Analysis and Control of Complex Dynamical Systems: Robust by Kazuyuki Aihara, Jun-ichi Imura, Tetsushi Ueta PDF

By Kazuyuki Aihara, Jun-ichi Imura, Tetsushi Ueta

ISBN-10: 4431550127

ISBN-13: 9784431550129

ISBN-10: 4431550135

ISBN-13: 9784431550136

This booklet is the 1st to record on theoretical breakthroughs on regulate of advanced dynamical platforms built via collaborative researchers within the fields of dynamical platforms idea and regulate thought. besides, its simple standpoint is of 3 sorts of complexity: bifurcation phenomena topic to version uncertainty, complicated habit together with periodic/quasi-periodic orbits in addition to chaotic orbits, and community complexity rising from dynamical interactions among subsystems. Analysis and keep an eye on of advanced Dynamical Systems deals a helpful source for mathematicians, physicists, and biophysicists, in addition to for researchers in nonlinear technological know-how and regulate engineering, letting them advance a greater basic knowing of the research and regulate synthesis of such advanced systems.

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Additional info for Analysis and Control of Complex Dynamical Systems: Robust Bifurcation, Dynamic Attractors, and Network Complexity

Example text

X j ∂ xn ∂λm ∂ x j ∂λm Here, ∂ xn∗ /∂λm , n = 1, 2, . . 6) with respect to λm . Then, we have ∂x∗ = ∂λm I− ∂g ∂x −1 x=x ∗ ∂g ∂λm x=x ∗ . A similar argument can be applied to the differential dynamics f . 11). 11). Therefore, the trajectories cannot pass through the set, according to the uniqueness of solutions for the initial value problem. This leads to the fact that, if we choose initial values (λ(0), ν(0)) with ν(0) > ρ(λ(0)), any solution (λ(τ ), ν(τ )) stays in the subspace ν > ρ(λ) for all τ > 0.

We take a local section Γ , that the solution crosses transversely, as follows: Γ = {ξ ∈ R N +1 | z(ξ ) = 0}, where z(ξ ) is a scalar valued function of ξ in R N +1 . Let us define h as a local coordinate of Γ h : Γ → Π ⊂ R N ; ξ → x = h(ξ ) and its inverse h −1 as an embedding map h −1 : Π → Γ ; x → ξ = h −1 (x), where ξ satisfies z(ξ ) = 0. Pick a point ξ ∈ Γ and let Π ⊂ Γ be some neighborhood of x = h(ξ ). Then the Poincaré map T is defined by the following composite map for a point x ∈ Π : T : Π → Π ; x → h(ψ(τ (h −1 (x)), h −1 (x), λ)), where τ denotes the time during which the trajectory emanating from a point ξ ∈ Γ hits the local cross-section Γ again.

Generic bifurcations are the tangent, perioddoubling, and Neimark-Sacker bifurcations, which correspond to the critical distriiθ bution √ of characteristic multiplier μ such that μ = +1, μ = −1, and μ = e with i = −1, respectively. Further, a pitchfork bifurcation can appear in a symmetric system as a degenerate case of the tangent bifurcation. 8) with a fixed μ depending on the bifurcation conditions to obtain unknown bifurcation sets x ∗ and λm . 7)). We denote this as a function of the parameters as follows ρ(λ) := μmax (D(x ∗ (λ), λ)).

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Analysis and Control of Complex Dynamical Systems: Robust Bifurcation, Dynamic Attractors, and Network Complexity by Kazuyuki Aihara, Jun-ichi Imura, Tetsushi Ueta

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