How to find eigen values of Fischer's equation?

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?

[Torsten]

You forgot to attach the file.

[simran]

Sorry, I've attached it now.

[Sam Chak]

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]

What is the solution to this state space system though?

[Sam Chak]

@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]

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]

Has the system in your problem been considered solved now?

[simran]

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

[simran]

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

[Sam Chak]

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]

But what are the eigen values of this ode?

[simran]

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]

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]

@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 .

x   = linspace(0, 1, 101);      % range of x
f   = @(x) x.*(1 - x).*x;       % nonlinear function
xop = 1/3;                      % operating point
Lin = 1/3*(x - xop) + f(xop);   % linear approximation around xop
plot(x, f(x), x, Lin), grid on
title('Linear approximation of f(\phi) at \phi = 1/3')
xlabel('\phi'), legend('Nonlinear', 'Linear', 'location', 'northwest', 'fontsize', 14)