By Yuri A. Kuznetsov

This can be a publication on nonlinear dynamical platforms and their bifurcations lower than parameter edition. It presents a reader with a stable foundation in dynamical structures concept, in addition to particular approaches for software of common mathematical effects to specific difficulties. specified consciousness is given to effective numerical implementations of the built thoughts. numerous examples from fresh examine papers are used as illustrations. The booklet is designed for complex undergraduate or graduate scholars in utilized arithmetic, in addition to for Ph.D. scholars and researchers in physics, biology, engineering, and economics who use dynamical platforms as version instruments of their stories. A reasonable mathematical historical past is believed, and, every time attainable, basically user-friendly mathematical instruments are used. This new version preserves the constitution of the first version whereas updating the context to include fresh theoretical advancements, specifically new and more desirable numerical equipment for bifurcation research. assessment of 1st version: "I comprehend of no different ebook that so truly explains the elemental phenomena of bifurcation theory." Math studies "The ebook is an exceptional addition to the dynamical structures literature. it's strong to work out, in our glossy rush to quickly ebook, that we, as a mathematical group, nonetheless have time to collect, and in any such readable and regarded shape, the vital effects on our subject." Bulletin of the AMS

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Specify now an autonomous system of ODEs on this manifold, called the suspension, by the equations t˙ = 1, x˙ = 0. 13) (viewed as subsets of R1 × Rn ) are straight lines inside the layer interrupted by “jumps” from its “top” face to the “bottom” face. 13) ϕT0 coincides on its invariant hyperplane {t = 0} with the map f . Let k > 0 satisfy the equation ekT0 = 2. 14). In both systems, x = 0 corresponds to a stable limit cycle of period T0 with the multiplier µ = − 12 . 14. Stable limit cycle on the M¨ obius strip.

1) starting at some point x: . . , f −1 (x), x, f (x), f 2 (x), . . 2) starting at a point y: . . , g −1 (y), y, g(y), g 2 (y), . . 1 Equivalence of dynamical systems 41 Topological equivalence implies that if x and y are related by the homeomorphism h, y = h(x), then the ﬁrst orbit is mapped onto the second one by this map h. Symbolically, f x −→ f (x) h↓ h↓ g y −→ g(y). Therefore, g(y) = h(f (x)) or g(h(x)) = h(f (x)) for all x ∈ Rn , which can be written as f (x) = h−1 (g(h(x))) since h is invertible.

Let S0 be a closed invariant set. 8 An invariant set S0 is called stable if (i) for any suﬃciently small neighborhood U ⊃ S0 there exists a neighborhood V ⊃ S0 such that ϕt x ∈ U for all x ∈ V and all t > 0; (ii) there exists a neighborhood U0 ⊃ S0 such that ϕt x → S0 for all x ∈ U0 , as t → +∞. If S0 is an equilibrium or a cycle, this deﬁnition turns into the standard deﬁnition of stable equilibria or cycles. Property (i) of the deﬁnition is called Lyapunov stability. If a set S0 is Lyapunov stable, nearby orbits do not leave its neighborhood.