On low speed travelling waves of the Kuramoto-Sivashinsky equation. Jeroen S.W. Lamb Joint with Jürgen Knobloch (Ilmenau, Germany) Marco-Antonio Teixeira (Campinas, Brazil) Kevin Webster (Imperial College London) Page 1
Outline: Motivation and travelling wave reduction Reversible fold-hopf bifurcation in R 3 Reversible heteroclinic cycle bifurcation Consequences for the KS equation Page 2
Kuramoto-Sivashinsky equation: u + uu t x + uxx + uxxxx = 0 Arises in context of several applications: eg thin liquid film down vertical plate, Belousov- Zhabotinskii, laminar flame fronts. Paradigm for nonlinear PDE with complex (chaotic?) behaviour. Page 3
Simulation of KS Page 4
Question: How can one describe and understand this complicated dynamics? In case of periodic boundary conditions, there exists an finite-dimensional inertial manifold. More modest aim: description of travelling waves and stationary states. Page 5
Travelling wave reduction Travelling wave solution u(x,t)=v(x-ct) Equation of motion for travelling wave in terms of w=v-c: 2 2 w! c ww' + w'' + w'''' = 0 " + w' + w''' = 0 2 Or with x=w, y=w and z=w : x! = y Michelson s system y! = z (Michelson 1986) 1 z! = c " x " y 2 2 2 Page 6
Steady-state reduction Steady-state solution u(x,t)=w(x) KS 2 w ww' + w'' + w'''' = 0! C " + w' + w''' = 0 2 w(x)=c solution C=c 2 /2 Michelson Steady-state and travelling wave reduction leads to the same equations! Page 7
Solutions with small wave speed c 0 At c=0, (0,0,0) is an equilibrium point (trivial solution of KS equation). Linear part has eigenvalues (0,±i): nonhyperbolic Hopf-zero (or fold-hopf) point. Question: What travelling wave solutions does the KS equation have for small wave speed c? Page 8
Local bifurcation theory Main objective: describe the solution set near a structurally unstable local solution (equilibrium or periodic solution) for small perturbations of the system. Important: to give meaning to the small perturbations mentioned above, it is important to 1. Choose a topology (often C k topology) 2. Describe the set of vector fields within which one intends to perturb (structure). Page 9
Example of structure: (time-) reversibility R is a time-reversal symmetry of a differential equation x =f(x) iff Fix R x(t) is a soln Rx(-t) is a soln Page 10
Reversibility of the Michelson system From the steady-state reduction, we may derive that the Michelson system is reversible due to a reflection symmetry of the KS equation: u(x,t) -u(-x,t) w(x) -w(-x) (x,y,z,t) (-x,y,-z,-t) in Michelson system Note: Michelson system has additional structure: volume preserving (divergence free vector field). 3 rd order differential equation Page 11
Approach: Study the (generic) reversible Hopf-zero bifurcation; this fits within a larger general programme concerning generic (Hamiltonian and non-hamiltonian) reversible (equivariant) bifurcation theory Verify later how/if the results can be applied to the Michelson system (which is a specific reversible dynamical system with additional structure). Page 12
Reversible system as an equivariant map Reversibility: f R=-Rf f can be thought of as a Z 2 -equivariant map f: V W with different representations of Z 2 on V and W : f ρ V (R)=ρ W (R)f with ρ V (R)=-ρ W (R) Useful property: equivariant maps preserve fixed point subspaces f:fix V R Fix W R Page 13
Reversible Hopf-zero bifurcation: linear approx Linear reversible systems: x =Lx λ eigenvalue of L - λ eigenvalue of L If dim Fix R dim Fix -R, L has forced zero eigenvalues, since if vector field L is R-reversible then L: Fix(R) Fix(-R). For instance, when R=diag(-1,1,-1) acting on R 3 then L has at least one zero eigenvalue. With eigenvalues (0,±i) one can choose local coordinates and rescale time such that L= " 0! 1 0# $ 1 0 0 % $ % $ & 0 0 0% ' Page 14
Reversible Hopf-zero bifurcation: affine approx If dim Fix(-R) > dim Fix R then generically there does not exist any symmetric (R-invariant) solution to f=0, since f: Fix(R) Fix(-R). In our case, a symmetric Hopf-zero equilibrium arises robustly in one-parameter families, ie such an equilibrium has codimension one. One needs to introduce an affine parameter: x! = C( µ ) + L( µ ) x In the present context (wlog) L can be chosen to be independent of µ. Page 15
(Birkhoff) Normal form theory Aim: perform coordinate transformations to simplify Taylor series expansion of vector field Normal form symmetry (Belitskii, Elphick et al): one can obtain expansion that commutes with S= {exp( sl) s! R}. Here, S=S 1. One can divide out the symmetry, consider vector field in R 3 /S 1 ~ R 2 with Z 2 equivariance (cylindrical coord. (r,z)) Page 16
