Mathematical Problems of Nonlinear Dynamics: A Tutorial

DOI10.1142/S0218127497001527zbMath0909.58008MaRDI QIDQ4214822

L. P. Shil'nikov

Publication date: 7 April 1999

Published in: International Journal of Bifurcation and Chaos (Search for Journal in Brave)

zbMATH Keywords

strange attractorsnonlinear systemscomplex dynamicsMorse-Smale systemshyperbolic attractorshomoclinic structuresNewhouse regions

Mathematics Subject Classification ID

Strange attractors, chaotic dynamics of systems with hyperbolic behavior (37D45) Dynamical systems with hyperbolic behavior (37D99) Morse-Smale systems (37D15) Smooth dynamical systems: general theory (37Cxx) Qualitative theory for ordinary differential equations (34Cxx)

Related Items (38)

BIFURCATION STRUCTURES GENERATED BY THE NONAUTONOMOUS DUFFING EQUATION ⋮ Hyperbolic chaos in self-oscillating systems based on mechanical triple linkage: testing absence of tangencies of stable and unstable manifolds for phase trajectories ⋮ Autonomous coupled oscillators with hyperbolic strange attractors ⋮ Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors ⋮ NOVEL NUMERICAL APPROACH TO SOLITARY–WAVE SOLUTIONS IDENTIFICATION OF BOUSSINESQ AND KORTEWEG–DE VRIES EQUATIONS ⋮ A “saddle-node” bifurcation scenario for birth or destruction of a Smale–Williams solenoid ⋮ Snap-back repellers in non-smooth functions ⋮ Noninvertible maps and their embedding into higher dimensional invertible maps ⋮ Chaotic dynamics of two Van der Pol-Duffing oscillators with Huygens coupling ⋮ Correlation analysis of dynamical chaos ⋮ BASIN FRACTALIZATIONS GENERATED BY A TWO-DIMENSIONAL FAMILY OF (Z₁–Z₃–Z₁) MAPS ⋮ Generation of Robust Hyperbolic Chaos in CNN ⋮ The paths of nine mathematicians to the realm of dynamical systems ⋮ Critical homoclinic orbits lead to snap-back repellers ⋮ From Geodesic Flow on a Surface of Negative Curvature to Electronic Generator of Robust Chaos ⋮ Hyperbolic chaos in a system of two Froude pendulums with alternating periodic braking ⋮ Mechanical Systems with Hyperbolic Chaotic Attractors Based on Froude Pendulums ⋮ Hyperbolic chaos in systems based on Fitzhugh-Nagumo model neurons ⋮ ORIGIN OF CHAOS IN A TWO-DIMENSIONAL MAP MODELING SPIKING-BURSTING NEURAL ACTIVITY ⋮ An autonomous system with attractor of Smale-Williams type with resonance transfer of excitation in a ring array of van der Pol oscillators ⋮ Complex dynamics in double-diffusive convection ⋮ A non-autonomous flow system with Plykin type attractor ⋮ Chaos and fractal properties induced by noninvertibility of models in the form of maps ⋮ Attractor of Smale-Williams type in an autonomous distributed system ⋮ TRANSITION TO CHAOS FROM QUASIPERIODIC MOTIONS ON A FOUR-DIMENSIONAL TORUS PERTURBED BY EXTERNAL NOISE ⋮ Robust Hyperbolic Chaos in Froude Pendulum with Delayed Feedback and Periodic Braking ⋮ REDUCIBLE FRACTIONAL HARMONICS GENERATED BY THE NONAUTONOMOUS DUFFING–RAYLEIGH EQUATION: POCKETS OF REDUCIBLE HARMONICS AND ARNOLD'S TONGUES ⋮ A METHOD TO CALCULATE BASIN BIFURCATION SETS FOR A TWO-DIMENSIONAL NONINVERTIBLE MAP ⋮ BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART I ⋮ ON THE ATTRACTORS OF TWO-DIMENSIONAL RAYLEIGH OSCILLATORS INCLUDING NOISE ⋮ CHAOTIC DYNAMICS OF A SPATIO-INHOMOGENEOUS MEDIUM ⋮ BEHAVIOR OF HARMONICS GENERATED BY A DUFFING TYPE EQUATION WITH A NONLINEAR DAMPING: PART II ⋮ Smale – Williams Solenoids in a System of Coupled Bonhoeffer – van der Pol Oscillators ⋮ Smale–Williams solenoids in autonomous system with saddle equilibrium ⋮ AUTOWAVES AND SOLITONS IN A THREE-COMPONENT REACTION–DIFFUSION SYSTEM ⋮ Plykin-type attractor in nonautonomous coupled oscillators ⋮ Explosions of chaotic sets ⋮ Plykin type attractor in electronic device simulated in <scp>MULTISIM</scp>

Cites Work

• Shift automorphisms in the Henon mapping
• The dynamics of the Hénon map
• Structural stability on two-dimensional manifolds
• Invariants of \(\Omega\)-conjugacy of diffeomorphisms with a nongeneric homoclinic trajectory
• A two-dimensional mapping with a strange attractor
• Arithmetic properties of topological invariants of systems with nonstructurally stable homoclinic trajectories
• Structural stability of Lorenz attractors
• The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms
• The structure of Lorenz attractors
• An ergodic closing lemma
• Diffeomorphisms with infinitely many sinks
• One-dimensional non-wandering sets
• Lorenz attractors through Šil'nikov-type bifurcation. Part I
• Homoclinic bifurcation to a transitive attractor of Lorenz type
• ON SYSTEMS WITH A SADDLE-FOCUS HOMOCLINIC CURVE
• Moduli of $ \Omega$-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour
• CHUA’S CIRCUIT: RIGOROUS RESULTS AND FUTURE PROBLEMS
• ON GEOMETRICAL PROPERTIES OF TWO-DIMENSIONAL DIFFEOMORPHISMS WITH HOMOCLINIC TANGENCIES
• NORMAL FORMS AND LORENZ ATTRACTORS
• Symmetries and regular behavior of Hamiltonian systems
• Deterministic Nonperiodic Flow
• Symbolic Dynamics for Hyperbolic Flows
• Differentiable dynamical systems
• Markov Partitions for Axiom A Diffeomorphisms
• A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE
• ON THREE-DIMENSIONAL DYNAMICAL SYSTEMS CLOSE TO SYSTEMS WITH A STRUCTURALLY UNSTABLE HOMOCLINIC CURVE. I
• Expanding attractors

This page was built for publication: Mathematical Problems of Nonlinear Dynamics: A Tutorial