Bifurcation and Chaos in Discontinuous and Continuous by Michal Fečkan

By Michal Fečkan

"Bifurcation and Chaos in Discontinuous and non-stop platforms" offers rigorous mathematical functional-analytical instruments for dealing with chaotic bifurcations in addition to distinctive and whole proofs including concrete functions awarded by way of many stimulating and illustrating examples. A huge number of nonlinear difficulties are studied regarding distinction equations, traditional and partial differential equations, differential equations with impulses, piecewise gentle differential equations, differential and distinction inclusions, and differential equations on endless lattices as well.

This publication is meant for mathematicians, physicists, theoretically prone engineers and postgraduate scholars both learning oscillations of nonlinear mechanical structures or investigating vibrations of strings and beams, and electric circuits via utilizing the fashionable thought of bifurcation equipment in dynamical systems.

Dr. Michal Fe?kan is a Professor on the division of Mathematical research and Numerical arithmetic at the college of arithmetic, Physics and Informatics on the Comenius college in Bratislava, Slovakia. he's engaged on nonlinear sensible research, bifurcation thought and dynamical platforms with purposes to mechanics and vibrations.

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Extra info for Bifurcation and Chaos in Discontinuous and Continuous Systems

Example text

The Poisson summation formula like in [4] could help to overcome this difficulty. 15. 20) x(i + 0) = x(i − 0) + ε a x(i − 0), ε , i ∈ Z , where f , a ∈ C2 (Rn+1 , Rn ), f (·, 0) has a hyperbolic fixed point x0 with a homoclinic orbit γ (·). Furthermore, assume that the adjoint variational equation 44 3 Chaos in Discrete Dynamical Systems ∗ v˙ = − Dx f γ (t), 0 v has only a unique (up to constant multiples) bounded nonzero solution u. 20) has the form M (t) = ∞ ∑ a γ (t + i), 0 , u(t + i) + i=−∞ ∞ −∞ Dε f γ (s), 0 , u(s) ds .

P. PALIS & W. DE MELO: Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York, 1982. M. GOLUBITSKY & V. GUILLEMIN: Stable Mappings and their Singularities, Springer-Verlag, New York, 1973. J. PALMER: Shadowing in Dynamical Systems, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 2000. C. IRWIN: Smooth Dynamical Systems, Academic Press, London, 1980. J. PALMER: Exponential dichotomies, the shadowing lemma and transversal homoclinic points, Dynamics Reported 1 (1988), 265–306.

13) where g1,2 ∈ C3 (R2 , R2 ), h1,2 ∈ C3 (R2 × R2 × R, R2 ), μ ∈R and z˙ = g1 (z), y˙ = g2 (y) are Hamiltonian systems. Let Ψ1 , Ψ2 be the 1-time Poincar`e mappings of z˙ = g1 (z), y˙ = g2 (y), respectively. Here z(i±) = lim z(s). We consider s→i± the mapping F(z, y, μ ) = Ψ1 (z) + μ h1 Ψ1 (z), Ψ2 (y), μ , Ψ2 (y) + μ h2 Ψ1 (z), Ψ2 (y), μ . 13). 1), we have x = (z, y) ∈ R2 × R2 , f (x) = Ψ1 (z), Ψ2 (y) h(x, μ , k) = μ h1 Ψ1 (z), Ψ2 (y), μ , μ h2 Ψ1 (z), Ψ2 (y), μ . 15) We suppose (a) g1,2 (0) = 0 and the eigenvalues of Dg1,2 (0) lie off the imaginary axis.

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