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How to find eigen values of Fischer's equation?

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simran 2024-4-30

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评论： Sam Chak 2024-5-1

Just like we have eigen values for heat equation as lambda=n*pi/l type. How can we find eigen values for the Fischer's equation. I'm attaching the file in which I've attempted to do so? But I'm not sure if it's correct?

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Torsten 2024-4-30

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You forgot to attach the file.

simran 2024-4-30

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Sorry, I've attached it now.

Sam Chak 2024-4-30

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I'm unfamiliar with the Fisher equation, but I recognize that the following state-space system

is a 2nd-order nonlinear differential equation

Where does the Fisher equation fit in this system?

simran 2024-4-30

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What is the solution to this state space system though?

Sam Chak 2024-4-30

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Ran in:

@simran, here's how you can solve the nonlinear system using the ode45 solver. However, I'm uncertain about finding the eigenvalues for the Fisher equation. Could you please explain the relationship between the Fisher equation and this nonlinear system?

tspan = [0 100]; % simulation time

x0 = [0.999 0]; % initial values

[t, x] = ode45(@odesys, tspan, x0); % call ode45 solver

plot(t, x), grid on, xlabel('t'), legend('\phi', '\psi')

function dxdt = odesys(t, x)

% initialization

dxdt = zeros(2, 1);

% definitions

phi = x(1);

psi = x(2);

% parameter

c = 2;

% differential equations

dxdt(1) = psi;

dxdt(2) = - (1 - phi)*phi^2 - c*psi;

end

simran 2024-5-1

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I have converted Fischer's equation into this system of ode as you can see in the file attached, so that finding eigen values is easier.

Sam Chak 2024-5-1

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Has the system in your problem been considered solved now?

simran 2024-5-1

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Could you explain what you've done in this code? Why have you taken c=2 in this?

simran 2024-5-1

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 I want the eigen values of this state space system of ode.

Sam Chak 2024-5-1

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I searched online for how to solve ODEs and copied the code from the ode45 link. Then, I arbitrarily chose c = 2. However, only you know the true value. What is the actual value of c?

simran 2024-5-1

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But what are the eigen values of this ode?

simran 2024-5-1

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Actually I have tried to convert fishers equation into a system of ode, and I have also found a way to linearize the system into another dynamical system.. So I want to compare that if there is any analogy between the eigen values of the system of ode and the linearised dynamical system.

Torsten 2024-5-1

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I don't know how eigenvalues of the nonlinear Fisher's equation are mathematically defined. Can you show us the equation with the "lambda" in it ?

Sam Chak 2024-5-1

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Ran in:

@simran, Sometimes we get confused when we first start to learn a new material. This is normal. What are the eigenvalues of the linearized dynamical system? Since the nonlinear Fisher equation contains only one nonlinear term, I visualized the linear approximation around a selected operating point at