UK Dynamical Systems Graduate School

Bifurcation, Symmetry and Pattern Formation

7-11th April 2008

Dept of Applied Mathematics, University of Leeds

Further information can be found on the home page.

The lecturers will be Mikhail Ivanchenko (MI), Paul Matthews (PCM), Alastair Rucklidge (AMR) and Thomas Wagenknecht (TW).

Programme

Details of course content

• Stability in time-continous and time-discrete systems.
• Characteristic roots, Lyapunov exponents, multipliers.
• Definition of bifurcations, codimension of bifurcation.
• Soft and hard bifurcations.
• Basics bifurcations in 2D time-continuous systems: saddle-node, transcritical, pitchfork, Hopf.
• Basic bifurcations in 1D time-discrete systems: tangent, pitchfork, period-doubling (flip).
• Bifurcations of time-continuous and related time-discrete systems in higher dimensions: Period doubling, Neimark, Bogdanov-Takens.
• Route to chaos via local bifurcations: period doubling.
• Center manifold theorem.
• Understanding basic bifurcations in 3D systems.
• (possibly) Local bifurcations in Hamiltonian systems. Krein signature.

• Homoclinic and heteroclinic bifurcations, heteroclinic cycles and networks, N-homoclinic orbits.
• Examples: Takens-Bogdanov, KdV equation, Shilnikov homoclinic example, homoclinic snaking.
• Conjugacy and equivalence. Invariant manifolds. Tangent spaces and transversality.
• Analytical techniques: return maps and linearisation theorems, Lin's method.
• Examples: blue sky catastrophe and Shilnikov homoclinics.
• Numerical methods for homoclinic bifurcations. codimension-two bifurcations: resonance, orbit-flip, inclination flip, T-points.
• Numerical methods: Simulation vs continuation; arclength continuation and BVPs; finding and continuing homoclinic orbits; test functions and detection of codim-2 points or, alternatively, computation of N-pulses.

• Stability theory: Real and imaginary eigenvalues. Normal mode solutions. Neutral stability curve. Dispersion relation.
• Weakly nonlinear theory: Swift-Hohenberg equation as an example. Solvability condition. Two-dimensional steady patterns on a lattice: rhombs, hexagons. Cross-coupling coefficient and the role of weakly damped modes.
• Centre manifold theorem for PDEs. Spectral gap.
• Periodic boundary conditions and lattices.
• Role of symmetry. Equivariance. Simple example: pitchfork.
• Equivariant bifurcation theory. Representations of common groups.
• Isotropy subgroups and fixed point subspaces.
• Equivariant branching lemma. Irreducible representations.
• Equivariant Hopf Theorem.
• Normal forms and stability of patterns.
• Examples (including examples with heteroclinic cycles).
• Pattern formation in large domains: Absence of the spectral gap.
• Long wave instabilities: Eckhaus, zig-zag.
• Ginzburg-Landau and Newell-Whitehead-Segel equations.
• Spiral Defect Chaos and the role of mean flow.
• Quasipatterns and small divisors.

• The symmetric group Sn and its natural representation
• How to construct equivariant normal forms
• Stationary bifurcation in all-to-all coupled systems
• Hopf bifurcation in all-to-all coupled systems
• Application to coupled nonlinear oscillators
• Matlab demonstrations

• Examples of additional continuous symmetries
• How continuous symmetries lead to large-scale neutral modes
• Effects of continuous symmetries on pattern formation
• Instant chaos in the Nikolaevskiy equation
• Localised patterns
• Brief introduction to numerical methods for pattern formation
• Matlab demonstrations

• The simpliest model: two coupled chaotic maps.
• Stability of synchronization.
• Geometrical approach to synchronization: transversal bifurcations of periodic trajectories; weak and strong synchronization; local and global riddling.
• Identical maps, general coupling: unidirectional coupling; asymmetrical local coupling; global coupling.
• Continuous-time systems.