Structures in Dynamics: Finite Dimensional Deterministic StudiesThe study of non-linear dynamical systems nowadays is an intricate mixture of analysis, geometry, algebra and measure theory and this book takes all aspects into account. Presenting the contents of its authors' graduate courses in non-linear dynamical systems, this volume aims at researchers who wish to be acquainted with the more theoretical and fundamental subjects in non-linear dynamics and is designed to link the popular literature with research papers and monographs. All of the subjects covered in this book are extensively dealt with and presented in a pedagogic form. These include the presentation of an environment for the route to chaos by quasi-periodicity (which is related to the Landau-Lifschitz and Ruelle-Takens scenario's concerning the onset of turbulence); the theories of 1-dimensional dynamics, singularities in planar vector fields, and quasi-periodicity in dissipative systems. |
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1-parameter family ax² basin behaviour blowing-up Cantor set centre manifold closed orbit codimension compact compare Chapter conjugacy consider contains coordinates defined definition differential equations dynamical systems eigenvalues entropy equivalent ergodic ergodic measures evolution example exists Figure fixed point fn(k follows function ƒ and g Hénon map homeomorphism homterval Hopf bifurcation hyperbolic I₁ Ij,n implies integrable invariant measures iterates Kupka Smale Lebesgue measure lemma linear map ƒ Math Morse Smale negative Schwarzian derivative neighbourhood Normal Form parameter period doubling bifurcation periodic attractor periodic orbit periodic point perturbation phase portrait phase space Poincaré mapping point attractor point of f probability measure proof quasi-periodic resp saddle node bifurcation Schwarzian derivative sequence singularities Springer-Verlag stationary structurally stable subsection Theorem topological torus turning point unstable manifolds vector fields wandering interval zero მე მყ