Lattice Structures for Attractors III

The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of ‘set-difference’ taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretical models of spectral, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations build order-theoretic models of dynamical systems, which are used to develop tools for computing global characteristics of a dynamical system.

[1]  R. Walker,et al.  The Stone-Cech Compactification , 1974 .

[2]  William D. Kalies,et al.  An Algorithmic Approach to Lattices and Order in Dynamics , 2018, SIAM J. Appl. Dyn. Syst..

[3]  D. Chillingworth.  THE GENERAL TOPOLOGY OF DYNAMICAL SYSTEMS , 1995 .

[4]  K. Mischaikow,et al.  Chaos in the Lorenz equations: a computer-assisted proof , 1995, math/9501230.

[5]  Konstantin Mischaikow,et al.  A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems , 2009, SIAM J. Appl. Dyn. Syst..

[6]  Marian Mrozek,et al.  Towards a formal tie between combinatorial and classical vector field dynamics , 2016 .

[7]  J. Aubin.  Set-valued analysis , 1990 .

[8]  Konstantin Mischaikow,et al.  Model Rejection and Parameter Reduction via Time Series , 2018, SIAM J. Appl. Dyn. Syst..

[9]  Konstantin Mischaikow,et al.  Lattice structures for attractors I , 2014 .

[10]  K. Mischaikow,et al.  Polygonal approximation of flows , 2007 .

[11]  Konstantin Mischaikow,et al.  Lattice Structures for Attractors II , 2013, Found. Comput. Math..

[12]  Konstantin Mischaikow,et al.  An Algorithmic Approach to Chain Recurrence , 2005, Found. Comput. Math..

[13]  Robert D. Franzosa.  The connection matrix theory for Morse decompositions , 1989 .

[14]  S. Vickers.  Topology via Logic , 1989 .

[15]  Joshua D. Reiss,et al.  Construction of symbolic dynamics from experimental time series , 1999 .

[16]  Richard McGehee,et al.  Attractors for closed relations on compact Hausdorff spaces , 1992 .

[17]  Konstantin Mischaikow,et al.  Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation , 2001, Found. Comput. Math..

[18]  M. Gameiro,et al.  Topological Horseshoes of Traveling Waves for a Fast–Slow Predator–Prey System , 2007 .

[19]  P. Halmos.  Lectures on Boolean Algebras , 1963 .

[20]  Konstantin Mischaikow,et al.  Combinatorial Representation of Parameter Space for Switching Networks , 2016, SIAM J. Appl. Dyn. Syst..

[21]  Thomas Wanner,et al.  Structure of the Attractor of the Cahn-hilliard equation on a Square , 2007, Int. J. Bifurc. Chaos.

[22]  William D. Kalies,et al.  Rigorous Computation of the Global Dynamics of Integrodifference Equations with Smooth Nonlinearities , 2013, SIAM J. Numer. Anal..

[23]  Robert D. Franzosa,et al.  Index filtrations and the homology index braid for partially ordered Morse decompositions , 1986 .

[24]  Konstantin Mischaikow,et al.  A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems , 2004, SIAM J. Appl. Dyn. Syst..

[25]  Konstantin Mischaikow,et al.  Efficient computation of Lyapunov functions for Morse decompositions , 2015 .

[26]  Konstantin Mischaikow,et al.  Global asymptotic dynamics of gradient-like bistable equations , 1995 .

[27]  L. Arnold.  Random Dynamical Systems , 2003 .

[28]  Garrett Birkhoff,et al.  The convexity lattice of a poset , 1985, Order.

[29]  C. Conley.  Isolated Invariant Sets and the Morse Index , 1978 .

[30]  Konstantin Mischaikow,et al.  Global Dynamics for Steep Nonlinearities in Two Dimensions. , 2017, Physica D. Nonlinear phenomena.

[31]  Brian A. Davey,et al.  Introduction to Lattices and Order: Frontmatter , 2002 .

[32]  Konstantin Mischaikow,et al.  On the global dynamics of attractors for scalar delay equations , 1996 .

[33]  MARIAN MROZEK.  Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes , 2017, Found. Comput. Math..

[34]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[35]  J. Mallet-Paret.  Morse Decompositions for delay-differential equations , 1988 .

[36]  M. Gameiro,et al.  Combinatorial-topological framework for the analysis of global dynamics. , 2012, Chaos.

[37]  Howard Weiss,et al.  Chaotic dynamics of a nonlinear density dependent population model , 2004 .

[38]  Steven Roman,et al.  Lattices and ordered sets , 2008 .

[39]  William D. Kalies,et al.  A computational approach to conley's decomposition theorem , 2006 .

[40]  Konstantin Mischaikow,et al.  Rigorous Numerics for Global Dynamics: A Study of the Swift-Hohenberg Equation , 2005, SIAM J. Appl. Dyn. Syst..