By Marat Akhmet, Ardak Kashkynbayev
This publication specializes in bifurcation thought for independent and nonautonomous differential equations with discontinuities of other kinds – people with jumps current both within the right-hand part, or in trajectories or within the arguments of recommendations of equations. the consequences bought should be utilized to varied fields, akin to neural networks, mind dynamics, mechanical platforms, climate phenomena and inhabitants dynamics. constructing bifurcation thought for numerous different types of differential equations, the e-book is pioneering within the box. It offers the most recent effects and offers a pragmatic consultant to utilising the speculation to differential equations with numerous forms of discontinuity. in addition, it bargains new how one can examine nonautonomous bifurcation situations in those equations. As such, it indicates undergraduate and graduate scholars how bifurcation idea will be constructed not just for discrete and non-stop structures, but additionally for those who mix those structures in very other ways. while, it bargains experts numerous strong tools built for the idea of discontinuous dynamical structures with variable moments of influence, differential equations with piecewise consistent arguments of generalized kind and Filippov systems.
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Extra info for Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities
67) systems dx = Ax, dt dz ˆ = bz, (x, z) ∈ / Γ0 , dt Δx |(x,z)∈Γ0 = B0 x, Δz |(x,z)∈Γ0 = c0 z. 68) dx = Ax + f (x, z), dt dz ˆ + g(x, z), (x, z) ∈ = bz / Γ (0), dt Δx |(x,z)∈Γ (0) = B(x, 0)x, Δz |(x,z)∈Γ (0) = c(z, 0)z. 69) and are D0 -system and D-system, respectively; (A4) Functions f˜, ν : G × (−μ0 , μ0 ) → R2 are analytic in x, z, and μ; (A5) f˜(0, 0, μ) = 0, ν(0, μ) = 0, uniformly for μ ∈ (−μ0 , μ0 ). 67) in the following form: dr = λ(μ)r + P(r, φ, z, μ), dφ dz = b(μ)z + Q(r, φ, z, μ), (r, φ, z) ∈ / Γ (μ), dφ Δr |(r,φ)∈ i (μ) = ki (μ)r + ω(r, φ, μ), Δφ |(r,φ)∈ i (μ) = θi (μ) + γ (r, φ, μ), ˜ μ).
1. 94) with the initial condition on Γ spirals in toward the origin, and if q > 1, then the solutions move away from the origin. On the other hand, when q = 1 we have the critical case. 94) with the initial condition on Γ may be periodic, and it may spiral in toward the origin or it may move away from the origin. In this study, as mentioned before, the case q = 1 is not of our interest and the critical case is investigated below. 93) again: ˜ = λ(μ)ρ + R(ρ, ϕ, μ), ϕ = φ0 (μ), Δρ|ϕ=φ0 (μ) = k(μ)ρ + W˜ (ρ, μ).
From now on, G denotes a neighborhood of the origin. The following assumptions will be needed: p (C1) Γ0 = i=1 Pi , p ∈ N, where Pi = i × R, i are half-lines starting at the origin defined by a i , x = 0 for i = 1, . . , p and a i = (a1i , a2i ) ∈ R2 are constant vectors; α −β (C2) A = , where β = 0; β α (C3) There exists a regular matrix Q ∈ R2×2 and nonnegative real numbers k and θ such that cos θ − sin θ 10 Q −1 − . B0 = k Q sin θ cos θ 01 For the sake of brevity, in what follows, every angle for a point or a line is considered with respect to the half-line of the first coordinate axis in x-plane.