Linear and nonlinear response of discrete dynamical systems

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We carry out a linear response theory for discrete dynanmical systems with periodic attractors. The symmetry properties of the susceptibility matrix are studied and its eigenvalues and eigenvectors are determined. Close to a period-doubling bifurcation where the susceptibility diverges, its half-width is related to the Lyapunov exponent. At the transition to chaos the susceptibility has some universal behaviour which is described by a critical exponent κ=1−(ln2/lnδ)=0.550193... At the bifurcation points where linear response theory becomes insufficient we also determine the nonlinear response.