Bifurcation without Parameters by Stefan Liebscher

Bifurcation without Parameters by Stefan Liebscher

By Stefan Liebscher

Targeted at mathematicians having not less than a easy familiarity with classical bifurcation thought, this monograph offers a scientific category and research of bifurcations with no parameters in dynamical structures. even if the tools and ideas are in brief brought, a previous wisdom of center-manifold savings and normal-form calculations may also help the reader to understand the presentation. Bifurcations with no parameters happen alongside manifolds of equilibria, at issues the place general hyperbolicity of the manifold is violated. the final concept, illustrated by means of many functions, goals at a geometrical realizing of the neighborhood dynamics close to the bifurcation points.

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The result of [39] goes farther and holds for inviscid profiles and—by perturbation—for viscous profiles alike. 1 ([39]) Let f W R3 ! e. u/x D 0 is strictly hyperbolic. 0/g there exists a C 5 vector field g W R3 ! e. 8) admits a Poincaré-Andronov-Hopf point in the sense of Chap. 5. 0/ to be nonzero and s D 0. We shall construct a particular source g with a straight line of equilibria. 0/. 7) holds. 8) is translated into a non-degeneracy condition on f . 7) imposed by the structure of one conservation law and two balance laws.

2 Consider a C k vector field F W Rn ! g. x1 ; 0; : : : ; 0/ Á 0. Assume that the kernel of the linearization at the origin is one-dimensional. Then, locally near the origin, there exists a cosymmetry L W Rn ! Rn of F , such that the origin is a non-cosymmetric equilibrium. 0/ ? x/ has a first column of zeros. x/i with adjoint K . x/ D ? Š Rn 1 . x/. e. we look for zeros of the map T W Rn ? Rn 1 ; Q ! x/. x; L/ Q C L/: Note that ? Š Rn 1 . 0/ of full rank. Again, the implicitfunction theorem yields the claim.

28) except for the line of equilibria at x D 0 and for the reversed flow direction for x < 0. 23) yields a time reversibility fQ. x; y/ g. x; y/ D . x; y/. 28) onto solutions with reversed time. 30). 0; Q 0/ D 0. 28). 33) that linearizes the vector field, due to Belitskii’s theorem [12]. x; Q y/ x. x; Q y/ C y. 34) 34 4 Transcritical Bifurcation commutes with the reversibility R and still linearizes the vector field, due to the reversibility of the vector field. e. 0; O y/ D 0. 26), in the hyperbolic case.

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