## An analysis of truncated fractals in the stability domains of transiently forced pendulum systems |

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2-tuples 89 Figure attractor behavior chaos chaotic Chapter computed Consider continuous time systems cosh(x cusps damping denoted differential equation dimension dimensional slice driving term dynamical system equilibrium point ergodic theory euclidean explicit solutions flow folding back forced systems forcing function formulated func global bifurcation gradient systems grid of initial hand side hence homoclinic orbit horse-shoe hyperbolic hyperbolic set implies increments infinite number initial condition set integration intersect invariant iterate limit cycle linear Lyapunov exponents Lyapunov function Melnikov function negative oo cosh(t pendulum system period doubling phase space Poincare map Poincare section power systems probability density saddle Section 4.6 self-affinity self-similar simulations singularities sinusoidally forced smooth stability boundary stability domains stable and unstable stable manifold stroboscopic swing equations three body problem tiling time-varying time-varying system tion trajectories TRANSIENTLY FORCED PENDULUM truncated fractal truncated-fractal Type 1 UEPs unstable fixed point V-function values vector field Wronskian xTf(x zero-crossings Zubov boundary