Dynamical Systems in Neuroscience:

Two-Dimensional Systems 107is called the Jacobian matrix of the system (4.5, 4.6) at the equilibrium (x 0 , y 0 ). Forexample, the Jacobian matrix of the system (4.3, 4.4) at the origin is( ) 0 −1. (4.9)−1 0It is important to remember that Jacobian matrices are defined for equilibria, andthat a nonlinear system can have many equilibria and hence many different Jacobianmatrices.4.2.3 Eigenvalues and eigenvectorsA non-zero vector v ∈ R 2 is said to be an eigenvector of the matrix L correspondingto the eigenvalue λ ifLv = λv (matrix notation) .For example, the matrix (4.9) has two eigenvectors( ) 1v 1 =and v12 =( 1−1corresponding to the eigenvalues λ 1 = −1 and λ 2 = 1, respectively. Any textbook onlinear algebra explains how to find eigenvectors and eigenvalues of an arbitrary matrix.It is important for the reader to get comfortable with these notions, since they are usedextensively in the rest of the book.Eigenvalues play important role in analysis of stability of equilibria. To find theeigenvalues of a 2×2-matrix L, one solves the characteristic equation( )a − λ bdet= 0 .c d − λThis equation can be written in the polynomial form (a − λ)(d − λ) − bc = 0 orwhereλ 2 − τλ + ∆ = 0 ,τ = tr L = a + d and ∆ = det L = ad − bcare the trace and the determinant of the matrix L, respectively.polynomial has two solutions of the form)Such a quadraticλ 1 = τ + √ τ 2 − 4∆2andλ 2 = τ − √ τ 2 − 4∆2(4.10)and they are either real (when τ 2 −4∆ ≥ 0) or complex-conjugate (when τ 2 −4∆ < 0).What can you say about the case τ 2 = 4∆?