339-(Nonlinear) Dynamics.sagews

(Nonlinear) Dynamics

Adam Getchell

Motivation

Why do we care about nonlinear systems? Well, first of all, because we can draw pretty pictures with fractals which captures public interest:

You can Google for many, many more examples, and there are tons of software packages dedicated just to creating beautiful images with fractals.

More importantly, many, many natural phenomena of interest occur that are essentially non-linear. In the linear world, you can break the system down into composite parts, solve each part separately, and then recombine the solutions to get the final answer.

(Un)Fortunately, the universe is quite a bit more complicated. Although we have made great progress in science using simplified models with linear properties, to understand more we need a new kind of ... we need extend our current toolset a bit, and incorporate all the advances of the past into a cohesive approach.

Dynamics, then, is the study of systems that change and evolve in time.

There are two types of dynamical systems: differential equations in continuous time and iterated maps (aka difference equations) in discrete time. When we solve them, often we know solutions for a particular set of initial conditions. We can then construct an abstract phase space describing the possible solution trajectories of the system. The goal of dynamics is to do the reverse: from the system, draw the trajectories from the phase space. This kind of geometric reasoning will often give good information about the solutions without actually solving the system!

A trick: most physical systems of interest have a time component. But by parametrizing time (set $x_{n+1} = t$ such that $\frac{dx_{n+1}}{dt} = 1$) we can turn an n-dimensional time-dependent system into an (n+1)-dimensional time independent system. (But don't tell the GR-string theory-quantum gravity crowd ;-)

Dynamics - A Capsule History (Table 1.1.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology)
1666	Newton	Invention of calculus, explanation of planetary motion
1700s		Flowering of calculus and classical mechanics
1800s		Analytical studies of planetary motion
1890s	Poincare	Geometric approach, nightmares of chaos
1920-1950		Nonlinear oscillators in physics and engineering, invention of radio, radar, laser
1920-1960	Birkhoff Kolmogorov Arnol'd Moser	Complex behavior in Hamiltonian mechanics
1963	Lorenz	Strange attractor in simple model of convection
1970s	Ruelle & Takens May Fiegenbaum Winfree Mandelbrot	Turbulence and chaos Chaos in logistic map Universality and renormalization, connection between chaos and phase transitions Experimental studies of chaos Nonlinear oscillators in biology Fractals
1980s		Widespread interest in chaos, fractals, oscillators, and their applications

Introduction

A major technique in the analysis of nonlinear systems is the use of pictures. The basic idea is to recast a differential equation as a vector field. This often gives insight into systems for which no closed-form solution exists.

Consider the following equation: $$\large \frac{dx}{dt} = \sin(x)$$ First, note that this equation is exactly solvable. Many systems of interest do not have closed form solutions. Anyway, by separating variables we get: $$\large dt = \frac{dx}{\sin(x)}$$ Sage can solve this very quickly:

integral(csc(x),x)

-log(csc(x) + cot(x))

Now, this is an indefinite integral, which means the actual solution is: $$\large t = -\log|\csc(x) +\cot(x)| + C$$ We have to find the constant C by the usual means, let $x = x_0 @ t = 0$. Then $C = log|csc(x_0) + cot(x_0)|$, and the full solution is: $$\large t = \log|\frac{\csc(x_0) + \cot(x_0)}{(\csc(x) + \cot(x)}|$$ But what does this mean exactly?

Quick, if $x_0=\pi/4$, what happens as $t\rightarrow\infty$? What happens as $t\rightarrow\infty$ for arbitrary $x_0$?

Let's think about this another way.

Instead, let's plot $\frac{dx}{dt}$ versus x, and think of the function as a vector field with fluid flowing steadily along the x-axis with velocity that varies according to the rule $\frac{dx}{dt} = \sin(x)$.

