Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems
HanĪ²mann, Heinz.
Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems [electronic resource] / by Heinz HanĪ²mann. - XVI, 242 p. 22 illus. online resource. - Lecture Notes in Mathematics, 1893 0075-8434 ; . - Lecture Notes in Mathematics, 1893 .
Bifurcations of Equilibria -- Bifurcations of Periodic Orbits -- Bifurcations of Invariant Tori -- Perturbations of Ramified Torus Bundles -- Planar Singularities -- Stratifications -- Normal Form Theory -- Proof of the Main KAM Theorem -- Proofs of the Necessary Lemmata.
Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems. While elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Hence, without the need for untypical conditions or external parameters, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system. The text moves gradually from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations have to be replaced by Cantor sets. Planar singularities and their versal unfoldings are an important ingredient that helps to explain the underlying dynamics in a transparent way.
9783540388968
10.1007/3-540-38894-X doi
Differentiable dynamical systems.
Differential Equations.
Global analysis.
Dynamical Systems and Ergodic Theory.
Ordinary Differential Equations.
Global Analysis and Analysis on Manifolds.
Theoretical, Mathematical and Computational Physics.
QA313
515.39 515.48
Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems [electronic resource] / by Heinz HanĪ²mann. - XVI, 242 p. 22 illus. online resource. - Lecture Notes in Mathematics, 1893 0075-8434 ; . - Lecture Notes in Mathematics, 1893 .
Bifurcations of Equilibria -- Bifurcations of Periodic Orbits -- Bifurcations of Invariant Tori -- Perturbations of Ramified Torus Bundles -- Planar Singularities -- Stratifications -- Normal Form Theory -- Proof of the Main KAM Theorem -- Proofs of the Necessary Lemmata.
Once again KAM theory is committed in the context of nearly integrable Hamiltonian systems. While elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Hence, without the need for untypical conditions or external parameters, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system. The text moves gradually from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations have to be replaced by Cantor sets. Planar singularities and their versal unfoldings are an important ingredient that helps to explain the underlying dynamics in a transparent way.
9783540388968
10.1007/3-540-38894-X doi
Differentiable dynamical systems.
Differential Equations.
Global analysis.
Dynamical Systems and Ergodic Theory.
Ordinary Differential Equations.
Global Analysis and Analysis on Manifolds.
Theoretical, Mathematical and Computational Physics.
QA313
515.39 515.48