Bifurcation formulae derived from center manifold theory

Hassard, Brian; Wan, Yieh-Hei

doi:10.1016/0022-247x(78)90120-8

Cited by 149 publications

(71 citation statements)

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“…It should be noticed that the above Υ is exactly the one derived by Hassard and Wan [10] by using center manifold reduction and normal form approach (for more details, see [7]). …”

Section: G(z α β) := P B(zq + Zq α β)supporting

confidence: 56%

“…In the standard Hopf bifurcation theory ( [9,10,13]), the central hypothesis is that at the critical value of the bifurcation parameter the infinitesimal generator has a complex conjugate pair of simple purely imaginary eigenvalues. The presence of symmetry may cause purely imaginary eigenvalues to arise with higher multiplicities, which causes the bifurcation problem to become more complicated; see for instance [6].…”

Section: ) Is Invariant Under the Transformation (X T) → ( (γ) T) mentioning

confidence: 99%

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Equivariant Hopf bifurcation for neutral functional differential equations

Guo

Lamb

2008

Proc. Amer. Math. Soc.

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Abstract. In this paper we employ an equivariant Lyapunov-Schmidt procedure to give a clearer understanding of the one-to-one correspondence of the periodic solutions of a system of neutral functional differential equations with the zeros of the reduced bifurcation map, and then set up equivariant Hopf bifurcation theory. In the process we derive criteria for the existence and direction of branches of bifurcating periodic solutions in terms of the original system, avoiding the process of center manifold reduction.

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“…It should be noticed that the above Υ is exactly the one derived by Hassard and Wan [10] by using center manifold reduction and normal form approach (for more details, see [7]). …”

Section: G(z α β) := P B(zq + Zq α β)supporting

confidence: 56%

Section: ) Is Invariant Under the Transformation (X T) → ( (γ) T) mentioning

confidence: 99%

Equivariant Hopf bifurcation for neutral functional differential equations

Guo

Lamb

2008

Proc. Amer. Math. Soc.

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“…Although theoretically the isochronous center problem can be solved by letting all period constants be zero, it is not the fact in practice which is due to the difficulty of computing the period constants. Questions relating to calculation of period constants have been studied by a number of authors (See [7,8,10,11]). However, only the first few ones can be given.…”

Section: Introductionmentioning

confidence: 99%

Calculation of generalized period constants via time-angle difference for complex analytic systems with resonant ratio

Wu¹

2016

J. Nonlinear Sci. Appl.

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In the case of a critical point being a center, the isochronicity problem (or linearizability problem) is far to be solved in general. A progressive way to find necessary conditions for isochronicity is to compute period constants. In this paper, we establish a new recursive algorithm of calculation of the so-called generalized period constants. Furthermore, we verify the new algorithm by the existing results for the Lotka-Volterra system with 3 : −2 resonance. Finally, the algorithm is applied to solve the linearizability problem for the Lotka-Volterra system in the ratio 4 : −5.

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“…where the /" are homogeneous polynomials of degree n, and then equating corresponding powers of ux we obtain a unique formal Taylor expansion of a; for a discussion of this procedure as well as the numerical implementation, see [15,16,23]. The number N can be taken as high as the differentiability of nt permits (i = 1,2).…”

mentioning

confidence: 99%

“…If we define the sequence of numbers km(m > 0) by (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) km = f~2\in^g2{r2(r, pj) dr, then (7.19) leads to…”

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confidence: 99%

Properties of center manifolds

Sijbrand¹

1985

Trans. Amer. Math. Soc.

134

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Abstract.The center manifold has a number of puzzling properties associated with the basic questions of existence, uniqueness, differentiability and analyticity which may cloud its profitable application in e.g. bifurcation theory. This paper aims to deal with some of these subtle properties.Regarding existence and uniqueness, it is shown that the cut-off function appearing in the usual existence proof is responsible for the selection of a single center manifold, thereby hiding the inherent nonuniqueness. Conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection. This intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium. In the case of nonuniqueness the difference between any two center manifolds will be less than c, exp(c2A_1) with c, and c2 constants, which explains why the formal Taylor expansions of different center manifolds are the same, while the expansions do not converge.The differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost.Center manifolds are usually not analytic; however, an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant. Using this manifold, a new and very simple proof of the Lyapunov subcenter theorem is given.

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Bifurcation formulae derived from center manifold theory

Cited by 149 publications

References 1 publication

Equivariant Hopf bifurcation for neutral functional differential equations

Equivariant Hopf bifurcation for neutral functional differential equations

Calculation of generalized period constants via time-angle difference for complex analytic systems with resonant ratio

Properties of center manifolds

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