Hopf Bifurcation in Impulsive Systems
This chapter is organized in the following manner. In the first section, we give the description of the systems under consideration and prove the theorem of existence of foci and centers of the nonperturbed system. The main subject of Sect. 2.1.2 is the f
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Hopf Bifurcation in Impulsive Systems
2.1 Hopf Bifurcation of a Discontinuous Limit Cycle This chapter is organized in the following manner. In the first section, we give the description of the systems under consideration and prove the theorem of existence of foci and centers of the nonperturbed system. The main subject of Sect. 2.1.2 is the foci of the perturbed equation. The noncritical case is considered. In Sect. 2.1.3, the problem of distinguishing between the center and the focus is solved. Bifurcation of a periodic solution is investigated in Sect. 2.1.4. The last section consists of examples illustrating the bifurcation theorem.
2.1.1 The Nonperturbed System 1
Denote by < x, y > the dot product of vectors x, y ∈ R2 , and ||x|| =< x, x > 2 , the norm of a vector x ∈ R2 . Moreover, let R be the set of all real-valued constant 2 × 2 matrices, and I ∈ R be the identity matrix. D0 -system. Consider the following differential equation with impulses dx = Ax, dt Δx|x∈ Γ0 = B0 x,
(2.1.1)
where Γ0 is a subset of R2 , and it will be described below, A, B0 ∈ R. The following assumptions will be needed throughout this chapter: p
(C1) Γ0 = ∪i=1 si , where p is a fixed natural number and half-lines si , i = 1, 2, . . . , p, are defined by equations < a i , x >= 0, where a i = (a1i , a2i ) are constant vectors. The origin does not belong to the lines (see Fig. 2.1). © Springer Nature Singapore Pte Ltd. and Higher Education Press 2017 M. Akhmet and A. Kashkynbayev, Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities, Nonlinear Physical Science, DOI 10.1007/978-981-10-3180-9_2
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2 Hopf Bifurcation in Impulsive Systems
Fig. 2.1 The domain of the nonperturbed system (2.1.1) with a vertex which unites the straight lines si , i = 1, 2, . . . , p
x2
l2 l1 x1
lp
(C2)
A=
α −β β α
,
where α, β ∈ R, β = 0; (C3) there exists a regular matrix Q ∈ R and nonnegative real numbers k and θ such that cos θ − sin θ 1 0 −1 Q − ; B0 = k Q sin θ cos θ 0 1 We consider every angle for a point with respect to the positive half-line of the first coordinate axis. Denote si = (I + B0 )si , i = 1, 2, . . . , p. Let γi and ζi be angles of si and si , i = 1, 2, . . . , p, respectively, B0 =
b11 b12 b21 b22
.
(C4) 0 < γ1 < ζ1 < γ2 < · · · < γ p < ζ p < 2π, (b11 + 1) cos γi + b12 sin γi = 0, i = 1, 2, . . . , p. If conditions (C1)–(C4) hold, then (2.1.1) is said to be a D0 -system. Exercise 2.1.1 Verify that the origin is a unique singular point of a D0 -system and (2.1.1) is not a linear system. Exercise 2.1.2 Using the results of the last chapter, prove that D0 -system (2.1.1) provides a B-smooth discontinuous flow. If we use transformation x1 = r cos(φ), x2 = r sin(φ) in (2.1.1) and exclude the time variable t, we can find that the solution r (φ, r0 ) which starts at the point (0, r0 ), satisfies the following system: dr = λr, dφ Δr |φ=γi (mod2π) = ki r,
(2.1.2)
2.1 Hopf Bifurcation of a Discontinuous Limit Cycle
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where λ = βα , the angle-variable φ is ranged over
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