The first step in the analysis of a dynamic
system is to derive its model.

Models may assume different forms, depending
on the particular system and the circumstances.

Mathematical model of a dynamic system can
often be expressed as a system of differential

(difference in the case of discrete-time systems)
equations.

The response of dynamic system to an input
may be obtained if these differential equations

are solved.

The differential equations can be obtained
by utilizing physical laws governing a particular

system, for example, Newtons laws for mechanical
systems, Kirchhoffs laws for electrical systems,

etc.

In obtaining a model, we must make a compromise
between the simplicity of the model and the

accuracy of results of the analysis.

In this lecture we learn how we can do that
by study two real world dynamical systems

but before that we need to learn and review
some concepts related to state space.

As we learned in pervious lecture, at any
given time, a dynamical system has a state

given by a vector that can be represented
as a point in a geometrical manifold.

What future states follow from the current
state is described by the evolution rule of

the dynamical system.

We also defined the set of all states of a
system of ordinary differential equations

as phase space.

If we plot the solution of equations of motion
in a phase space then we have a phase curve.

The phase curve is dependent on initial state
if we plot single- or multiple phase curves

corresponding to different initial conditions
in the same phase plane, then we have a phase

portrait.

To fully understand the behaviour of a dynamical
system, we need to know how a system move

from one position to another and this is described
by trajectories through state space.

A trajectory or path is set of positions in
state space through which system might pass

successively.

Now lets look at a real world dynamical system,
A biological system containing two species

– predators (foxes) and prey (rabbits).

There are literally hundreds of examples predator
prey relations in an ecosystem, predation

is a biological interaction where an organism
that hunting feeds on its prey.

There is a continuous tussle between predators
and their prey and an inverse relationship

between the number of predators and prey.

If we take fox – rabbit as an example then

Rabbits, left on their own, will reproduce
with velocity dr/dt= 100 r, foxes, without

rabbits, will starve and their population
will decline with velocity dw/dt= -50f.

When brought into the same environment, foxes
will catch and eat rabbits.

Loss to the rabbit population will be proportional
to number of foxes f and number of rabbits

r.

The predator–prey model was initially proposed
by Lotka in the theory of autocatalytic chemical

reactions in 1910 and Volterra developed his
model independently from Lotka and used it

to explain d'Ancona's observation his son
in law regarding increasing the fauna population

during world war I in Adriatic.

Lotka–Volterra model are a pair of first-order
nonlinear differential equations, frequently

used to describe the dynamics of biological
systems in which two species interact, one

as a predator and the other as prey.

The populations change through time according
to the pair of equations you see here:

One of the motion you might encounter and
we would like to model here is periodic motion,

for example, the motion of the planets around
the sun is periodic.

This type of periodic motion is very predictable
and we can predict far out into the future

and way back into the past when eclipses happen.

In these systems small disturbances are often
rectified and do not increase to alter the

systems trajectory very much in the long run.

The change in traffic lights or particle motion,
under a force, linear in the displacement

is are also example of periodic motion.

A differential equation of motion, usually
identified as some physical law and applying

definitions of physical quantities, is used
to set up an equation for the problem.

Solving the differential equation will lead
to a general solution with arbitrary constants,

which results in to a family of solutions.

A particular solution can be obtained by setting
the initial values, which fixes the values

of the constants.

21: now if we solve the equation for Particle
motion, under a force then we can see that

the motion is simple harmonic in each of the
two dimensions.

Both oscillations have the same frequency
but (in general) different amplitudes.

The amplitudes A, B, & the phases α, β are
determined by initial conditions.

So different initial conditions gives different
solution.

22: If we

want to plot phase portrait for this system
we need to find all of its paths.

An equation for path is obtained by eliminating
t in the solution.

And trick for this system is defining new
parameter δ [Symbol] α – β then we can

see that except for special cases, the general
path is an ellipse like!

Therefore, the phase portrait for this system
is a family of ellipses, each of which is

a separate phase curve for different initial
conditions.

Studying such diagrams gives insight into
the physics of the particle motion.

As homework, you can try to create phase portrait
of this oscillator in Mathematica.