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algebraic algorithm amplitude analysis Applied Mathematics assume assumption asymptotic expansion bifurcation theory boundary conditions bounded calculated cluster coefficients compute consider constant convex convolutional code Corollary corresponding curve defined denote density depends derived determined deterministic differential equations dominating set dynamic edge eigenvalues equilibrium solution equivalent example exists exponentially finite fixed-point property flame function given graph Hence Hopf bifurcation implicit function theorem implies initial integral invariant Largest Remainder Method leading order Lemma linear lower quota Math matrix node nonlinear nonnegative null space obtain operator optimal ordinary differential equations oscillations p-center p-median p-radius parabolic parameter periodic solution perturbation phase phase plane polynomial population positive recursions resonance secondary bifurcation sequence space stability stochastic subtree sufficiently small theory Toeplitz operators transition tree unique University unstable value problem variable vector velocity vertex vertex-unweighted vertices zero