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.

Best differential equations books

Boundary Value Problems: And Partial Differential Equations

Boundary price difficulties is the best textual content on boundary price difficulties and Fourier sequence for execs and scholars in engineering, technology, and arithmetic who paintings with partial differential equations. during this up-to-date variation, writer David Powers presents a radical review of fixing boundary price difficulties regarding partial differential equations by means of the equipment of separation of variables.

Invertible Point Transformations and Nonlinear Differential Equations

The invertible element transformation is a strong device within the learn of nonlinear differential and distinction questions. This ebook offers a entire advent to this method. traditional and partial differential equations are studied with this process. The booklet additionally covers nonlinear distinction equations.

Dynamical systems and numerical analysis

This publication unites the examine of dynamical structures and numerical answer of differential equations. the 1st 3 chapters include the weather of the speculation of dynamical platforms and the numerical answer of initial-value difficulties. within the final chapters, numerical tools are formulted as dynamical structures and the convergence and balance houses of the tools are tested.

Extra resources for Bifurcation without Parameters

Example text

The result of  goes farther and holds for inviscid profiles and—by perturbation—for viscous profiles alike. 1 () 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 . 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.