Dynamical Systems in Neuroscience:

108 Two-Dimensional SystemsIn general, 2 × 2-matrices have two eigenvalues with distinct (independent) eigenvectors.In this case a general solution of the linear system has the form( ) u(t)= cw(t) 1 e λ1t v 1 + c 2 e λ2t v 2 ,where c 1 and c 2 are constants that depend on the initial condition. This formula is validfor real and complex-conjugate eigenvalues. When both eigenvalues are negative (orhave negative real parts), u(t) → 0 and w(t) → 0, meaning x(t) → x 0 and y(t) → y 0 ,so that the equilibrium (x 0 , y 0 ) is exponentially (and hence asymptotically) stable. It isunstable when at least one eigenvalue is positive or has a positive real part. We denotestable equilibria as filled circles • and unstable equilibria as open circles ◦ throughoutthe book.4.2.4 Local equivalenceAn equilibrium whose Jacobian matrix does not have zero eigenvalues or eigenvalueswith zero real part is called hyperbolic. Such an equilibrium can be stable or unstable.The Hartman-Grobman theorem states that the vector-field and hence the dynamics ofa nonlinear system, e.g., (4.5, 4.6) near such a hyperbolic equilibrium is topologicallyequivalent to that of its linearization (4.7, 4.8). That is, the higher-order terms that areneglected when (4.5, 4.6) is substituted by (4.7, 4.8) do not play any qualitative role.Thus, understanding and classifying the geometry of vector-fields of linear systemsprovides an exhaustive description of all possible behaviors of nonlinear systems nearhyperbolic equilibria.A zero eigenvalue (or eigenvalues with zero real parts) arise when the equilibriumundergoes a bifurcation, e.g., as in Fig. 4.14b, and such equilibria are called nonhyperbolic.Linear analysis cannot answer the question of stability of a nonlinearsystem in this case, since small nonlinear (high-order) terms play a crucial role here.We denote equilibria undergoing a bifurcation as half-filled circles, e.g., .4.2.5 Classification of equilibriaBesides defining the stability of an equilibrium, the eigenvalues also define the geometryof the vector field near the equilibrium, as we illustrate in Fig. 4.15 and ask the readerto prove in Ex. 4. (The proof is a straightforward consequence of (4.10)). There arethree major types of equilibria:Node (Fig. 4.16): The eigenvalues are real and of the same sign. The node is stablewhen the eigenvalues are negative, and unstable when they are positive. Thetrajectories tend to converge to or diverge from the node along the eigenvectorcorresponding to the eigenvalue having the smallest absolute value.Saddle (Fig. 4.17): The eigenvalues are real and of opposite signs. Saddles are alwaysunstable, since one of the eigenvalues is always positive. Most trajectories