Planar bifurcation (Takens) Normal form: r! = arz (a R, b =±1) (fin. det.) z! = µ + br 2 " z 2 Parameters in Michelson system: a>0, b =-1 Local phase portraits through bifurcation µ<0 Page 17 µ =0 µ >0
Interpretation of the phase portrait Equilibria (non-symmetric) Periodic soln (symmetric) 1D heteroclinic 2D heteroclinic Page 18
Local dynamics beyond normal form approx In general dissipative systems: codimension one local bifurcations give (branches of) hyperbolic solutions and simple local dynamics (no cycles) In reversible systems, more complicated dynamics may arise: for instance here due to heteroclinic cycles in normal form approximation. Page 19
Normal form approximation and perturbation S 1 normal form symmetry valid up to arbitrarily high order S 1 -symmetry breaking perturbations (should) break degenerate (nontransversal) 1D and 2D connections 2D connection: intersects Fix R transversally in two points at least two connections remain (typically transversal and isolated) Page 20
Flat perturbations of 1D connection Consider X R µ : space of one parameter families of reversible vector fields with Hopf-zero bifurcation at µ=0, endowed with C -topology. By means of an explicit flat perturbation (small in C -topology) one can create sequences of homoclinic and heteroclinic orbits accumulating to µ=0. (cf Broer, van Strien, Vegter) Page 21
Theorem 1 (Abundance of one-round cycles) There exists an open subset U X R µ which is determined by the 2-jet of the vector field at (0,0) R 3 x R, such that the set of vector fields for which in a neighbourhood of the origin in R 3 x R there exists a countable infinity of one-round homoclinic orbits and heteroclinic cycles between the saddle-foci, is residual in U. (residual = countable intersection of open and dense sets = generic ) Page 22
Dynamical consequences of Theorem 1 Unfolding of homoclinic cycles follows Shilnikov scenario: when normal form coefficient 0<a<2 there are horseshoes (shift dynamics). [true in Michelson system] Question: What happens near the heteroclinic cycles? Page 23
Heteroclinic cycle bifurcation between saddle-foci [H1] two non-symmetric equilibria [H2] of saddle-focus type [H3] h(t) transversal symmetric heteroclinic [H4] generic unfolding at λ=0 Page 24
Theorem 2 (Dynamics near heteroclinic cycle) Generically, the following statements hold: At λ=0 for all n>1 there is a countably infinite number of symmetric and non-symmetric (n>2) transverse n-2d heteroclinic orbits accumulating to the symmetric heteroclinic cycle. For each heteroclinic cycle there exists a countable infinity of periodic solutions accumulating to the heteroclinic cycle. For each n>1 there exists a countably infinite set of parameters λ k (n) converging exponentially to 0 as k such that at λ = λ k (n) there exists a n-1d heteroclinic orbit, converging to the one-round heteroclinic cycle. Similar for (pairs of asymmetric) homoclinic orbits. At λ=0 there exists an indecomposable R-invariant non-uniformly hyperbolic invariant set containing a countably infinite number of horseshoes, whose dynamics is topologically conjugate to a full shift on an infinite number of symbols. For small nonzero λ an R-invariant uniformly hyperbolic invariant set remains, whose dynamics is topologically conjugate to a full shift on a finite number of symbols. Page 25
Properties of first-hit maps Return map(s) F near symmetric heteroclinic cycle are reversible, ie R F = F -1 R. Use C 1 linearized transition map near saddle-foci. Diffeomorphisms near remaining connections. A curve intersecting W s (p 0 ) transversally in Σ 0 (near intersection with h(t)) is mapped to a logarithmic spiral in Σ 1 A logarithmic spiral in Σ 1 is mapped to a countably infinite set of lines accumulating to W u (p 1 ) in Σ 0. Page 26
Topological horseshoes Does not require a Shilnikov type condition on eigenvalues of saddle foci Page 27
Hyperbolicity: additional hypothesis [H5] No tangencies of the spiral traces of 2D stable and unstable manifolds in Σ 1 Bifurcation: unfolding of tangency between spirals Page 28
