PH316 Non-Linear Phenomena and Chaos.

Examples and Exercises

NOTE: The numbering of these corresponds to the same numbered section of notes.

Exercises and Examples 1.

1. What is the linear deficiency for [ d (y + z)/ dx ]^2 ?

2. Learn to recognise nonlinear differential and difference equations. Which of the following equations are nonlinear?

(i) x^2 y'' + x y' (x^2-n^2)y = 0

(ii) dy/dx = y^2 + x

(iii) (x^2 + y^2)dy/dx = xy

(iv) dx/dt=a(x-xy), dy/dt=-c(y-xy)

(v) dy/dx + Q(x)y + R(x)y^2 = P(x)

(vi) x_(n+1)=ax_(n)(1-x_(n))

(vii) x_(n+1)=1-ax_(n)^2 + by_(n), y_(n+1)=x_(n)

3. Can a linear behavior (function) be obtained by successive nonlinear actions? More precisely, can you obtain a linear function L(x) = f(g(x)) through function composition of nonlinear functions f(x) and g(x)?
Can you suggest any systems, natural or artificial, which produces linear behavior by combining nonlinear mechanisms?

Exercises and Examples 2.

The first two examples illustrate the determination of a collection of projected variables whose dynamics can be approximately be described by a deterministic system of either linear or nonlinear equations.

1. Noting that a real system may contain more than 10^25 particles, which may typically be projected down to a few variables, plus assorted constants; list a set of dynamic variables and a set of related physical parameters (which are usually deterministic) in simple commonly occurring electrical circuits.

2. A mechanical system consists of a solid wheel, which can rotate about a central axle. An off-centre mass is attached to the wheel and another mass hangs from a rope wrapped around the wheel. Provide a set of (hopefully) deterministic dynamic variables, and associated parameters.
State any major assumptions about these 10^25 particles. Determine the equilibrium configuration of this system (a nonlinear property).
A linear property is to determine the frequency of small oscillations about its equilibrium configuration. Try to determine an expression for the angular frequency.

3. It is interesting and instructive to perform the following tasks:

Calculate and plot the first 15 iterates of the logistic map

x_{n+1} = r x_{n} (1 - x_{n})

starting from x_0 = 1/2 for
(a) r = 2.5
(b) r = 3.1
(c) r = 3.8

Comment on any differences you may observe in these three sets of results?

[ Those of you who like computing may wish to automate this procedure, although it can also be done with a pocket calculator and pencil and paper without too much trouble. ]

Determine for this map the (two) fixed points for the case r > 1 for several values of r. What, if any, is the relationship between the fixed points and the value of r ?

Exercises and Examples 3.

1. Find all the fixed points of the flow: x' = sin x .
At which points of x does the flow have greatest velocity to the right? What is the flow's acceleration as a function of x?

2. Find all the fixed points for x'= x^2 - 1, and classify their stability.

3. Sketch the phase portrait corresponding to x'= x - cos x, and determine the stability of all the fixed points.

4. Analyse the following equations graphically:

- Sketch the vector field on the real line,
- find all the fixed points,
- classify their stability and
- sketch the graphs of x(t) for different initial conditions.

[ Then, without spending too much time, it might be of some interest for you to obtain if you can, the analytical solutions x(t), using either Maple or even your own brain power.]

(i) x'= 4 x^2 - 16
(ii) x'= x - x^3
(iii) x'= exp(x) - cos x
(iv) x' = r x (1 - x/K)

Exercises and Examples 5.

1. Convert the following equations into the standard form of a set of coupled first order ODE's:

(i) x'' + ax' + b sin x = c cos (f t) where a,b,c,f are constants.

(ii) x''' = - b x^2 where b is constant.

2. Use linear stability analysis to classify the fixed points of the following systems.
If linear stability analysis fails, because f'(x*)=0, use a graphical method to determine the stability.

(i) x' = x(1 - x)

(ii) x' = x(1 - x)(2 - x)

(iii) x' = 1 - exp(-x^2)

(iv) x' = ln x .

3. Graph the potential for x' = x - x^3 and identify all equilibrium points.

4. Use linear stability analysis to study the dynamical behaviour of the 1-D system

x' = a x - b x^3 (a, b constant)

for both cases (i) a < 0 and (ii) a > 0.

NB. this example models Landau's theory of 2nd order phase transitions such as occurs in the paramagnetic -> ferromagnetic transition. [ see eg. the excellent discussion on this in Chapts 2 and 3 of Rowlands ]

5. Repeat the analysis of Question 4 for the equation:

x' = a x - b x^2 when a and b are +ve constants.

