By Marat Akhmet, Ardak Kashkynbayev
This publication makes a speciality of bifurcation thought for self sufficient and nonautonomous differential equations with discontinuities of other kinds – people with jumps current both within the right-hand aspect, or in trajectories or within the arguments of strategies of equations. the consequences acquired may be utilized to numerous fields, corresponding to neural networks, mind dynamics, mechanical platforms, climate phenomena and inhabitants dynamics. constructing bifurcation thought for numerous varieties of differential equations, the ebook is pioneering within the box. It provides the most recent effects and offers a pragmatic consultant to utilising the speculation to differential equations with quite a few varieties of discontinuity. additionally, it deals new how one can study nonautonomous bifurcation situations in those equations. As such, it exhibits undergraduate and graduate scholars how bifurcation conception may be constructed not just for discrete and non-stop platforms, but additionally for those who mix those platforms in very alternative ways. while, it bargains experts numerous strong tools constructed for the idea of discontinuous dynamical platforms with variable moments of effect, differential equations with piecewise consistent arguments of generalized sort and Filippov systems.
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Additional info for Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities
We will apply the results of current chapter to prove the existence of a stable periodic motion of the model, in the perturbed system corresponding to this model. 2, we will handle a more complicated case of two coupled oscillators where one of the oscillators is subdued to the impacts modeled by the Newton’s law of restitution. We strictly believe that results of the present section can be applied to other mechanical, electrical, as well as biological problems if one adopts the models by special transformations to the considered case.
Then, we have γ2 j = η2 j − θ (μ) + Θ(r (η2 j ), η2 j , μ), r (γ2 j ) = (1 + k(μ))r (η2 j ) + R(r (η2 j ), η2 j , μ). Throughout this section, [a, b] denotes the oriented interval for any a, b ∈ R. That is, it denotes [a, b] when a < b, and it denotes [b, a] otherwise. Denote by r1 (φ) the solution of dr = λ(μ)r + R(r, φ, μ) dφ with the initial condition r1 (γ2 j ) = r (γ2 j ). 3 Periodic Solutions of the Van der Pol Equation 49 and on the interval [γ2 j , φ0 (μ) − θ (μ)], we have φ r1 (φ) = r (γ2 j ) + γ2 j [λ(μ)r1 (s) + R(r1 (s), s, μ)]ds.
It is rather reasoned by specifically arranged interaction of continuous and discontinuous stages of the process. To be precise, we use a generalized eigenvalue to evaluate which we apply a characteristic of the impact as well as of the continuous process between moments of discontinuity. This approach when continuous and discontinuous stages are equally participated in creating a certain phenomena is common for the theory of differential equations with impulses [1, 216]. Our results are, rather, close to those, which obtained for systems where continuous flows and surfaces of discontinuity are transversal [1, 6, 39, 109, 150].