By Stefan Liebscher
Targeted at mathematicians having not less than a uncomplicated familiarity with classical bifurcation concept, this monograph offers a scientific class and research of bifurcations with no parameters in dynamical platforms. even though the equipment and ideas are in brief brought, a previous wisdom of center-manifold discount rates 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 idea, illustrated through many purposes, goals at a geometrical realizing of the neighborhood dynamics close to the bifurcation points.
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Additional resources for Bifurcation without Parameters
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. 28). s; 0/ on the y-axis. e. the harmonic oscillator, for finite time, and thus hits the y-axis at least twice. Every orbit intersecting the fixed-point space of the reversibility R, on the other hand, is a reversible periodic orbit. In particular it is mapped by R onto itself.
Chapter 3 Cosymmetries Cosymmetries have been introduced by Yudovich and Kurakin to study limit cycles near manifolds of equilibria via Lyapunov-Schmidt reduction [49, 50]. They turn out to be equivalent to the existence of manifolds of equilibria, provided some nondegeneracy conditions are satisfied. Given a vector field F W Rn ! Rn , a cosymmetry is any other vector field orthogonal to F . L W Rn ! 1) A non-cosymmetric equilibrium is any zero of F where the cosymmetry does not vanish. x0 / has a nontrivial kernel, too.
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. In particular, the constructed cosymmetry has normalized projection onto the kernel of the adjoint of the linearization. t u Both theorems can be extended to sets of m simultaneous cosymmetries and m-dimensional manifolds of equilibria. x0 / D m. 4) 3 Cosymmetries 23 The condition on the kernel of the linearizations is in fact consistent with the non-degeneracy conditions of our bifurcations discussed in the following chapters, as long as no additional symmetries are considered.
Bifurcation without Parameters by Stefan Liebscher