Digression: on the nature of homoclinic bifurcations Alternatively to the geometric approach, using return maps, the nonwandering dynamics can also be found by Liapunov-Schmidt reduction (via Lin s method) In the context of the heteroclinic cycle the resulting bifurcation equation is at lowest (determining) order equivalent to the problem of finding the intersection of two logarithmic spirals. The bifurcation point µ =0 corresponds to the case that the centres of the spirals coincide. Unfolding corresponds to moving one of the spirals. The logarithmic spirals are naturally parametrized by two parameters indicating times along orbits between hitting successive Poincaré sections. Page 29
Intersections of two logarithmic spirals Bifurcations are dense in µ-parameter space (infinite moduli, van Strien). Let ρ 1 and ρ 2 denote the contraction rates of the spirals. Then, if ρ 1 /ρ 2 is irrational, there exist a countable infinity of transverse intersections. In Michelson case, due to symmetry, ρ 1 =ρ 2. Despite exceptional cases, typically there exist a countable infinity of transverse intersections. But no longer dense set of bifurcations! Open question: what is the nature of the exceptional set, without an infinity of transversal intersections? Finite? Page 30
Intersections of logarithmic spirals Page 31
Theorem 3 (Dynamics near reversible Hopf-zero) There exists an open subset U Ì X R µ which is determined by the 2-jet of the vector field at (0,0) e R 3 x R, such that the set of vector fields for which in a neighbourhood of the origin in R 3 x R there exists for each n N a countable infinity of n-homoclinic orbits a countable infinity of symmetric n-heteroclinic orbits a countable infinity of non-symmetric n-heteroclinic orbits (n>2) a countable infinity of n- periodic orbits (accumulating to n- heteroclinic cycles) is residual in U. The subset of U for which in the neighbourhood of the origin in R 3 x R there exists a nontrivial hyperbolic basic set (horseshoe) is open and dense. Page 32
What do these results imply for the Michelson system? General problem: how to apply generic results to explicit examples. Some hypotheses cannot be verified: [H3,H5]: transversality of intersections of 2D (un)stable manifolds in sections Σ 0 and Σ 1. [H4]: generic unfolding of 1D connection. By examining carefully the arguments, we can relax these hypotheses without losing our main conclusions: [H3,H5] isolated intersections [H4] isolated 1D connection at λ=0 Page 33
Verifying modified hypotheses: Some results from the literature: (Adams et al 03, Yang 97) No one-round heteroclinic cycle near c=0. (But two-round heteroclinic cycles are abundant.) Exactly two 1-2D connections for small c. Consequences of analyticity: Isolated intersections and connections Transversal intersections near tangencies ( hyperbolicity) Page 34
Theorem 4 (Michelson system) Consider the Michelson system with parameter c. In every parameter interval (0,δ] with δ>0, there exists for each n N a countable infinity of n-homoclinic orbits, a countable infinity of n-heteroclinic cycles n>1, a countable infinity of n-periodic orbits, accumulating to each n-heteroclinic and n-homoclinic cycles), For all c sufficiently small there exist nontrivial hyperbolic basic sets (horseshoes). Page 35
Michelson system special in C -topology No one-round heteroclinic cycles near c=0. Argument used smooth flat non-analytic perturbation! Question: Is the behaviour of the Michelson system typical in the C ω -topology? Page 36
What this talk has aimed to illustrate: Structure is important. Analysis may require combination of techniques (local and global bifurcations may be intimately intertwined). Insight into generic dynamics is useful when studying explicit equations. Topology (eg smooth versus analytic) may be an issue when considering generic dynamics. There are important (but not generally well understood) geometrical aspects of homoclinic bifurcations. Page 37
Outlook: Local bifurcation theory: Extensions to Hopf n -zero bifurcation (with Buzzi & Teixeira) More general understanding of complex dynamics near local bifurcations in Hamiltonian and/or reversible (equivariant) systems. Homoclinic and heteroclinic bifurcation theory in the presence of symmetry: In progress with Jukes, Homburg, Knobloch and Webster Dynamics of the KS equation. Page 38
Publications: Jeroen S.W. Lamb, Marco-Antonio Teixeira and Kevin N. Webster. Heteroclinic cycle bifurcations near Hopf-zero bifurcation in reversible vector fields in R 3. To appear in J. Differential Equations. Kevin N. Webster. Bifurcations in reversible systems with applications to the Michelson system. PhD thesis, Imperial College London (2003). Jürgen Knobloch, Jeroen S.W. Lamb and Kevin N. Webster. Shift dynamics near T-point heteroclinic cycles. In preparation. Page 39