Exercises and Examples 6.

1. Find and determine the stability of the fixed points of x' = r - x^2 for when the control parameter r < 0, r = 0 & r > 0.

2. Show that the system x' = r - x - exp(-x) undergoes a saddle-node bifurcation as r is varied and find the value of r at the bifurcation point.
[ Hint: adopt the geometric approach ]

3. Show that the system x' = x(1 - x^2) - a(1 - exp(-bx)) undergoes a transcritical bifurcation at x = 0 when the parameters a, b satisfy a certain condition; determine this condition.
Find an approximate expression for the fixed point that bifurcates from x = 0 assuming that the parameters (a,b) are close to the bifurcation curve.

4. Plot the potential for x' = r x - x^3 for the cases r < 0, r = 0 & r > 0. [ Recall that the potential V(x) for x'=f(x) is defined as f(x)=-dV/dx ]

5. Equations of type x' = - x + b tanh(x) arise in statistical mechanical models of magnetism and neural networks.
Show that this equation undergoes a supercritical pitchfork bifurcation as b is varied.
Construct a plot of fixed points (x*) vs b , for the range b = 0 to 4.

Exercises and Examples 7.

1. To demonstrate that the origin of the square root scaling law is due to a saddle-node type bottleneck:

estimate the period of

(theta)' = w - a sin(theta) in the limit a -> w-

(ie. as a approaches w from below)

[ Hint: convert the eqn to the appropriate normal form and separate the variables ]

2. Construct representative phase plots for the eqns:

(i) x'' + w^2 x = 0 w constant
(ii) x'' - w^2 x = 0 w constant

3. Find the equilibrium points of the following systems of eqns and determine the Jacobian of the system for these steady states:

(i) x' = x^2 - y^2
y' = x(1 - y)

(ii) x' = y - xy
y' = xy

(iii) x' = x - x^2 -xy
y' = y(1-y)

Exercises and Examples 8.

1. Sketch the phase plane behaviour of the following systems of linear eqns and classify the stability characteristic of the steady state at (0,0):

(a) x' = -2y
y' = x

(b) x' = 3x + 2y
y' = 4x + y

(c) x' = 5x + 8y
y' = -3x -5y

(d) x' = -4x - 2y
y' = 3x - y

(e) x' = 2x + y
y' = x + 2y

(f) x' = x - 4y
y' = x + y

2. Suppose we were asked to explore the behaviour of trajectories of the following (coupled) nonlinear system:

dx/dt = xy - y

dy/dy = xy - x .

A general procedure to adopt is to determine any equilibrium points and their character and then use that information to construct a corresponding phase plane portrait to which trajectories can be added.
This suggests the following strategy:
first we find the equilibrium points from the eqns of the system; then determine the general Jacobian matrix for the system; then evaluate the Jacobian matrix at each of the fixed points. (This has effectively linearized the system in the neighbourhood of the fixed points and allows us to use linear stability theory.) This now allows us to find the nature of each of the fixed points and thus enables us to start to sketch appropriate trajectories in the phase plane.

[ An alternative procedure would be to make use of the Maple facility for plotting phase portraits and direction fields with DEtools. See Assignment 8 (above). ]

3. Find all the fixed points of the following nonlinear systems and use linearization to classify them:

(a) x' = y
y' = y - x^2 + 2x

(b) x' = y exp(y)
y' = 1 - x^2

(c) x' = x + x^2 - 2xy
y' = xy - y

[ If you have time it is of interest to sketch the phase portraits for these, either by hand or using Maple. ]

Exercises and Examples 9.

1. Plot the nullclines for the system
x' = x + y^2

y' = x + y

by examining: x = + , along y = -x and x = - , along x = -y^2 and then making use of the non-vanishing Jacobian short-cut to complete the picture.

2. Explore the phase plane behaviour of the van der Pol eqn:

x'' + rx'(x^2-1) + x = 0 for the case r >> 1.

Hint: Writing F(x)=x^3/3 - x and w = x' + rF(x) allows one to readily plot the so-called cubic nullcline.

3. Express the undamped Duffing eqn:

x'' + ax + bx^3 = 0 a, b constant

in the standard form of a system of two first order eqns.

Determine the fixed points and explore the phase-plane behaviour ie. sketch a selection of typical trajectories as 'a' -> +ve -> -ve for +ve 'b' values.

4. Consider the Duffing eqn of the previous example; what happens to the dimension of the system if we were to add a forcing (or driving) term ( of the form p*cos(wt) ) to the right hand side?

