By Stefan Liebscher
Targeted at mathematicians having at the very least a uncomplicated familiarity with classical bifurcation idea, this monograph presents a scientific category and research of bifurcations with out parameters in dynamical platforms. even supposing the tools and ideas are in brief brought, a previous wisdom of center-manifold mark downs and normal-form calculations may help the reader to understand the presentation. Bifurcations with out parameters happen alongside manifolds of equilibria, at issues the place common hyperbolicity of the manifold is violated. the final concept, illustrated via many purposes, goals at a geometrical figuring out of the neighborhood dynamics close to the bifurcation points.
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Extra info for Bifurcation without Parameters
Chapter 6 Application: Decoupling in Networks Networks are an important structure in many applications ranging from chemistry and biology to engineering. Pattern formation in networks has caught an ever growing interest in recent years . The main focus is usually the synchronization of the cells of the network. Here, we study the converse phenomenon: under suitable symmetry assumptions, networks can decouple and continua of states emerge where all couplings cancel out each other and several pairs of cells can have arbitrary phase differences.
See Fig. 1b. The formulation of the assumptions of the above theorem can be simplified: we first restrict to the three dimensional center manifold and assume that this manifold is flat. Then we take coordinates in direction of the real, generalized eigenvectors of the linearization at the Hopf point. 0; y/ Á 0. 0; 0/ > 0. 0; 0/ ¤ 0; D @2x1 C @2x2 . The first condition is our structural assumption, (ii) describes our bifurcation point, and (iii,iv) are non-degeneracy assumptions fulfilled generically.
0; y. /; / Á 0 along a curve. Without loss of generality, we took this curve to be the -axis. e. under small perturbations of F respecting (i) there is a point near the origin satisfying (ii–v) for the perturbed system. From the point of view of singularity theory, (ii,iv) define a singularity of codimension two, which is unfolded versally by the coordinate y along the line of trivial equilibria and the parameter . 2) with smooth FQ . 3) > 0. e. has no purely imaginary eigenvalues. 4) for determinant and trace, we therefore have ı ¤ 0, and ¤ 0 if ı > 0.