TY - JOUR
AU - Dellnitz, Michael
AU - Hohmann, Andreas
AU - Junge, Oliver
AU - Rumpf, Martin
ID - 16552
JF - Chaos: An Interdisciplinary Journal of Nonlinear Science
SN - 1054-1500
TI - Exploring invariant sets and invariant measures
ER -
TY - JOUR
AB - Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.
AU - Dellnitz, Michael
AU - Junge, Oliver
ID - 16535
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Almost Invariant Sets in Chua's Circuit
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Hohmann, Andreas
ID - 16533
SN - 9783034875202
T2 - Nonlinear Dynamical Systems and Chaos
TI - The Computation of Unstable Manifolds Using Subdivision and Continuation
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Field, Michael
AU - Golubitsky, Martin
AU - Ma, Jun
AU - Hohmann, Andreas
ID - 16550
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Cycling Chaos
ER -
TY - JOUR
AU - Dellnitz, M
AU - Heinrich, C
ID - 16532
JF - Nonlinearity
SN - 0951-7715
TI - Admissible symmetry increasing bifurcations
ER -
TY - JOUR
AB - Spiral patterns have been observed experimentally, numerically, and theoretically in a variety of systems. It is often believed that these spiral wave patterns can occur only in systems of reaction–diffusion equations. We show, both theoretically (using Hopf bifurcation techniques) and numerically (using both direct simulation and continuation of rotating waves) that spiral wave patterns can appear in a single reaction–diffusion equation [ in u(x, t)] on a disk, if one assumes "spiral" boundary conditions (ur = muθ). Spiral boundary conditions are motivated by assuming that a solution is infinitesimally an Archimedian spiral near the boundary. It follows from a bifurcation analysis that for this form of spirals there are no singularities in the spiral pattern (technically there is no spiral tip) and that at bifurcation there is a steep gradient between the "red" and "blue" arms of the spiral.
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Hohmann, Andreas
AU - Stewart, Ian
ID - 16551
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Spirals in Scalar Reaction–Diffusion Equations
ER -
TY - CHAP
AU - Golubitsky, Martin
AU - Marsden, Jerrold
AU - Stewart, Ian
AU - Dellnitz, Michael
ID - 16611
SN - 9780821803264
T2 - Normal Forms and Homoclinic Chaos
TI - The constrained Liapunov-Schmidt procedure and periodic orbits
ER -
TY - JOUR
AB - In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.
AU - Aston, Philip J.
AU - Dellnitz, Michael
ID - 16510
JF - International Journal of Bifurcation and Chaos
SN - 0218-1274
TI - Symmetry Breaking Bifurcations of Chaotic Attractors
ER -
TY - JOUR
AU - Dellnitz, M
AU - Melbourne, I
ID - 16542
JF - Nonlinearity
SN - 0951-7715
TI - A note on the shadowing lemma and symmetric periodic points
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Nicol, Matthew
ID - 16549
SN - 0066-5452
T2 - Trends and Perspectives in Applied Mathematics
TI - Symmetry of Attractors and the Karhunen-Loève Decomposition
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Scheurle, Jürgen
ID - 16544
SN - 9789401044134
T2 - Dynamics, Bifurcation and Symmetry
TI - Eigenvalue Movement for a Class of Reversible Hamiltonian Systems with Three Degrees of Freedom
ER -
TY - JOUR
AU - Dellnitz, Michael
AU - Melbourne, Ian
ID - 16541
JF - Journal of Computational and Applied Mathematics
SN - 0377-0427
TI - Generic movement of eigenvalues for equivariant self-adjoint matrices
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17014
JF - Schlaglichter der Forschung: Zum 75. Jahrestag der Universität Hamburg
TI - Collisions of chaotic attractors
ER -
TY - JOUR
AB - AbstractWe obtain normal forms for infinitesimally symplectic matrices (or linear Hamiltonian vector fields) that commute with the symplectic action of a compact Lie group of symmetries. In doing so we extend Williamson's theorem on normal forms when there is no symmetry present.Using standard representation-theoretic results the symmetry can be factored out and we reduce to finding normal forms over a real division ring. There are three real division rings consisting of the real, complex and quaternionic numbers. Of these, only the real case is covered in Williamson's original work.
AU - Melbourne, Ian
AU - Dellnitz, Michael
ID - 16633
JF - Mathematical Proceedings of the Cambridge Philosophical Society
SN - 0305-0041
TI - Normal forms for linear Hamiltonian vector fields commuting with the action of a compact Lie group
ER -
TY - JOUR
AU - Barany, Ernest
AU - Dellnitz, Michael
AU - Golubitsky, Martin
ID - 16518
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
TI - Detecting the symmetry of attractors
ER -
TY - JOUR
AU - Melbourne, Ian
AU - Dellnitz, Michael
AU - Golubitsky, Martin
ID - 16634
JF - Archive for Rational Mechanics and Analysis
SN - 0003-9527
TI - The structure of symmetric attractors
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17013
JF - Lectures in Applied Mathematics
TI - The equivariant Darboux theorem
VL - 29
ER -
TY - JOUR
AU - Dellnitz, M
AU - Melbourne, I
AU - Marsden, J E
ID - 16548
JF - Nonlinearity
SN - 0951-7715
TI - Generic bifurcation of Hamiltonian vector fields with symmetry
ER -
TY - JOUR
AU - Dellnitz, Michael
ID - 17012
IS - 3
JF - IMA Journal of Numerical Analysis
TI - Computational bifurcation of periodic solutions in systems with symmetry
VL - 12
ER -
TY - CHAP
AU - Dellnitz, Michael
AU - Golubitsky, Martin
AU - Melbourne, Ian
ID - 16546
SN - 9783034875387
T2 - Bifurcation and Symmetry
TI - Mechanisms of Symmetry Creation
ER -