Determine the stability of a fixed point of oscillations

In summary, the student has been able to solve the differential equations and has determined a stationary state. He is trying to determine which points are near the stationary state and sketching the isoclines.
  • #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 xn+1=f(xn) =xn 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 xn=x*+en and xn+1=x*+en+1 where |ei|<<1. Using a Taylor expansion this can be written as x*+enf'(x*). Substituting for xn+1 we get that the derivative f'(x*)=en+1/en. If |f'(x*)|>1 the trajectory will diverge from x* since en+1>en. 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: xn+1=xn+Δt*dxn/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 xn, 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).
 
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  • #2
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, xn.

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.
 
Last edited:
  • #3
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?
 

Related to Determine the stability of a fixed point of oscillations

1. What is a fixed point of oscillations?

A fixed point of oscillations is a point in a system where the oscillating motion remains constant. This means that the system does not move away from this point over time.

2. How do you determine the stability of a fixed point?

The stability of a fixed point can be determined by analyzing the behavior of the system near the fixed point. If the system returns to the fixed point after being slightly perturbed, it is considered stable. If the system moves away from the fixed point, it is considered unstable.

3. What factors affect the stability of a fixed point?

The stability of a fixed point can be affected by various factors such as the system's initial conditions, the parameters of the system, and external forces. These factors can change the behavior of the system and its tendency to return to the fixed point.

4. Why is it important to determine the stability of a fixed point?

Determining the stability of a fixed point is important in understanding the behavior of a system and predicting its future movements. It also helps in designing and controlling systems to ensure that they remain in a stable state.

5. What are some methods for determining the stability of a fixed point?

There are various methods for determining the stability of a fixed point, such as linearization, phase plane analysis, and Lyapunov stability analysis. These methods involve mathematical calculations and analysis of the system's equations to determine the stability of the fixed point.

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