By Stefan Liebscher
Targeted at mathematicians having no less than a easy familiarity with classical bifurcation idea, this monograph offers a scientific class and research of bifurcations with out parameters in dynamical platforms. even supposing the equipment and ideas are in short brought, a previous wisdom of center-manifold rate reductions and normal-form calculations can assist the reader to understand the presentation. Bifurcations with no parameters take place alongside manifolds of equilibria, at issues the place general hyperbolicity of the manifold is violated. the overall idea, illustrated by way of many functions, goals at a geometrical realizing of the neighborhood dynamics close to the bifurcation points.
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Additional info for Bifurcation without Parameters
3) A truncated normal form is then given by zP D . 4) in complex notation, z 2 C with complex coefficient c 2 C. This normal form is also called Stuart-Landau oscillator. Assuming non-degenerate real part of c, c
6 Rescaling and Blow Up 19 Good choices for the scaling ˛ are given by the Newton polyhedron. 13) ˇ2Nn be the Taylor expansion of the vector field. 0; : : : ; 0; 1; 0; : : : ; 0/ is the k-th unit vector. 11). Every outer facet F of N , facing the origin, yields a viable scaling ˛ ? 15) The time rescaling ˛ is then given by the distance of F from the origin. 12, D 0) contains exactly the monomials to points of N in F . 12) for 0 < 1 and kxk Q D1 as spherical coordinates near the origin. Then the boundary D 0 is the blow up of the singularity x D 0 to a sphere kxk Q D 1.
Then, the rescaling of time t D ˛ tQ yields the system xQ k0 D ˛k ˛ fk . 11) with respect to starts with terms of order 0 in . Summarized: the coordinates are rescaled by (positive) powers of the rescaling parameter . Then the resulting vector field is divided by to the largest possible power, without introducing a singularity in . 12, D 0) corresponds to a desingularized vector field of the blown-up singularity x D 0. Regular perturbation theory can be applied to obtain results for ' 0, describing the dynamics in a neighborhood of x D 0.
Bifurcation without Parameters by Stefan Liebscher