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Linearizing system of ODEs, Minimizing a functional

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Question 3:

A dynamical system is governed by two equations

x4 y 2x,

yy 8 y 12x.

(a) Show that the critical points of this system are P(0, 0) and Q(1, 2).

(b) Using linearation of the system in the neighbourhood of each critical point, determine the nature of the critical points.

(c) Draw qualitatively and reasonably careful these critical points and corresponding trajectory diagrams.

Question 4:

Minimise the functional
such that y(0) = y(1) = 0 .

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Solution Summary

The solution shows how to linearize a system of first order differential equations, how to find the equilibrium points and how to categorize them.
It then goes on and show how to find the minimal value of a functional, by finding the minimizing integration path.

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1.
The general system of equations is
(1.1)
A critical point, also known as equilibrium point, is a point where the variables and do not change. in other words, it is a point where the derivatives of and with respect to t are zero.
So we can see that at the equilibrium point
(1.2)
Now we expand the functions and into their respective Taylor series keeping only the linear term (hence the "linearization"):
(1.3)
 
But around the equilibrium point we have , so the system (1.1) becomes
(1.4)
Therefore, around the equilibrium point we can write the system in a matrix form:
(1.5)
Where the matrix (the Jacobian) is given by:
(1.6)
where the subscript indicates that the derivatives are evaluated at the equilibrium point.
Such a system has the general solution
(1.7)
Where are the eigenvalues of the Jacobian matrix and are their respective eigenvectors ( are just constants to be determined from initial conditions).
• We see that if and the solution will diverge away from the equilibrium. This is an unstable equilibrium also called a "source" since the field lines ...

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