵ and A t = 1. a α = 1.
Fig. 4 Phase-plane diagrams for the autonomous linear system (29)–(30) for various representative values of α, ϵ and A t. a α = 1.
For the autonomous linear fractional differential systems with order, the necessary and sufficient conditions on stability and asymptotic stability are given, which are almost the same as those with the fractional derivative order.
This feature is concluded for solutions to the autonomous linear transport equation with "less regular" coefficients.
H. Balakrishnan, I. Hwang, J. S. Jang, C. J. Tomlin, Inference Methods for Autonomous Stochastic Linear Hybrid Systems, Springer-Verlag Lecture Notes in Computer Science (LNCS 2993), Alur and Pappas (Eds)., pp. 64-79, March 2004.
This is the familiar vector matrix state differential equation of an autonomous linear system.
We extend the autonomous linear equations in Chitnis et al. (2008, (5)) to a system of θ p -periodic linear nonhomogeneous difference equations.
Below, in the next paragraph, we formulate, in convenient for us form, a result about the existence and structure of the solution for general non-autonomous linear systems of ODEs.
Linearizing about the time-dependent solution of the unperturbed equation ((mathbf{h}equiv 0)) leads to the following (non-autonomous) linear equation for the perturbed solution (u_{alpha}(t)=u_{alpha}^{h}(t -u_{alpha}^{0}(t -u_{alpha}{d u_{alpha}}{dt}= -u_{alpha}+sum_{beta}w_{alpha beta }F' bigl(u_{beta}^{0}bigr)u_{beta} +h_{alpha}(taufrac{d
