Dynamical Systems in Neuroscience:

110 Two-Dimensional Systemsv 2 v 2v 1v 1stable nodeunstable nodeFigure 4.16: Node equilibrium occurs when both eigenvalues are real and have the samesign, e.g., λ 1 = −1 and λ 2 = −3 (stable) or λ 1 = +1 and λ 2 = +3 (unstable). Mosttrajectories converge to or diverge from the node along the eigenvector v 1 correspondingto the eigenvalue having the smallest absolute value.v 2v 2v 1v 1saddlesaddleFigure 4.17: Saddle equilibrium occurs when two real eigenvalues have opposite signs,e.g., λ 1 = +1 and λ 2 = −1. Most trajectories diverge from the equilibrium along theeigenvector corresponding to the positive eigenvalue (in this case v 1 ).stable focusunstable focusFigure 4.18: Focus equilibrium occurs when the eigenvalues are complex-conjugate,e.g., λ = −3 ± i (stable) or λ = +3 ± i (unstable). The imaginary part (here 1)determines the frequency of rotation around the focus.