By Yuri A. Kuznetsov

This can be a publication on nonlinear dynamical structures and their bifurcations lower than parameter version. It presents a reader with a stable foundation in dynamical structures conception, in addition to particular approaches for program of common mathematical effects to specific difficulties. designated cognizance is given to effective numerical implementations of the constructed ideas. a number of examples from contemporary study papers are used as illustrations. The ebook is designed for complicated 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 experiences. A reasonable mathematical history is believed, and, each time attainable, basically simple 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 better numerical tools for bifurcation research. assessment of 1st version: "I understand of no different ebook that so truly explains the elemental phenomena of bifurcation theory." Math reports "The publication is a superb addition to the dynamical platforms literature. it really is stable to work out, in our smooth rush to fast e-book, that we, as a mathematical group, nonetheless have time to compile, and in this type of readable and thought of shape, the very important effects on our subject." Bulletin of the AMS

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In the linear case the theorem is obvious from the Jordan normal form. 2, being applied to the N0 th iterate f N0 of the map f at any point of the periodic orbit, also gives a suﬃcient condition for the stability of an N0 -cycle. Another important case where we can establish the stability of a ﬁxed point of a discrete-time dynamical system is provided by the following theorem. 3 (Contraction Mapping Principle) Let X be a complete metric space with distance deﬁned by ρ. Assume that there is a map f : X → X that is continuous and that satisﬁes, for all x, y ∈ X, ρ(f (x), f (y)) ≤ λρ(x, y), with some 0 < λ < 1.

The matrix M (T0 ) is called a monodromy matrix of the cycle L0 . The following Liouville formula expresses the determinant of the monodromy matrix in terms of the matrix A(t): det M (T0 ) = exp T0 0 tr A(t) dt . 6 The monodromy matrix M (T0 ) has eigenvalues 1, µ1 , µ2 , . . , µn−1 , where µi are the multipliers of the Poincar´e map associated with the cycle L0 . 30 1. 14) near the cycle L0 . Consider the map ϕT0 : Rn → Rn . Clearly, ϕT0 x0 = x0 , where x0 is an initial point on the cycle, which we assume to be located at the origin, x0 = 0.

This angle equals zero at ρ0 = 1 and increases as ρ0 → 0. 9), preserving time direction. Thus, the two systems are topologically equivalent within U . However, the homeomorphism h is not diﬀerentiable in U . More precisely, it is smooth away from the origin but not diﬀerentiable at x = 0. To see dy this, one should evaluate the Jacobian matrix dx in (x1 , x2 )-coordinates.