- #1

peadar2211

- 1

- 0

## 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

_{n}f'(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+X

^{2}Y and Y'=BX-X

^{2}Y

## 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).