Renormalization-group analysis for the transition to chaos in Hamiltonian systems

DOI10.1016/S0370-1573(01)00094-1zbMath0994.70013MaRDI QIDQ1348790

H. R. Jauslin, Cristel Chandre

Publication date: 14 May 2002

Published in: Physics Reports (Search for Journal in Brave)

zbMATH Keywords

classical mechanicscanonical coordinate changeinvariant torus break-up

Mathematics Subject Classification ID

Dynamical systems in classical and celestial mechanics (37N05) Stability problems for problems in Hamiltonian and Lagrangian mechanics (70H14) Canonical and symplectic transformations for problems in Hamiltonian and Lagrangian mechanics (70H15)

Related Items (14)

Numerical Computation of Critical Surfaces for the Breakup of Invariant Tori in Hamiltonian Systems ⋮ Renormalization-group analysis for the transition to chaos in Hamiltonian systems ⋮ Computation of the breakdown of analyticity in statistical mechanics models: Numerical results and a renormalization group explanation ⋮ A remark on the onset of resonance overlap ⋮ On a new fixed point of the renormalization group operator for area-preserving maps ⋮ Renormalization group operators for maps and universal scaling of universal scaling exponents ⋮ Reducing or enhancing chaos using periodic orbits ⋮ Existence of critical invariant tori ⋮ Renormalization and destruction of 1/γ2 tori in the standard nontwist map ⋮ A renormalization group approach to quasiperiodic motion with Brjuno frequencies ⋮ Control of chaos in Hamiltonian systems ⋮ The critical renormalization fixed point for commuting pairs of area-preserving maps ⋮ Renormalization of vector fields and Diophantine invariant tori ⋮ A study on the statistical properties of the prime numbers using the classical and superstatistical random matrix theories

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states
• Universal properties of the transition from quasi-periodicity to chaos in dissipative systems
• A renormalization approach to invariant circles in area-preserving maps
• Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori
• An approximation to the critical commuting pair for breakup of noble tori.
• Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
• Converse KAM: theory and practice
• External field induced chaos in an infinite square well potential
• Destruction of KAM tori in Hamiltonian systems: Link with the destabilization of nearby cycles and calculation of residues
• Boundary circles for area-preserving maps
• Universal small-scale structure near the boundary of Siegel disks of arbitrary rotation number
• An obstruction method for the destruction of invariant curves
• Mather sets for plane Hamiltonian systems
• Construction of analytic KAM-surfaces and effective stability bounds
• Exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant tori
• Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. I: Theory
• A rigorous partial justification of Greene's criterion
• Critical Behavior of a KAM surface. I: Empirical results
• An approximate KAM-renormalization-group scheme for Hamiltonian systems
• Renormalization-group analysis for the transition to chaos in Hamiltonian systems
• The Brjuno functions and their regularity properties
• Hyperbolicity of renormalization of critical circle maps
• Renormalization and periodic orbits for Hamiltonian flows
• Existence of quasi-periodic orbits for twist homeomorphisms of the annulus
• Regular and stochastic motion
• Scaling law for the critical function of an approximate renormalization
• Converse KAM theory for symplectic twist maps
• Recent Developments in the Theory of Hamiltonian Systems
• Fractal bifurcation sets, renormalization strange sets and their universal invariants
• Higher-order fixed points of the renormalisation operator for invariant circles
• Greene's residue criterion
• Existence, nonexistence and universal breakdown of dissipative golden invariant tori. I. Golden critical circle maps
• On the standard map critical function
• Locally most robust circles and boundary circles for area-preserving maps
• New Directions in Dynamical Systems
• A renormalization group for Hamiltonians: numerical results
• Scaling for a Critical Kolmogorov-Arnold-Moser Trajectory
• A renormalization group for Hamiltonians, with applications to KAM tori
• Scaling law in the standard map critical function. Interpolating Hamiltonian and frequency map analysis
• A version of Thirring’s approach to the Kolmogorov–Arnold–Moser theorem for quadratic Hamiltonians with degenerate twist
• PROOF OF A THEOREM OF A. N. KOLMOGOROV ON THE INVARIANCE OF QUASI-PERIODIC MOTIONS UNDER SMALL PERTURBATIONS OF THE HAMILTONIAN
• Canonical transformations depending on a small parameter
• Determination of the threshold of the break-up of invariant tori in a class of three frequency Hamiltonian systems

This page was built for publication: Renormalization-group analysis for the transition to chaos in Hamiltonian systems