Download Bifurcation and Chaos in Discontinuous and Continuous by Michal Fečkan PDF

By Michal Fečkan

"Bifurcation and Chaos in Discontinuous and non-stop structures" offers rigorous mathematical functional-analytical instruments for dealing with chaotic bifurcations in addition to detailed and entire proofs including concrete functions offered via many stimulating and illustrating examples. A vast number of nonlinear difficulties are studied concerning 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 e-book is meant for mathematicians, physicists, theoretically vulnerable engineers and postgraduate scholars both learning oscillations of nonlinear mechanical platforms or investigating vibrations of strings and beams, and electric circuits by way of using the trendy concept of bifurcation tools 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 practical research, bifurcation concept and dynamical platforms with purposes to mechanics and vibrations.

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2) has a hyperbolic fixed poitnt 0, that is, the eigenvalues of gx (0) are non-zero and they lie off the unit circle. 2) has a one-parameter family of ho∞ moclinic solutions γ (α ) = {γn (α )}∞ −∞ , α ∈ R connecting 0. That is, {γn (α )}−∞ is a non-zero sequence of C3 -smooth vector functions satisfying γn+1 (α ) = g(γn (α )) and lim γn (α ) = 0 uniformly with respect to bounded α . The set n→±∞ ∪n∈Z ∪α ∈R {γn (α )} is bounded. (H4) gx (γn (α )) is invertible, and g−1 x (γn (α )) is uniformly bounded on Z.

5) is embedded into a Cr -smooth family. So g j are Cr -smooth satisfying g j (0, 0, 0, 0) = 0 and Dx j gk (0, 0, 0, 0) = 0 c for j, k = s, u, c. 6) is a graph c Wloc, ε = {(hs (xc , ε ), hu (xc , ε ), xc ) | xc ∈ Uc } for hs;u ∈ Cr (Uc ×V, E s;u ) and V is an open neighborhood of ε = 0. Moreover, it holds hs;u (0, 0) = 0 and Dxc hs;u (0, 0) = 0. e. 6) staying in Us ×Uu ×Uc for all t ∈ R. 7). 5) has a first integral H : Rn → R if H ◦ φt = H for any t ∈ R. 5 Global Center Manifolds Let Cbk (Rm , Rn ) be the Banach space of Ck functions from Rm to Rn which are bounded together with their derivatives, endowed with the usual sup-norm.

Consequently, the Melnikov vector mapping has the form M(α , β ) = (M1 (α , β ), M2 (α , β )) where M1 (α , β ) = 2 M2 (α , β ) = 2 ∞ ∑ rn (α )rn+1 (α ) − 2β 2 ∑ rn (α )rn (α ) − 2β 2 n=−∞ ∞ n=−∞ ∞ ∑ rn+1 (α )2 rn (α ) , n=−∞ ∞ ∑ rn (α )3 . 2 Transversal Homoclinic Orbits 53 ∞ ∞ ∑ rn (0)2 rn (0) = sinh3 w ∑ sech7 nw sinh2 nw(cosh2 nw − 2), n=1 n=1 ∞ ∑ ∞ ∑ rn (0)rn (0) = sinh2 w n=−∞ ∞ ∑ rn (0)3 = sinh3 w n=−∞ ∞ sech3 nw sinh nw = 0, n=−∞ ∞ ∑ sech6 nw sinh3 nw = 0, n=−∞ ∞ ∂ M2 (0, β ) = 2 ∑ (rn (0)rn (0) + rn (0)2 ) − 12β 2 ∑ rn (0)2 rn (0) = A3 (w, β ) .

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