To plot the diagram of a Hopf bifurcation for a three-dimensional nonlinear fractional-order system in MATLAB, you can use the following steps:

1. Define the fractional-order system: Define the fractional-order system as a set of differential equations using the Caputo fractional derivative operator. For example, a three-dimensional nonlinear fractional-order system can be defined as follows:

```matlab

function dx = fractional_system(t, x, a, b, c, alpha, beta, gamma)

dx = zeros(3,1);

dx(1) = ((a - x(1)) - b*x(2)^alpha)/(1 + abs(x(2))^beta);

dx(2) = (x(1) - c*x(2) + gamma*x(3)^alpha)/(1 + abs(x(3))^beta);

dx(3) = -x(2)^alpha;

end

```

2. Calculate the equilibrium point: Calculate the equilibrium point of the system by setting the derivatives equal to zero. The equilibrium point is (a, 0, 0) for the system defined above.

3. Compute the Jacobian matrix: Compute the Jacobian matrix of the system evaluated at the equilibrium point. This can be done using the symbolic toolbox in MATLAB or by hand. The Jacobian matrix for the system defined above is:

```matlab

syms x1 x2 x3

J = jacobian([(a - x1 - b*x2^alpha)/(1 + abs(x2)^beta);...

(x1 - c*x2 + gamma*x3^alpha)/(1 + abs(x3)^beta);...

-x2^alpha], [x1, x2, x3]);

J_eq = double(subs(J, [x1 x2 x3], [a 0 0]));

```

4. Compute the eigenvalues of the Jacobian matrix: Compute the eigenvalues of the Jacobian matrix evaluated at the equilibrium point. The eigenvalues will determine the stability of the equilibrium point and whether a Hopf bifurcation occurs. You can use the `eig` function in MATLAB to compute the eigenvalues:

```matlab

eig_vals = eig(J_eq);

```

5. Determine the critical parameter value: The critical parameter value for the Hopf bifurcation can be determined by finding the value of the parameter that causes the imaginary part of the eigenvalues to go from negative to positive. This can be done by varying the parameter value and computing the eigenvalues at each point. The critical parameter value is the point at which the imaginary part of the eigenvalues crosses zero. For example, if the parameter `b` is the critical parameter, you can vary its value and compute the eigenvalues as follows:

```matlab

b_values = linspace(0, 2, 100);

eig_vals_imag = zeros(size(b_values));

for i = 1:length(b_values)

b = b_values(i);

J = jacobian([(a - x1 - b*x2^alpha)/(1 + abs(x2)^beta);...

(x1 - c*x2 + gamma*x3^alpha)/(1 + abs(x3)^beta);...

-x2^alpha], [x1, x2, x3]);

J_eq = double(subs(J, [x1 x2 x3], [a 0 0]));

eig_vals = eig(J_eq);

eig_vals_imag(i) = imag(eig_vals(1));

end

```

6. Plot the Hopf bifurcation diagram: Plot the imaginary part of the eigenvalues as a function of the critical parameter value. The Hopf bifurcation occurs when the imaginary part of the eigenvalues crosses zero. You can use the `plot` function in MATLAB to create the plot:

```matlab

plot(b_values, eig_vals_imag)

xlabel('b')

ylabel('Imaginary part of eigenvalues')

```

This will plot the Hopf bifurcation diagram for the three-dimensional nonlinear fractional-order system as a function of the parameter `b`. You can modify the code to plot the diagram as a function of a different parameter if desired.

Credit: mainly AI Tools

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