Numerical techniques: Deterministic Dynamical Systems Henk Dijkstra Institute for Marine and Atmospheric research Utrecht, Department of Physics and Astronomy, Utrecht, The Netherlands
Transition behavior from (proxy) data: Oxygen Isotope Ratio (ice cores) time (kyr)
Transition behavior from (model) data: FAMOUS model Equilibrium Simulations Control Simulation Hosing Simulation Depth/m 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 30 20 10 0 10 20 30 40 50 60 Latitude 15 10 5 0 5 10 Time series of the MOC (in Sv, 1 Sv = 10 6 m 3 s -1 ) at 26N and a depth 1000m in the Atlantic
Elementary bifurcations (can be obtained with variation of one parameter) Saddle-node bifurcation (limit point, turning point) Transcritical bifurcation Pitchfork bifurcation Hopf bifurcation
Bifurcation diagram for dx dt = λ x 2 # degrees of freedom: d = 1 Attracting fixed points trajectories (partial) bifurcation diagram
Exercise 1: Saddle node Determine all fixed points of the dynamical system: dx dt = λ x 2 and next their linear stability
Steady solutions and their stability dx dt = λ x 2 steady 0 = λ x _ 2 stable Linear Stability saddle node T unstable attracting repelling Bifurcation diagram
Other elementary (co-dim 1) bifurcations dx dt = x x2 transcritical dx dt x = λx x3 pitchfork λ Solution for all values of the parameter (Reflection) Symmetry in the problem
Hopf bifurcation x = λx ωy x(x 2 + y 2 ) y = λy + ωx y(x 2 + y 2 ) y supercritical # degrees of freedom: d = 2 x λ 2 y 1.5 y 2 1.5 1 1 0.5 0.5 0 example 0-0.5-0.6-0.4-0.2 0 0.2 0.4 x -0.5-0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 x steady state limit cycle
Numerical Bifurcation Theory System of PDEs: Operators containing parameters Discretization (N) Dynamical system: x: state vector
Exercise 2: Burgers equation @u @t + u@u @x = @2 u @x 2 u(0,t) = 1; @u @x (1,t)=0 Use central differences to derive the corresponding dynamical system. What is the state vector?
Demonstration MatCont = 360 ; µ 2 =6.25
Autonomous systems: fixed points Arclength parametrization
Euler-Newton continuation Starting Point: Compute initial tangent: Solve Extended system: With Euler guess:
The initial tangent Differentiate: to s: If is not a bifurcation point, then this matrix has rank N
The Newton - Raphson process G(x) = 0 Scalar function: G(x) Newton-Raphson x G(x) = 0 G (xk ) xk+1 = G(xk ) y = G0 (xk )x + b and hence y = G0 (xk )(x Then: G(xk ) = G0 (xk )xk + b xk ) + G(xk ) 0 = G0 (xk )(xk+1 xk ) + G(xk )
Exercise 3: The Newton - Raphson process Formulate the NR process for the extended system: Solve Extended system:
Detection of bifurcation points 1. Direct indicators f(s) det( x (s)) = 0 =0 2. Solve linear stability problem Use secant iteration:
Branch switching 1. Orthogonal to tangent 2. Use imperfections
Determining isolated branches (d) Residue continuation use homotopy parameter
Linear stability Dynamical system: Exercise 4: Show that the linear stability problem of a fixed point leads to a generalized eigenvalue problem Why is B often singular?
Numerical linear algebra Solution methods:! 1. QZ 2. Jacobi-Davidson QZ 3. Arnoldi 4. Simultaneous Iteration
Cayley Transform C =(A B) 1 (A µb) Cx = x = µ = µ = 100
Linear stability = r + i i ; x =ˆx r + iˆx i How to detect bifurcation points? Transcritical, Saddle-node, Pitchfork: A single real eigenvalue crosses the imaginary axis Hopf: A complex conjugated pair of eigenvalue crosses the imaginary axis Periodic orbit near Hopf bifurcation? (t) =e rt (ˆx r cos i t ˆx i sin i t)
Exercise 5 Formulate a generalized eigenvalue problem as a fixed point problem to trace branches of eigenvalues of a linear stability problem.!
Computation of Periodic Orbits (autonomous systems) 1. Boundary value problem 2. Fixed points of Poincare map Poincare section
Stability of Periodic Orbits: I Fixed point (periodic orbit) Linear stability Example: 1 Quasi-periodic behavior C Period doubling B 0 Additional periodic orbits A
Stability of Periodic Orbits: II A Cyclic Fold C Cyclic Pitchfork Naimark-Sacker (Torus) Period Doubling B
Useful tools auto, http://indy.cs.concordia.ca/auto/! xppaut www.math.pitt.edu/~bard/xpp/xpp.html Solves ODEs,DDEs,also AUTO built in! winpp Windows version of xppaut but used LOCBIF instead of AUTO! matcont allserv.rug.acbe/~ajdhooge/research.html Continuation software in Matlab July 9 th 2004 (lastest version)!! DDE-BIFTOOL Matlab package for numerical bifurcation analysis of delay equations