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.