Singularity confinement and chaos in discrete systems

Abstract

We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the analysis of the complexity (``algebraic entropy'') of the map using the growth of the degree of its iterates: integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems.

4 pages, revtex, 2 PostScript-figures

References in corpus (1)

• Discrete soliton equations and convergence acceleration algorithms

Cited by in corpus (78)

• Continuous Symmetries of Difference Equations
• Integrable lattice equations and their growth properties
• Diophantine Integrability
• Algebraic entropy and the space of initial values for discrete dynamical systems
• Singularity confinement and algebraic entropy: the case of the discrete Painlevé equations
• Algebraic Entropy for lattice equations
• Complexity spectrum of some discrete dynamical systems
• Searching for integrable lattice maps using factorization
• Discrete dynamical systems associated with root systems of indefinite type
• A geometric approach to singularity confinement and algebraic entropy
• Linearisable Mappings and the Low-Growth Criterion
• Investigation into the role of the Laurent property in integrability
• Integrable systems without the Painlevé property
• Laurent Polynomials and Superintegrable Maps
• Three Dimensional Integrable Mappings
• On the algebraic structure of rational discrete dynamical systems
• Singularity confinement for maps with the Laurent property
• Multiple-scale analysis of discrete nonlinear partial difference equations: the reduction of the lattice potential KdV
• Solvable Chaos
• Some integrable maps and their Hirota bilinear forms
• Singularity confinement and full-deautonomisation: a discrete integrability criterion
• The redemption of singularity confinement
• Deautonomisation by singularity confinement: an algebro-geometric justification
• Third-order integrable difference equations generated by a pair of second-order equations
• Fiber-dependent deautonomization of integrable 2D mappings and discrete Painlevé equations
• Calculating algebraic entropies: an express method
• Exact Localized Solutions of Quintic Discrete Nonlinear Schrödinger Equation
• Singularity confinement and algebraic integrability
• Space of initial conditions and geometry of two 4-dimensional discrete Painlevé equations
• Diophantine non-integrability of a third order recurrence with the Laurent property
• Singularity confinement as an integrability criterion
• Singularity confinement for matrix discrete Painleve Equations
• Irreducibility and co-primeness as an integrability criterion for discrete equations
• Singularity confinement in delay-differential Painlevé equations
• Singularity confinement and chaos in two-dimensional discrete systems
• Calculating the algebraic entropy of mappings with unconfined singularities
• Algebraic entropy of an extended Hietarinta-Viallet equation
• On some classes of discrete polynomials and ordinary difference equations
• Invariant varieties of periodic points for some higher dimensional integrable maps
• Algebraic entropy computations for lattice equations: why initial value problems do matter
• Integrability criterion in terms of coprime property for the discrete Toda equation
• A Characterization of Discrete Time Soliton Equations
• C-Integrability Test for Discrete Equations via Multiple Scale Expansions
• Full-deautonomisation of a lattice equation
• Integrability of Discrete Equations Modulo a Prime
• Algebraic entropy for semi-discrete equations
• A systematic method for constructing discrete Painlevé equations in the degeneration cascade of the E${}_8$ group
• Height growth of solutions and a discrete Painlevé equation
• Algebraic entropy for face-centered quad equations
• Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation
• Discrete mappings with an explicit discrete Lyapunov function related to integrable mappings
• An exercise in experimental mathematics: calculation of the algebraic entropy of a map
• On the Coprimeness Property of Discrete Systems without the Irreducibility Condition
• Nonlinear forms of coprimeness preserving extensions to the Somos-$4$ recurrence and the two-dimensional Toda lattice equation --investigation into their extended Laurent properties--
• The Schwarzian octahedron recurrence (dSKP equation) II: geometric systems
• From integrable equations to Laurent recurrences
• Degree growth of lattice equations defined on a 3x3 stencil
• Do All Integrable Evolution Equations Have the Painlevé Property?
• Singularities and growth of higher order discrete equations
• Algebraic entropy for hex systems
• Algebraic entropy for systems of quad equations
• Coprimeness-preserving discrete KdV type equation on an arbitrary dimensional lattice
• Generators of rank 2 cluster algebras of affine types via linearization of seed mutations
• Möbius Symmetry of Discrete Time Soliton Equations
• A family of integrable and non-integrable difference equations arising from cluster algebras
• Super-QRT and 4D-mappings reduced from the lattice super-KdV equation
• A two dimensional lattice equation as an extension of the Heideman-Hogan recurrence
• Detecting discrete integrability: the singularity approach
• Multiplicative equations related to the affine Weyl group E${}_8$
• Algebraic entropy of a multi-term recurrence of the Hietarinta-Viallet type
• Singularity confinement and proliferation of tau functions for a general differential-difference Sawada-Kotera equation
• Integrability of one degree of freedom symplectic maps with polar singularities
• Complete integrable systems with unconfined singularities
• Toda type equations over multi-dimensional lattices
• A lemma on the difference quotients
• General solution of functional equations defined by generic linear-fractional mappings F_1: C^N \to C^N and by generic maps birationally equivalent to F_1
• Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property
• The nature of manifolds of periodic points for higher dimensional integrable maps