''' Plot vector field picture of f
    Later, this should calculate arrows from zeroes of f automatically
'''

plotrange = 3
x = var('x')
f = sin(x)

# Add plot
g = plot(f, -plotrange * pi, plotrange * pi)

# Add arrows
startx = []
endx = []
arrow_len = 1
    
def create_arrow_endpoints(start, end, arrow_length, n, parity):
    if parity:
        start.append(n*pi - arrow_length)
        start.append(n*pi + arrow_length)
        end.append(n*pi)
        end.append(n*pi)
    else:
        start.append(n * pi)
        start.append(n * pi)
        end.append(n*pi - arrow_length)
        end.append(n*pi + arrow_length)

for i in range(-plotrange, plotrange+1):
    odd = (True if i % 2 != 0 else False)
    create_arrow_endpoints(startx, endx, arrow_len, i, odd)

y = [0 for y in range(len(startx))]
arrowstart = zip(startx, y)
arrowend = zip(endx, y)

def arrow_gen(start, stop):
    ''' Creates an arrow starting at start and ending at stop
        with color and line width
    '''
    return arrow(start, stop, rgbcolor=(1,0,0), width=0.05)

for i in range(len(arrowstart)):
    g += arrow_gen(arrowstart.pop(), arrowend.pop())

g.show()

When $\frac{dx}{dt}$ is positive, flow is to the right (+x direction). When $\frac{dx}{dt}$ is negative, flow is to the left (-x direction). Where $\frac{dx}{dt}$ is zero, there is no flow, and these are called fixed points. Notice that the fixed points can be classified in terms of their stability:

• Stable fixed points (attractors)
• Unstable fixed points (repellors)
• Saddle points

This classification carries over to higher-dimensional systems, with additions. For now, note that stability is a local property only; if you perturb the system far enough, you will get different behavior.

If we plot the actual solution to the equation (now x is the x-axis, t is the y-axis), we get:

@interact
def f(c = slider(-10, 10, .25, label = 'x0')):
    print c
    show(plot(log(abs((((csc(c)+cot(c))/(csc(x)+cot(x))))))))

Inspecting this, just like in the previous diagram we can see that for $\frac{\pi}{4}$ the system tends to accelerate at first towards the right, but then slow down as it approaches a limiting value. Of course, often times we don't have the luxury of an exact equation to plot!

In the previous case, we found the fixed points of the system, which helped us identify the behavior of the equation for various domains of interest. Next, we'll consider adding a parameter r to the system. We'll start off with a simple equation, $y = x^2 + r$:

@interact
def f(r = slider(-3,3,.25, label = 'r')):
    show(plot(r + x**2, -3, 3))

Notice that the location and classification of our fixed points depends upon the value of r.

Now let's look at a deceptively simple equation, the Logistic equation: $$\large y = r*x*(1-x)$$

@interact
def f(r = slider(0, 5,.1, label='r')):
    show(plot(r * x * (1.0 - x), 3, -3))

Something strange is going on here; we are getting different values. One tool we can use to look at its stability is a CobWeb diagram.

Now lets go onto 2D systems. Let's consider the system with the parameter a: $$\large \frac{dx}{dt} = -y + ax(x^2+y^2)$$ $$\large \frac{dy}{dt} = x + ay(x^2+y^2)$$

a=-1
plot_vector_field((lambda x,y:-y+a*x*(x^2+y^2),lambda x,y:x+a*y*(x^2+y^2)), (x,-3,3),(y,-3,3))

Bifurcation and Period doubling

%cython

'''Plot a bifurcation diagram for the Logistic Map
    x_n+1 = r*x_n*(1 - x_n)
'''

from numpy import *
from pylab import *
import math

# Define LogisticMap

def LogisticMap(double r, double x):
    #return r * x * (1.0 - x)
    return r * x - x*x*x

# Setup parameter range
rlow  = 1
rhigh = 4.0

# Setup the plot
# Stuff parameter range into a string via the string formating commands.
TitleString  = 'Logistic map, f(x) = r*x*(1 - x), '
TitleString += 'bifurcation diagram for r in [%g,%g]' % (rlow,rhigh)
title(TitleString)
# Label axes
xlabel('Control parameter r')
ylabel('{X(n)}')
# Set the initial condition used across the different parameters
ic = .01
# Establish the arrays to hold the set of iterates at each parameter value
# The iterates we'll throw away
nTransients = 200
# This sets how much the attractor is filled in
nIterates = 250
# This sets how dense the bifurcation diagram will be
nSteps = 500
# Sweep the control parameter over the desired range
rInc = (rhigh-rlow)/float(nSteps)
for r in arange(rlow,rhigh,rInc):
    # Set the initial condition to the reference value
    state = ic
    # Throw away the transient iterations
    for i in xrange(nTransients):
        state = LogisticMap(r,state)
    # Now store the next batch of iterates
    rsweep = [ ]   # The parameter value
    x = [ ]        # The iterates
    for i in xrange(nIterates):
        state = LogisticMap(r,state)
        rsweep.append(r)
        x.append( state )
    plot(rsweep, x, 'k,') # Plot the list of (r,x) pairs as pixels

