PH316 Non-Linear Phenomena and Chaos.

Summer 1997

TWO hours ...Answer THREE questions.
Describe, using an illustrative example of your own choice, what is meant by a dynamical system. Indicate clearly its two main attributes.

Write illustrated notes describing the three broad categories of dynamical systems: flows, maps and cellula automata. Explain why different mathematical techniques must be adopted for the study of each of these.

In the case of cellula automata list the possible behaviour patterns which are associated with the following dynamical features: a fixed point, a limit cycle, an attractor (or repellor), a basin of attraction, stability (or instability) and chaos.


Explain the basics of linear stability analysis for one-dimensional flows by cone solutions of the equation:  of the form  in the vicinity of the solution at the fixed point  .

Show that e(t) grows exponentially if the Lyapunov exponent  and decays exponentially if  .
Show how a graphical method may be used to determine stability in the case when  .

Indicate how such an analysis can lead to the provision of characteristic time scales.

Analyse the stability of the following system: f(x)=x(1-x)(3-x).

Defining a potential function V(x) by f(x)=-dV/dx determine and sketch V(x) for the one-dimensional system  and characterize all of the equilibrium points.


Write down the equation of motion for an Euler strut and show how, when it is overdamped, it can be used to illustrate bistability in the form of a supercritical pitchfork bifurcation.

Provide suitably labelled sketches of the control space diagram and the phase-plane behaviour as the control parameter passes therough the critical point.

Describe briefly subcritical pitchfork bifurcations.

By transforming to Cartesian coordinates and evaluating the Jacobian at the origin show that the non-linear system:

gives rise to a Hopf bifurcation as  , the control parameter, changes sign.

Describe briefly the role played by such bifurcations in the various routes to chaos.

Explain carefully how the stability of a linearized two-dimensional system may be classified from the nature of the roots of its characteristic equation.

Provide an outline strategy of how to obtain global solutions of two-dimensional non-linear flows.

Locate and classify the fixed points of the following nonlinear system:


Plot the local phase portrait near each equilibrium point and indicate the global behaviour.

Use cobweb diagrams to describe the changes in behaviour of the iterates of the logistic map:

as the control parameter L is increased from the value minus one, through zero to positive values.

Explain with the aid of suitable diagrams how period-2 and period-3 points arise as L increases beyond the vaule three.
What did Feigenbaum discover about this map?
What role did this system play in exploring the various routes to chaos?

Answer TWO of the following:

(a) Specify the circumstances for which invertible maps, non-invertible maps and systems of first order ordinary differential equations can exhibit chaotic behaviour.
Provide examples for each one of these.

(b) Express the non-linear equation:

as a system of first order equations.
Describe in geometric terms how its dynamics changes as a changes sign whilst b remains positive.
State briefly the effect on the dynamics when a time dependent driving term is added to the original equation.


Describe how von Koch curves can be used to illustrate the concept of fractal dimension. Show why they have infinite length.


Define what is meant by the fractal dimension of a phase space attractor.
Describe how Lyapunov exponents can be used to determine the fractal dimension of a strange attractor in a dissipative system.

Mike Wilson

Wed Jan 7 11:45:14 GMT 1998