A survey of boundedness, stability, asymptotic behaviour of by Andrea Bacciotti, Lionel Rosier PDF

By Andrea Bacciotti, Lionel Rosier

ISBN-10: 3540213325

ISBN-13: 9783540213321

This e-book provides a latest and self-contained therapy of the Liapunov approach for balance research, within the framework of mathematical nonlinear keep watch over thought. a selected concentration is at the challenge of the life of Liapunov services (converse Liapunov theorems) and their regularity, whose curiosity is principally prompted by way of purposes to automated keep an eye on. Many contemporary ends up in this region were amassed and offered in a scientific means. a few of them are given in prolonged, unified types and with new, easier proofs. within the second version of this profitable ebook a number of new sections have been extra and outdated sections were more advantageous, e.g concerning the Zubovs approach, Liapunov features for discontinuous structures and cascaded platforms. Many new examples, factors and figures have been additional making this profitable publication available and good readable for engineers in addition to mathematicians.

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Extra resources for A survey of boundedness, stability, asymptotic behaviour of differential and difference equs

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By regarding an autonomous system which admits a bifurcation of this type as an asymptotically autonomous system, one sees that the autonomous situation fits well into our context. Interesting books on the topic of autonomous bifurcation theory are Chow & Hale [46], Guckenheimer & Holmes [72], Hale & Koc ¸ ak [78], Kuznetsow [101] and Luo & Wang & Zhu & Han [109]. For a brief introduction, see also Crawford [60]. 2 Topological Skew Product Flows In the bifurcation theory of nonautonomous dynamical systems where the base set is supposed to have a certain topological structure, one distinguishes between attractor-repeller bifurcations and bifurcations of solutions.

A solution µ : (τ, ∞) → RN is called Lyapunov-stable if for all t0 > τ and ε > 0, there exists a δ = δ(t0 , ε) > 0 with λ t, t0 , Uδ (µ(t0 )) ⊂ Uε (µ(t)) for all t ≥ t0 . Furthermore, a solution µ : (τ, ∞) → RN is called Lyapunov-attractive if for all t0 > τ , there exists an η = η(t0 ) > 0 such that 40 Chapter 2: Notions of Attractivity and Bifurcation lim t→∞ λ(t, t0 , x) − µ(t) = 0 for all x ∈ Uη (µ(t0 )) . , ¨ [30, p. 59]). 16, p. 325] and Bhatia & Szego However, if a solution is both Lyapunov-stable and Lyapunov-attractive, we call this solution Lyapunov-asymptotically stable.

Moreover, it is shown in Chapter 7 that the classical bifurcation scenarios of saddle node, pitchfork, transcritical or Hopf type can be transferred to asymptotically autonomous equations. By regarding an autonomous system which admits a bifurcation of this type as an asymptotically autonomous system, one sees that the autonomous situation fits well into our context. Interesting books on the topic of autonomous bifurcation theory are Chow & Hale [46], Guckenheimer & Holmes [72], Hale & Koc ¸ ak [78], Kuznetsow [101] and Luo & Wang & Zhu & Han [109].

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A survey of boundedness, stability, asymptotic behaviour of differential and difference equs by Andrea Bacciotti, Lionel Rosier


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