Natural tangent dynamics with recurrent biorthonormalizations: A geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets

Abstract This article develops a geometric method, referred to as Natural Tangent Dynamics with Recurrent Biorthonormalizations (NTDRB), that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold (associated with an equilibrium point), or exhibiting nontrivial limit sets (limit cycles and chaotic attractors). For the first class of systems the main focus relies on the identification of the slow/fast timescales, while for the latter (dynamics within nontrivial attractors) the central issue is the temporal dichotomy between unstable/stable components as described by the structure of the Lyapunov spectrum. The method is based on vector and covector dynamics in the tangent and cotangent bundles (induced by the phase-space flow), and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable/slow subspaces. The NTDRB method provides the most convenient and natural vector and covector bases for: (i) the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold; (ii) the identification of the stable and unstable components of the dynamics on a limit set; (iii) the construction of a low-dimensional reduced model reproducing the asymptotic behavior of the original systems.

[1]  G. Nicolis,et al.  Sustained oscillations and threshold phenomena in an operon control circuit. , 1976, Biophysical chemistry.

[2]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[3]  Irving R. Epstein,et al.  Experimental and modeling study of oscillations in the chlorine dioxide-iodine-malonic acid reaction , 1990 .

[4]  U. Maas,et al.  Intrinsic low-dimensional maniforlds of heterogeneous combustion processes , 2000 .

[5]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[6]  Ulrich Parlitz,et al.  Comparison of Different Methods for Computing Lyapunov Exponents , 1990 .

[7]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[8]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[9]  Robert W. Dibble,et al.  Combustion: Physical and Chemical Fundamentals, Modelling and Simulation, Experiments, Pollutant Formation , 1996 .

[10]  T. Matsumoto.  Chaos in electronic circuits , 1987, Proceedings of the IEEE.

[11]  Michael J. Davis,et al.  Geometric investigation of low-dimensional manifolds in systems approaching equilibrium , 1999 .

[12]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[13]  Ulrich Maas,et al.  Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds , 1992 .

[14]  A. Iserles,et al.  On the Implementation of the Method of Magnus Series for Linear Differential Equations , 1999 .

[15]  S. Lam,et al.  The CSP method for simplifying kinetics , 1994 .

[16]  Christopher K. R. T. Jones,et al.  Invariant manifolds and singularly perturbed boundary value problems , 1994 .

[17]  Marc R. Roussel,et al.  On the geometry of transient relaxation , 1991 .

[18]  Bruno Sportisse,et al.  Solving reduced chemical models in air pollution modelling , 2003 .

[19]  K. Ramasubramanian,et al.  A comparative study of computation of Lyapunov spectra with different algorithms , 1999, chao-dyn/9909029.

[20]  S. H. Lam,et al.  Using CSP to Understand Complex Chemical Kinetics , 1993 .

[21]  Hans G. Kaper,et al.  Asymptotic analysis of two reduction methods for systems of chemical reactions , 2002 .

[22]  U. Maas,et al.  A general algorithm for improving ILDMs , 2002 .

[23]  Chaotic oscillations of cool flames , 1989 .

[24]  A. N. Gorban,et al.  Constructive methods of invariant manifolds for kinetic problems , 2003 .

[25]  Stephen Wiggins,et al.  Identification of low order manifolds: Validating the algorithm of Maas and Pope. , 1999, Chaos.

[26]  Massimiliano Giona,et al.  Exterior Algebra-Based Algorithms to Estimate Liapunov Spectra and stretching Statistics in High-Dimensional and Distributed Systems , 2002, Int. J. Bifurc. Chaos.

[27]  Bruno Sportisse,et al.  Partitioning techniques and lumping computation for reducing chemical kinetics: APLA: an automatic partitioning and lumping algorithm , 2002 .

[28]  J. Changeux,et al.  ON THE NATURE OF ALLOSTERIC TRANSITIONS: A PLAUSIBLE MODEL. , 1965, Journal of molecular biology.

[29]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[30]  J. Seinfeld,et al.  Atmospheric Chemistry and Physics: From Air Pollution to Climate Change , 1997 .

[31]  L. Chua,et al.  The double scroll family , 1986 .

[32]  A Goldbeter,et al.  Unified mechanism for relay and oscillation of cyclic AMP in Dictyostelium discoideum. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[33]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[34]  Massimiliano Giona,et al.  Modal reduction of PDE models by means of Snapshot Archetypes , 2003 .

[35]  Michael Kirby,et al.  Minimal dynamical systems from PDEs using Sobolev eigenfunctions , 1982 .

[36]  Jack K. Hale,et al.  Infinite dimensional dynamical systems , 1983 .

[37]  Lyapunov exponents and subspace evolution , 1987 .

[38]  L. Sirovich.  Chaotic dynamics of coherent structures , 1989 .

[39]  C. Law,et al.  Complex CSP for chemistry reduction and analysis , 2001 .

[40]  S. Scott,et al.  Period doubling and chaos during the oscillatory ignition of the CO + O2 reaction , 1990 .

[41]  Epaminondas Mastorakos,et al.  Global reduced mechanisms for methane and hydrogen combustion with nitric oxide formation constructed with CSP data , 1999 .

[42]  Stephen B. Pope,et al.  Computationally efficient implementation of combustion chemistry using in situ adaptive tabulation , 1997 .

[43]  L. Tsimring,et al.  The analysis of observed chaotic data in physical systems , 1993 .

[44]  Celso Grebogi,et al.  Unstable dimension variability: a source of nonhyperbolicity in chaotic systems , 1997 .

[45]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[46]  Ulrich Maas,et al.  Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space , 1992 .

[47]  P. Holmes,et al.  Turbulence, Coherent Structures, Dynamical Systems and Symmetry , 1996 .

[48]  B. Hess,et al.  Oscillatory phenomena in biochemistry. , 1971, Annual review of biochemistry.

[49]  L. Olsen,et al.  Chaos in an enzyme reaction , 1977, Nature.

[50]  Claudio Tebaldi,et al.  Sequences of infinite bifurcations and turbulence in a five-mode truncation of the Navier-Stokes equations , 1979 .

[51]  Samuel Paolucci,et al.  On slow manifolds of chemically reactive systems , 2002 .

[52]  Marc R. Roussel,et al.  Geometry of the steady-state approximation: Perturbation and accelerated convergence methods , 1990 .

[53]  Adele Cutler,et al.  Archetypal analysis of spatio-temporal dynamics , 1996 .

[54]  J. Griffiths.  Reduced kinetic models and their application to practical combustion systems , 1995 .