MATH3397/5398 Nonlinear Dynamics
5. Pattern Formation: summary
Patterns can be found in many dierent systems. Examples are uid ows, such as thermally driven convection,
and in nature the stripes of a zebra or the pattern of a ower. Pattern formation is co
MATH3397/5398 Nonlinear Dynamics
4. Limit Cycles and Co-Dimension Two Bifurcations: summary
1(A) Poincar-Linstedt method
e
This is a method for nding the amplitude and frequency of limit cycles generated at a Hopf bifurcation
near the Hopf Bifurcation. We
MATH3397/5398 Nonlinear Dynamics
3. Centre Manifolds: summary
1(A) Finding centre manifolds
(i) Orbits close to an equilibrium point can be understood by nding the eigenvectors of the linearised
equations about the equilibrium. If we include generalized e
MATH3397/5398 Nonlinear Dynamics
2. Local Bifurcations in n-dimensions: summary
1(A) Linear theory
(i) Equations are
x1 = f1 (x1 , x2 , , xn ),
x2 = f2 (x1 , , xn ),
,
xn = fn (x1 , , xn ).
We have a stationary solution X at which f1 (X) = f2 (X) = = fn (
MATH3397/5398 Nonlinear Dynamics
1. Bifurcations in 1-Dimensional systems: summary
1(A) Bifurcation types for 1D equations x = G(x, )
(i) Stationary solutions at G = 0. Growth/decay according to Gx > 0, Gx < 0. Bifurcation at x = x0 if
G = Gx = 0 there.
(
1
MATH3397/5398 Nonlinear Dynamics
Lecture 2
1.2 Bifurcations in one-dimensional systems
A one-dimensional system can be written x = G(x, r). Here r is a parameter, xed during the time
integration. G is any continuous and dierentiable function of x and r.
1
MATH3397 (CAJ, November 8, 2011)
MATH3397/5398 Nonlinear Dynamics
Example Sheet 5: Pattern Formation
Please hand Q. 1 -3 in by Thursday 8th December 2011
1. A chemical system with two components with concentrations U and V satises
U
2U
U
= 3 1+D 2,
t
V
1
MATH3397 (CAJ, November 8, 2011)
MATH3397/5398 Nonlinear Dynamics
Example Sheet 4: Limit cycles and Co-dimension 2 Bifurcations
Please hand in questions 1, 3 and 4 by
Thursday 24th November 2011
1. (a) Show that the system of dierential equations
dx
= x
1
MATH3397 (CAJ, October 24, 2011)
MATH3397/5397 Nonlinear Dynamics
Example Sheet 3: Centre Manifolds
Please hand in by Thursday 10th November 2011
1. (i) For the system
x = 2x + y x2 ,
y = xy x2
write down the linearised equations about x = y = 0 and nd
1
MATH3397 (CAJ, October 12, 2011)
MATH3397/5397 Nonlinear Dynamics
Example Sheet 2: Local Bifurcation theory in n-dimensions
Please hand in by Thursday 27th October 2011
1. Which two of these four linear systems of ODEs has a hyperbolic point at the orig
1
MATH3397/5398 (CAJ, September 22, 2011)
MATH3397/5398 Nonlinear Dynamics
Example Sheet 1: One-dimensional Local Bifurcation theory
Please hand in Q.1-5 by Thursday 13th October 2011
1. For the system x = 1 + rx + x2 , where r is a parameter, nd the stat
includegraphicshomoclinic.eps
Figure 1: A homoclinic orbit
MATH5397 Advanced Nonlinear Dynamics
Global Bifurcation Theory
1(A) Basic ideas
It is possible for a trajectory to leave an equilibrium point along its unstable manifold, and then to return
to tha
1
MATH5397 (CAJ, October 26, 2011)
MATH5397 Advanced Nonlinear Dynamics
Example Sheet on Global Bifurcation theory
Please hand in by Thursday 24th November 2011
1. (a) Show that the system of dierential equations
dx
= x y 2x3 ,
dt
dy
= x + y 3y 3
dt
has a
MAGIC042 lecture 2
Grant Lythe
Leeds
Oct 08
Grant Lythe (Leeds)
MAGIC042 lecture 2
Oct 08
1 / 21
Oct 08
2 / 21
Part I
Random walks and Markov chains
1
Gamblers ruin
2
Tossing an unfair coin
3
Discrete-time Markov chains
4
Continuous-time Markov chains
5
B