Non-linear Dynamics: A Primer by Marji Lines Alfredo Medio

By Marji Lines Alfredo Medio

A sophisticated undergraduate and graduate textbook at the idea of nonlinear dynamical platforms.

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If det(A) = 0, the unique equilibrium, for which x˙ = y˙ = 0, is x = y = 0. The characteristic equation is 0 = det a11 − λ a21 a12 a22 − λ = λ2 − (a11 + a22 ) λ + (a11 a22 − a12 a21 ) = λ2 − tr(A) λ + det(A). 3 Continuous systems in the plane 35 where ∆ ≡ [tr(A)]2 − 4 det(A) is called the discriminant. 29) the discriminant is ∆ = (a11 + a22 )2 − 4(a11 a22 − a12 a21 ) = (a11 − a22 )2 + 4a12 a21 . 29) can be described in terms of the two eigenvalues of the matrix A, which in planar systems can be completely characterised by the trace and determinant of A.

The solution corresponding to the zero eigenvalue is a constant (c1 eλ1 t = c1 ) and the dynamics are determined by the sign of tr(A) as for cases (i) or (ii). In this case, however, there is no longer a unique equilibrium, but a continuum of equilibria whose locus is the straight line through the origin defined by y/x = −(a11 /a12 ) = −(a21 /a22 ). 36 Review of linear systems y y x (a) x (b) y y x (c) x (d) y y x (e) x (f) Fig. 1 Equilibrium types in the plane equilibrium is called a stable node, the term ‘node’ referring to the characteristic shape of the ensemble of orbits around the equilibrium.

If that eigenvalue or eigenvalue pair is repeated, the equilibrium is unstable. As for continuous-time systems, there are criteria that can be applied to the coefficient matrix B to ensure stability. In this case, stability conditions guarantee that all eigenvalues have moduli inside the unit circle of the complex plane. 5). We proceed as we did for the continuous-time case. 37). We can verify that Xn = B n X 0 where Xn = X(n) and X0 = X(0). Then xn = Xn c = B n x0 where c = (c1 , c2 , . . 39) and xn = x(n), x0 = x(0).

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