Fixed Points and Stability

A fixed point (or equilibrium) of a dynamical system is a state that does not change over time. Understanding their stability tells us whether the system returns to equilibrium after a small perturbation.

1D Systems

For $\dot{x} = f(x)$, a fixed point $x^*$ satisfies $f(x^*) = 0$.

Stability criterion:

• $f'(x^*) < 0$ → stable (perturbations decay)
• $f'(x^*) > 0$ → unstable (perturbations grow)

We estimate the derivative numerically using the central difference:

$f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}$

2D Systems

For $\dot{x} = f(x, y)$ and $\dot{y} = g(x, y)$, stability is determined by the Jacobian matrix at the fixed point:

$J = \begin{pmatrix} \partial f/\partial x & \partial f/\partial y \\ \partial g/\partial x & \partial g/\partial y \end{pmatrix}$

The eigenvalues of $J$ determine the behaviour. From the characteristic polynomial $\lambda^2 - \tau\lambda + \Delta = 0$ (where $\tau = J_{11}+J_{22}$ is the trace and $\Delta = J_{11}J_{22} - J_{12}J_{21}$ is the determinant):

$\lambda_{1,2} = \frac{\tau \pm \sqrt{\tau^2 - 4\Delta}}{2}$

Your Task

Implement:

• numerical_derivative(f_func, x, h=1e-6) — central difference approximation of $f'(x)$
• is_stable_1d(f_func, x_star) — returns True if the fixed point is stable
• jacobian_eigenvalues(J11, J12, J21, J22) — returns (lambda1, lambda2) as real floats (the real parts, sorted descending); for complex eigenvalues return the real part $\tau/2$