The Hopf Bifurcation and Its Applications

“…We state the results for ordinary differential equations although they apply, via standard reduction techniques (see Marsden and McCracken [1976]), to certain partial differential equations as well. Consider the ordinary differential equation…”

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confidence: 99%

Hopf bifurcation in the presence of symmetry

Golubitsky¹,

Stewart²

1984

Bull. Amer. Math. Soc.

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In this note we state a generalization of the Hopf bifurcation theorem to differential equations with symmetry. We state the results for ordinary differential equations although they apply, via standard reduction techniques (see Marsden and McCracken [1976]), to certain partial differential equations as well. Consider the ordinary differential equationwhere F(0, A) = 0 and F commutes with an (orthogonal) action of a compact Lie group T on V = R n ; that is, F(^x, A) = *)F(x, A) for 7 G I\ Assume dF| 0 ,o has pure imaginary eigenvalues. The symmetry can force these eigenvalues to have high multiplicity and the standard Hopf theorem does not apply. Despite this degeneracy, the symmetry can also force the occurrence of a branch of periodic solutions to (1 all consider the example T = O(2). Our approach differs from these by emphasizing the general role of isotropy subgroups in determining the occurrence of branches.These ideas prove useful in studying Taylor For x G V define the isotropy group E x = {a G T\ax = x}. Let SÇT and define the fixed-point subspace V^ = {y G V\ay = y for all a G £}. Notice that F maps V^ to itself.In order for dF|o,o to have pure imaginary eigenvalues, the representation of r on V must satisfy certain conditions. There are two 'simplest' cases:(a) The action of T on V is irreducible but not absolutely irreducible.(b) V = W 0 W, where T acts absolutely irreducibly on W and by the diagonal action on W © W.Henceforth we assume (b) holds. If c/F|o,o has pure imaginary eigenvalues then we may assume, without loss of generality, that dF|o,o = (? V)-The eigenvalues of <£F|O,A are a(X) ± i(A), each of multiplicity dimVT = n/2.

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“…There are many results on the Hopf bifurcation, probably the more classical one can be found in the book of Marsden and McCracken [7], see also [1]. But for studying the Hopf bifurcation here we shall use the averaging theory of fourth order which is rarely used in this context.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%