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Let us start getting round up with dynamics
and nonlinear systems. Now let start with
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the word dynamics, here is a simple definition,
it is the study of systems that evolve in
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time. Now such systems may eventually settle
down to some equilibrium or they may keep
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repeating in cycles or actually do something
much more complicated. Let us now visualise
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some dynamics, x is the depended variable,
time is the independent variable.
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There is an equilibrium value and their trajectories
settled to that equilibrium. You can have
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another scenario, where they do not settled
equilibrium; they actually just keep repeating
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themselves in cycles. The third scenario is
where you neither have an equilibrium nor
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a cycle but you keep doing something which
is fairly complicated.
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Now let us take a historical perspective on
dynamics the story, really started in mid
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1600s with Newton. What he did was he invented
calculus and he worked on differential equations.
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He then discovered his laws of motion and
universal gravitation and he combined them
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to explain Kepler’s laws of planetary motion.
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Now essentially what he did was solved two
body problem, which is calculating the motion
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of the earth around the Sun given the inverse
square law of gravitational attraction between
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them. Now what proved to be rather difficult
was the three body problem. That is the problem
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of the Sun, the Earth and the Moon, this turned
out to be very difficult problem to solve,
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in the sense of trying to obtain explicit
formulas for the motions of Sun, the Earth
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and the Moon.
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Progress then happened in late 1800s with
the work of Poincare. What he really did was
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introduce a brand new point of view, what
he said was the following: Let focus on qualitative
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rather than quantitative questions. Here is
an example of a quantitative question, can
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we find the exact positions of the planets,
at all times.
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Here is a qualitative question, will our solar
system be stable for ever, in other words
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might some of the planets suddenly decide
to leave the solar system and go off to another
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galaxy. Now what Poincare did was developed
a geometric approach to answer such qualitative
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questions.
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Now let us move on to the first half of 20th
century. Dynamics was largely concerned with
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nonlinear oscillators with applications in
physics and in engineering. Now this led to
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the development of the Radio, Radar, Phase
Locked Loop and Lasers. In the second half
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of the 20th century computers started to allow
us to visualise complex dynamics. Kepler Lorenz
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in his famous 1963 paper observed chaotic
motion in a simple model for atmospheric dynamics.
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Later in the 1970s, working in mathematical
biology, Winfree worked on nonlinear oscillators
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like heart rhythms. In physics Feigen Baum
went on to establish connection between chaos
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and phase transitions. And in the 1980s there
was a famous book by Kuramoto called chemical
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oscillations, waves and turbulence. Now moving
on to the beginning of the 21st century, the
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key words that one often hear are complex
systems and network systems.
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Now usually such systems are large scale and
highly nonlinear and their applications all
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over science and engineering.
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Now we start talking about dynamical systems.
There are two main types of dynamical systems,
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one is differential equations where the evolution
of the system happens in continuous time.
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The other variant is difference equations,
where time is discreet. We focus on differential
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equations, as these are widely used in science
and engineering. Very general frame work for
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ordinary differential equations abbreviated
as ODEs is x1 dot is equal to f1 x1 to x n
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all the way up to x n dot is equal to fn x1
to x n the over dot’s represent differentiation
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with the respect to time t so the xi dot is
equal to dxi dt.
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The variables x1 to x n could be concentrations
of chemicals within a reactor or there could
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be populations of different species within
an ecosystem and so on. The functions f1 to
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fn are determined by the specific problem
that had.
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As an example, let us consider Damped Harmonic
oscillator here is the differential equations
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that is m d square x dt squared plus b dx
dt plus kx is equal to zero, where m b and
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k are greater than zero. Now this is an example
of an ordinary differential equation. What
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we do is we introduce new variables x1 is
equal to x and x2 is equal to x dot. Then
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x1 dot is equal to x2 and x2 dot is equal
to x double dot which is equal to minus b
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on m x dot minus k on m x which is minus b
on m x2 minus k on m x1.
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The equivalent system when turns out be x1
dot is equal to x2 and x2 dot is equal to
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minus b on m x2 minus on k on m x1. Now recall
the general form of the ordinary differential
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equation that is x1 dot is equal to f1is the
function of x1 and x2 and x2 dot is equal
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to f2 is the function of x1 and x2 . So if
f1 is x2 and f2 is minus b on m x2 minus k
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on m x1 then we essentially just turn the
Damped Harmonic oscillator into general form
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of the ODEs that we had earlier.
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And here are some notes the above system is
linear, as all the xi’s on the right hand
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side are to the first power only. For a nonlinear
system, typically the terms are products,
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powers and functions of the xi’s. For example,
x1 x2, x1 squared sine of x1 etc are all examples
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of nonlinear terms.
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Now let us consider the example of a swinging
pendulum the differential equation is x double
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dot plus g on L sine x is equal to zero. Where
x is the angle of the pendulum from vertical,
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g is the acceleration due to gravity and L
is the length of the pendulum. The equivalent
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system is x1 dot is equal to x2 and x2 dot
is equal to minus g on L sine of x1. Now we
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get to this equivalent system using exactly
the same trick as the previous example.
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This system is nonlinear because of the sine
x1 term. This makes the equation very difficult
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to solve analytically. Now we can use the
small angle approximation that is sine x is
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approximately equal to x for x much less than
one, which will turn the equation into linear
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one. But we will also try to extract information
about the original nonlinear system using
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geometric methods.
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The rough idea is as follows now suppose that
for a particular initial condition, we just
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happen to know the solution to the pendulum
system. The solution could be a pair of functions
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x1 and x2 representing the position and velocity
of the pendulum. Now if we construct an abstract
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space x1, x2 then the solution x1 t and x2
t will correspond to a point moving along
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the curve in this space.
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Now let just go ahead and plot this we have
an initial condition x1 zero and x2 zero we
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go ahead and plot curve. Now this curve is
called a trajectory. Space is called the Phase
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Space. Now the Phase Space is actually filled
with trajectories, as each point can actually
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serve as an initial condition.
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Now we will outline the Poincare’s geometric
perspective. Now given the system we want
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to draw the trajectories and there by extract
information about the solutions. Now in numerous
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cases such geometric reasoning actually allows
us to draw trajectories without actually solving
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the system. Here are some notes the phase
space for the general system is the space
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with coordinates x1 to xn.
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So n represents the dimension of the phase
space, so the space is n dimensional. Now
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phase portrait is collection of all the qualitatively
different trajectories in the system.
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Now let us have brief discussion on linear
versus nonlinear systems. When you look at
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linear systems they can be broken into parts
then each part can be solved separately and
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then recombined to get to final answer. Now
they are equal to the sum of their parts and
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utilise methods like Laplace transforms and
Fourier analysis. So natural question to ask
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is given that we have such rich theory for
linear systems and very well established methods
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and they behave nicely then why not just model
using linear systems.
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The answer as follows when parts of the system
interfere, cooperate and compete. We very
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naturally get nonlinear interactions. Additionally,
most systems in science and engineering do
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not really act like linear systems.
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Now here is one way to look at world of dynamics.
On one axis we write down the dimension of
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the phase space and on the other axis we classify
system as linear or nonlinear. Then our current
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focus will be on this regime where we will
be dealing with one two dimensional nonlinear
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systems.