5. Consider the system of eqns containing a parameter r:

x' = y

y' = -y^3 + r*y - x

Show that a Hopf bifurcation occurs as r is varied.

[ Hint: Determine the fixed point, the Jacobian at the fixed point, the eigenvalues of the Jacobian matrix at the fixed point. ]

Exercises and Examples 10.

1. What happens if we apply a periodic driving force to a self-excited system such as a van der Pol oscillator?

2. The phase portrait of the quasi-periodic oscillation:

x = b1 cos (w1 t) + b2 cos (w2 t)

is not closed when w1 and w2 are incommensurate.

Show that if x(t) is sampled stroboscopically with a period 2*pi/w1 the points (x_n, x'_n) generated describe an ellipse in the Poincare map.

What does the Poincare map look like for the case of w1/w2 rational ?

Exercises and Examples 11.

1. Suppose one tries to predict the future state of a chaotic system to within a tolerance e = 10^(-3).

Given that our estimate of the initial state is uncertain to within ||o || = 10^(-7) , for about how long can we predict the state of the system, while remaining within the tolerance?

If we were to improve our initial error to be a million times better ie. ||o || = 10^(-13) , estimate for how much longer we may safely predict the state of the system.

2. Show that the map

x(n+1) = 1 + (1/2) sin[x(n)]

has a unique fixed point and determine its stability.

Exercises and Examples 12.

1. Show that the Verhulst type eqn:

p(n+1) = p(n) + r*p(n)[1-p(n)]

is equivalent to: x(n+1) = a*x(n)[1-x(n)]

where x(n)= [r*p(n)]/(r+1) and a = r + 1.

2. Consider the logistic eqn in the form

x(n+1) = L x(n)[1 - x(n)] = f(x(n)).

By considering the second-iterate map (ie. f(f(x))) show why a 2-cycle dosn't exist for L < 3.

3. Consider the tent map defined by the function:

r*x, 0 < x < 1/2 f(x) = r-r*x, 1/2 < x < 1

for 0 < r < 2 and 0 < x < 1 .

Show that for r < 1 the only fixed point is an attractor at the origin.

Show that for r > 1 there are two unstable fixed points.

Show that for r = 2 these two repellors are at x* = 0 and 2/3.

Show that the Lyapunov exponent lamda = ln r (ie. independent of the initial condition x(0) ).

[ This suggests that the tent map has chaotic solutions for all r > 1. Note that in Chaos Demos r = L/2.]

Exercises and Examples 13.

1. Use the divergence criterion to show whether the following systems are dissipative or not:

(a) x'' + x = 0

(b) x'' + x' + x = 0

2. (a) The Lorenz eqns describe a crude model of convective fluid flow; use the divergence criterion to decide whether or not they are dissipative.

x' = -s x + s y y' = -x z + r x - y z' = x y - b z where s, r, b are constants.

(b) Henon-Heiles studied stellar orbits in a galaxy using a 2-D model with potential:

V(x,y) = (x^2 + y^2)/2 + x^2 y - y^3/3.

This leads to a system of eqns in 4-D space of:

p'_x = - x - 2 x y p'_y = - y -x^2 + y^2 x' = p_x y' = p_y.

Is this system dissipative?

3. Estimate the Lyapunov exponents from the given figure and then use them in the Kaplan-Yorke formula to deduce the fractal dimension.

4. For a particular mapping it is found that an attractor has the following values of Lyapunov exponents:

lambda 1 = 0.67 lambda 2 = -0.70 and lambda 3 = -1.36.

Use the Kaplan-Yorke formula to deduce the dimension of the attractor. Comment on the result.

Exercises and Examples 14.

1. Show that the Henon map f(x,y): R^2 -> R^2

where f(x,y) = ( 1 + y -a x^2, b x) where a, b are constants

is invertible if b = 0.

Write down the transformation that describes the inverse mapping.

By constructing the determinant of the Jacobian matrix show that the Henon map contracts area if |b| < 1.

Sketch the effect of the Henon transform on a rectangle in the x-y plane.

Exercises and Examples 15.

1. Universality of approaches to chaos in different systems have been noted. Underlying this fact is the self-similarity (under re-scaling) of the specific process. Explain the reasons as to why period-doubling is self-similar. Explain the reasons as to why intermittency is self-similar.

2. Consider the cat map, T, of Arnold:

x(n+1) = x(n) + y(n) mod x = 1, mod y = 1. y(n+1) = x(n) + 2*y(n)

Show that T is area-preserving.
Show that the product of the eigenvalues of T is unity.
Note the effect of T on a cat after just two iterations.