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Brown, R. [2016] IDETheoryPart03 Brown, R. [2016} FormalTheoryIDE05 Brown, R., [2016] IDEVectorCalculus Brown, R., [2015] Theory of IDEs

A homoclinic tangle is the trellis of intersections of the stable and unstable manifold of a nonlinear dynamical system. Homoclinic tangles were first identified by H. Poincare in the late 1800's as the source of the complexity in the three body problem. In the 1960's S. Smale gave the first formal proof that homoclinic tangles were the source of complexity in dynamical systems generally, although his theorem only addressed "transverse tangles" like those seen above. In the 1980's J. Yorke coined the term "chaos" to refer to the complexity seen in these dynamical systems and the name has stuck ever since.

While homoclinic tangles are mathematically significant in understanding complex nonlinear dynamical systems, they, and their associated unstable manifolds, may have the status of art in their geometry. This website presents images of unstable manifolds from the "twist and flip" equations derived by R. Brown in the early 1990's. In these examples, the stable manifolds are the mirror image of the unstable manifolds, and the homoclinic tangle is the intersection of the image seen and its reflection. In most cases, this is too complex to see on a computer due to the low resolution of the computer graphics, hence we only present the unstable manifolds.

For an introduction to chaos see R. Devaney. For a popularized account see J.Gleick.

All images are copyright R. Brown but may be used for illustrations in research papers and text books. Equations are available upon request for research only. No commercial use is permitted.

Selected Publications

Brown, R., [1992] “Generalizations of the Chua Equations,” IEEE Transactions on Circuits and Systems 40(11).
Brown, R., Chua, L. [1992] Chaos or Turbulence
Chua, L., Brown, R. & Hamilton, N. [1993] “Fractals in the Twist-and-Flip Circuit,” Proceedings of the IEEE” Special Issue on Fractals in Circuits, October 1993. (Front Cover)
Brown, R. and Chua, L., [1993] “Dynamical Synthesis of Poincare Maps,” International Journal of Bifurcation and Chaos 3(5).
Brown, R. [1995] “Horseshoes in the Measure Preserving Henon Map,” Ergodic Theory and Dynamical Systems.
Brown, R. and Chua, L., [1996] “Clarifying Chaos: Examples and Counterexamples” “International Journal of Bifurcation and Chaos 6(2).
Brown, R. and Chua, L., [1996] “From almost periodic to chaotic: The fundamental map” “International Journal of Bifurcation and Chaos 6(6). (Front Cover)
Brown, R. and Chua, L., [1997] “Chaos: Generating complexity from simplicity” “International Journal of Bifurcation and Chaos 7(7).
Brown, R. and Chua, L., [1998] “Clarifying Chaos II: Bernoulli Chaos” “International Journal of Bifurcation and Chaos 8(2). (Front Cover)
Brown, R. and Chua, L., [1999] “Clarifying Chaos III: Stochastic Processes” “International Journal of Bifurcation and Chaos 9(5).
Brown, R. [1999], On Solving Nonlinear Functional, Finite Difference, Composition, and Iterated Equations, Fractals 1999.
Brown, R., Berezdivin, R., and Chua, L., [2001] “Chaos and Complexity”, International Journal of Bifurcation and Chaos 11(1)
Brown, B., Brown, R, Shlesinger, M., [2003] “Solution of Doubly and higher order iterated equations”, J. of Stat. Physics, Vol. 110, Nos. 3-6
Brown, R., Jain, V. [2009] "A New Approach to Chaos" Dynamics of Continuous, Discrete and Impulsive Systems (Figures for "A New Approach to Chaos")
Brown, R. [2014] "The Hirsch Conjecture", to appear in Dynamics of Continuous, Discrete and Impulsive Systems
Brown, R. [2014] "Infinitesimal Stretching and Folding I", to appear.
Brown, R. [2014] "Infinitesimal Stretching and Folding II", to appear in Dynamics of Continuous, Discrete and Impulsive Systems .
Brown, R. [2016] "Proving a Neurodynamical Theory" Chapter 18, Cognitive Phase Transitions in the Cerebral Cortex- Enhancing the the Neuron Doctrine in the Springer series Studies in Systems, Decision and Control
Brown, R., and Hirsch, M. W., [2016] "Stretching and Folding in the KIII Neurodynamical Model" Chapter 17, Cognitive Phase Transitions in the Cerebral Cortex- Enhancing the the Neuron Doctrine in the Springer series Studies in Systems, Decision and Control 2016
Brown, R. [2015] "The Theory of Infinitesimal Diffeomorphisms." Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms 22 (2015) 199-222
Brown, R. [2015] "Henon Transitions"Dynamics of Continuous, Discrete and Impulsive Systems Series B :Applications & Algorithms 22 (2015) 255-262

Table of ContentsPublications/TOC.pdf