# Use this to save figure as a bitmap png file
savefig('LogisticBifn', dpi=100)

# Turn on matplotlib's interactive mode.
#ion()

# Display plot in window
show()

Bifurcation for Cosine map

Universality

It's remarkable how similiar these two bifurcation plots look, even for systems with quite different character. The resemblance is more than skin deep. Due to a 1973 theorem by Metropolis et. al., the periodic attractors always occur in the same sequence! This sequence is called the U-sequence:

$$\large 1, 2, 2x2, 6, 5, 3, 2x3, 5, 6, 4, 6, 5, 6 ...$$

Thus, for systems in the low range of chaotic behavior, the form of $f(x)$ is irrelevant!

Furthermore, they all period-double at same rate! This rate is given by the Feigenbaum constant, $\delta = 4.699$.

Amazingly, regardless of the details of the physical system, they exhibit bifurcations (period-doubling) in the same order, at the same rate!

Chaos

Chaos is defined as aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions (positive Liapunov exponent).

Chaos is not equivalent to instability. For example, $\frac{dx}{dt}=x$ is deterministic and shows separation of nearby trajectories. It's not chaotic because the trajectories are repelled to inifinity and never return. So $\infty$ acts like an attracting fixed point, whereas true chaotic behavior is aperiodic (which excludes fixed points as well as periodic behavior).

The canonical example is the Lorenz equations: $$\frac{dx}{dt} = \sigma(y-x)$$ $$\frac{dy}{dt} = rx - y -xz$$ $$\frac{dz}{dt} = xy - bz$$

In sum, there's a lot more we don't know about dynamical systems and chaos. Hopefully many of you will help find answers!

Table 1.3.1 from Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology)
	Number of variables $\rightarrow$
	n=1	n=2	n>=3	n>>1	Continuum
Linear	Growth, decay, or equilibrium	Oscillations		Collective phenomena	Wave and patterns
	Exponential growth RC circuit Radioactive decay	Linear oscillators Mass and spring RLC circuit 2-body problem (Kepler, Newton)	Civil engineering, structures Electrical engineering	Coupled harmonic oscillators Solid-state physics Molecular dynamics Equilibrium statistical mechanics	Elasticity Wave equations Electromagnetism (Maxwell) Quantum mechanics (Schrodinger, Heisenberg, Direc) Heat and diffusion Acoustics Viscous fluids
Nonlinear	Fixed points Bifurcations Overdamped systems, relaxational dynamics Logistic equation for single species	Pendulum Anharmonic oscillators Limit cycles Biological oscillators (neurons, heart cells) Predator-prey cycles Nonlinear electronics (van der Pol, Josephson)	Chaos Strange attractors (Lorenz) 3-body problem (Poincare) Chemical kinetics Iterated maps (Feigenbaum) Fractals (Mandelbrot) Forced nonlinear oscillators (Levinson, Smale) Practical uses of chaos Quantum chaos?	Coupled nonlinear oscillators Lasers, nonlinear optics Nonequilibrium statistical mechanics Nonlinear solid-state physics (semiconductors) Josephson arrays Heart cell synchronization Neural networks Immune system Ecosystems Economics	Spatio-temporal complexity Nonlinear waves (shocks, solitons) Plasmas Earthquakes General Relativity (Einstein) Quantum field theory Reaction-diffusion, biological and chemical waves Fibrillation Epilepsy Turbulent fluids (Navier-Stokes) Life

Thanks to Professor James Crutchfield and his PHY 250 Nonlinear Dynamics class

References:

Strogatz, Steven H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology. Studies in Nonlinearity. Westview Press, 2001.