By Igor Chueshov

This ebook is dedicated to heritage fabric and lately constructed mathematical equipment within the examine of infinite-dimensional dissipative structures. the speculation of such structures is encouraged via the long term target to set up rigorous mathematical types for turbulent and chaotic phenomena. the purpose here's to provide normal tools and summary effects bearing on basic dynamical platforms homes relating to dissipative long-time habit. The ebook systematically provides, develops and makes use of the quasi-stability procedure whereas considerably extending it by means of together with for attention new periods of versions and PDE structures bobbing up in Continuum Mechanics. The ebook can be utilized as a textbook in dissipative dynamics on the graduate level.

**Igor Chueshov** is a Professor of arithmetic at Karazin Kharkov nationwide college in Kharkov, Ukraine.

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**Sample text**

15. 0/ D 0. In relation to this example, it is interesting to mention the following simple assertion concerning uniqueness for 1D equations. 16. Let f W R 7! R be a continuous function. x0 / ¤ 0. Proof. 2 in the Appendix). The same theorem yields the uniqueness in the Lipschitz case. x0 / ¤ 0 only. x/ is continuous and preserves the sign. x/ D dt Z x x0 d : f. t// D t for all t > 0 small enough. t/ is locally unique. 17. 15 is locally unique for every initial datum x0 ¤ 0. t/ outside a small neighborhood of this x0 is also unique.

6, v is a Lyapunov stable fixed point. v; w/ Ä ı. St w/ ! 0 as t ! C1. 3 yields that St w ! 0. Hence, v is asymptotically Lyapunov stable. 7 Complete theory of 1D continuous systems Now we describe all possible scenarios in one-dimensional systems with continuous time. We start with several simple observations by including them in the following exercises. 1. Describe the dynamics in the models generated by the following equations: (a) xP D x, (b) xP D x, (c) xP D jxj, (d) xP D 1. Show that the corresponding dynamical systems cannot be topologically equivalent.

X/ for all x; t 2 R: We also have the following assertion. 10. R; St / with continuous reversible time is conjugate with a system generated by some ordinary differential equation. Proof. We use the argument given in SIBIRSKY [212]. x/ D 0 for every x 2 N . a; b/: We perform this similarly in the case b D 1. x/ generates a dynamical system with continuous reversible time. St ; R/. t u We do not know whether a similar result holds in higher dimensions. 11. R; SQ t / with reversible time have finite numbers of equilibria N D fx1 < x2 < : : : < xN g and NQ D fQx1 < xQ 2 < : : : < xQ NQ g: D C1.