By Shangjiang Guo

ISBN-10: 1461469910

ISBN-13: 9781461469919

ISBN-10: 1461469929

ISBN-13: 9781461469926

This publication presents a crash path on numerous tools from the bifurcation conception of practical Differential Equations (FDEs). FDEs come up very certainly in economics, existence sciences and engineering and the learn of FDEs has been a massive resource of thought for development in nonlinear research and limitless dimensional dynamical structures. The ebook summarizes a few functional and normal methods and frameworks for the research of bifurcation phenomena of FDEs looking on parameters with chap. This good illustrated booklet goals to be self contained so the readers will locate during this ebook all appropriate fabrics in bifurcation, dynamical platforms with symmetry, useful differential equations, common kinds and heart manifold aid. This fabric used to be utilized in graduate classes on useful differential equations at Hunan collage (China) and York college (Canada).

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10) for all t ∈ [t0 ,t0 + A). 10) for t ∈ (t0 , β ), the maximal interval of existence of the solution xϕ . Furthermore, if β < ∞, then there exists a sequence tk → β − such that |xϕ (tk )| → ∞ as k → ∞. For further results on existence, uniqueness, continuation, and continuous dependence of solutions for DDEs, see, for example, [18, 30, 51, 70, 120, 144–147, 154, 206, 208, 300, 302]. 11) and the following DDE with discrete delay x(t) ˙ = h(x(t), x(t − τ1 ), . . 12) where τ = max{τ1 , . . , τk }, g: [−τ , 0] × Rn → Rn , and h: Rn × · · · × Rn (= Rn(k+1) ) → Rn are continuous.

Suppose a > 0. 44): (i) There are two equilibria E1 and E2 for μ ∈ J11 , where E1 is a saddle and E2 is a source. (ii) There are three equilibria E1 , E2 , and E3 for μ ∈ J12 , where E1 is a sink, E2 is a source, E3 = (y2 , ρ2 ) satisfying y2 > 0 and ρ2 > 0 is a saddle. (iii) There are two equilibria E1 and E2 for μ ∈ J13 , where E1 is a sink and E2 is a saddle. (iv) There are two equilibria E1 and E2 for μ ∈ J14 , where E1 is a source and E2 is a saddle. (v) There are three equilibria E1 , E2 , and E3 for μ ∈ J15 , where E1 is a source, E2 is a sink, E3 = (y2 , ρ2 ) satisfying y2 < 0 and ρ2 > 0 is a saddle.

1. Suppose a > 0. 45) has no solution (respectively, one solution (y2 , ρ2 ) with y2 > 0, one solution (y2 , ρ2 ) with y2 < 0) for parameters μ in J \ (J12 ∪ J15 ) (respectively, J12 , J15 ). Proof. We distinguish two cases: Case 1: μ1 < 0. Then y2 + μ1 y is negative if 0 < y < − μ1 and positive otherwise. If μ ∈ J11 , then y1 < y2 < 0, and hence ρ j = −y2j − μ1 y1 < 0, j = 1, 2. If μ ∈ J12 , then y1 < 0 < y2 < − μ1 , and hence ρ1 < 0 and ρ2 > 0. If μ ∈ J13 , then y1 < 0 < −μ1 < y2 , and hence ρ j = −y2j − μ1 y1 < 0, j = 1, 2.

### Bifurcation Theory of Functional Differential Equations by Shangjiang Guo

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