Determine the stability of a fixed point of oscillations

[peadar2211]

Homework Statement

I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the stability of the fixed point of each equation analytically and compare them to the numerically determined values. The exact wording is:



The steady state solution for the rate equations may be termed a fixed point(x*) since x_n+1=f(x_n) =x_n or x*=f(x*) where f(x) represents the left side of the expression for the Euler algorithm.
To determine the stability of a fixed point we let x_n=x*+e_n and x_n+1=x*+e_n+1 where |e_i|<<1. Using a Taylor expansion this can be written as x*+e_nf'(x*). Substituting for x_n+1 we get that the derivative f'(x*)=e_n+1/e_n. If |f'(x*)|>1 the trajectory will diverge from x* since e_n+1>e_n. The opposite is true for |f'(x*)|<1 and the fixed point is stable. Investigate the stability of the fixed point analytically and compare with your numerical results.

Homework Equations

Euler's Method: x_n+1=x_n+Δt*dx_n/dt
The differential equations I solved, although I don't think they're relevant to this part are
X'=A-(B+1)X+X²Y and Y'=BX-X²Y

The Attempt at a Solution

I know that I need to investigate f'(x*) so I've tried differentiating Euler's method with respect to x_n, but then I have 1+(something) where the something is either really complicated because it would be d/dx(dx/dt)Δt, or it would be 0. Either way, it doesn't seem like something I could use to either graph or solve analytically to see when it has a value above 1 (As it is constant in one of those cases).

 

[epenguin]

As far as I can make out you have a problem with just two variables, X and Y, and are quoting a text elaborating the formulation for n variables, x_n.

IMO you would do better to just considered two for now.

You have been able to solve the differential equations, exceptional situation, lucky you! However even without solving them you can quite easily obtain what the steady state is. Or are, because in a non-linear case there may be more than one. In this case?

When you know a stationary state the next thing Is to see how the differential equation looks like just near it - in first place what are signs of X' and Y' near the s.p.? Helpful to sketch the isoclines (X' = 0 and Y' = 0) and put in little arrows showing which way sample points in each sector are moving.

 

[epenguin]

When student posts a problem in his first and only post and then never comes back even to the site in over a week we can count the problem as solved? - what is the practice here?