linear stability analysis

You have found a fixed point; now you want to know whether it is a valley or a peak, whether a small nudge dies away or grows. Linear stability analysis answers this by zooming in so close to the fixed point that the curved flow looks straight, then reading off whether the straightened flow points inward or outward.

Write the state as x = x* + eta, where eta is a small perturbation. Substituting into dx/dt = f(x) and keeping only the term linear in eta gives d(eta)/dt = J eta, where J is the Jacobian matrix, J_ij = partial f_i / partial x_j evaluated at x*. The perturbation therefore grows or decays like a sum of exponentials e^(lambda t), where the lambda are the eigenvalues of J. The fixed point is (linearly) stable if every eigenvalue has negative real part, and unstable if any eigenvalue has positive real part. In one dimension this collapses to a single number: stable if f'(x*) < 0, unstable if f'(x*) > 0.

The method is powerful but local and linear: it describes only the immediate neighborhood, and only to first order. When an eigenvalue has exactly zero real part the linear terms cancel and the test is inconclusive; the fate of the fixed point is then decided by the nonlinear terms you threw away, and this borderline case is precisely where bifurcations happen. Linear stability tells you nothing about behavior far from the fixed point, such as whether a distant limit cycle exists.