Computability and Computational Complexity of the Evolution of Nonlinear Dynamical Systems

Nonlinear dynamical systems abound as models of natural phenomena. They are often characterized by highly unpredictable behaviour which is hard to analyze as it occurs, for example, in chaotic systems. A basic problem is to understand what kind of information we can realistically expect to extract from those systems, especially information concerning their long-term evolution. Here we review a few recent results which look at this problem from a computational perspective.

[1]  C. M. Place,et al.  Ordinary Differential Equations , 1982 .

[2]  Olivier Bournez.  Achilles and the Tortoise Climbing up the Hyper-Arithmetical Hierarchy , 1999, Theor. Comput. Sci..

[3]  Daniel Silva Graça,et al.  Some recent developments on Shannon's General Purpose Analog Computer , 2004, Math. Log. Q..

[4]  Cristopher Moore,et al.  Closed-for Analytic Maps in One and Two Dimensions can Simulate Universal Turing Machines , 1999, Theor. Comput. Sci..

[5]  W. Tucker.  The Lorenz attractor exists , 1999 .

[6]  Daniel S. Graça,et al.  Effective Computability of Solutions of Differential Inclusions The Ten Thousand Monkeys Approach , 2009, J. Univers. Comput. Sci..

[7]  P. Odifreddi.  Classical recursion theory , 1989 .

[8]  Ning Zhong.  Computational unsolvability of domains of attraction of nonlinear systems , 2009 .

[9]  Michael S. Branicky,et al.  Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems , 1995, Theor. Comput. Sci..

[10]  Norbert Th. Müller,et al.  Uniform Computational Complexity of Taylor Series , 1987, ICALP.

[11]  Ning Zhong,et al.  Computability in planar dynamical systems , 2010, Natural Computing.

[12]  Giulio Bisconcini,et al.  Sur le problème des trois corps , 1906 .

[13]  A. Kolmogorov.  On conservation of conditionally periodic motions for a small change in Hamilton's function , 1954 .

[14]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[15]  Amaury Pouly,et al.  On the complexity of solving initial value problems , 2012, ISSAC.

[16]  Ning Zhong,et al.  Computability, noncomputability and undecidability of maximal intervals of IVPs , 2009 .

[17]  Keijo Ruohonen.  An Effective Cauchy-Peano Existence Theorem for Unique Solutions , 1996, Int. J. Found. Comput. Sci..

[18]  George Birkhoff.  Physical aspects of dynamical systems , 1927 .

[19]  J. Hadamard,et al.  Les surfaces a courbures opposees et leurs lignes geodesique , 1898 .

[20]  Piotr Sankowski,et al.  Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings , 2011, MFCS.

[21]  J. Hubbard,et al.  Differential Equations: A Dynamical Systems Approach , 2013 .

[22]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[23]  A. Kolmogorov,et al.  Preservation of conditionally periodic movements with small change in the Hamilton function , 1979 .

[24]  M. L. Cartwright.  On non-linear differential equations of the second order , 1949 .

[25]  Eugene Asarin,et al.  Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy , 1998, J. Comput. Syst. Sci..

[26]  Daniel S. Graça,et al.  Computability with polynomial differential equations , 2008, Adv. Appl. Math..

[27]  Norbert Th. Müller,et al.  Making big steps in trajectories , 2010, CCA.

[28]  Thomas Ottmann.  Automata, Languages and Programming , 1987, Lecture Notes in Computer Science.

[29]  S. Smale.  Mathematical problems for the next century , 1998 .

[30]  Ning Zhong,et al.  Computability, noncomputability, and hyperbolic systems , 2011, Appl. Math. Comput..

[31]  Arthur G. Werschulz,et al.  Computational complexity of one-step methods for systems of differential equations , 1980 .

[32]  Robert M. Corless,et al.  A New View of the Computational Complexity of IVP for ODE , 2002, Numerical Algorithms.

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

[34]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[35]  Warren D. Smith.  Church's thesis meets the N-body problem , 2006, Appl. Math. Comput..

[36]  J. Shepherdson.  Computational Complexity of Real Functions , 1985 .

[37]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[38]  Klaus Weihrauch,et al.  Topological Complexity of Blowup Problems , 2009, J. Univers. Comput. Sci..

[39]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[40]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[41]  Amaury Pouly,et al.  Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains , 2011, MFCS.

[42]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.