We consider first a deterministic dynamical system governed by the ordinary differential equation x = f(x,#alpha#), x(0) = x0 in not an element of R~d, (1) where f(x, #alpha#) is a continuously differentiable function with respect to its arguments. Suppose the equation possesses a steady-state solution x_s(t, #alpha#). As the parameter #alpha# is varied it is possible that this solution may become unstable at some value #alpha#_c and a further bifucating solution emerges for #alpha# in the vicinity of #alpha#_c. The stability of the original solution and of the bifurcating solution is usually examined by investigating the stability of the trivial solution of the linearisation of equation (1) around each of these slutions. The standard procedure for this purpose is the evaluation of the maximal Lyapunov exponent which determines the exponential rate of growth or decay of some norm of the solution; the sign of the maximal Lyapunov exponent determines the stability of the trivial solution.
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