## Chaos Bifurcations and Fractals Around Us: A Brief IntroductionDuring the last twenty years, a large number of books on nonlinearchaotic dynamics in deterministic dynamical systems haveappeared. These academic tomes are intended for graduate students andrequire a deep knowledge of comprehensive, advanced mathematics. |

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7-periodic analog computer attractors exist ball basin boundary basins of attraction bifurcation curve bifurcation diagram boundary crisis Cantor set chaotic motion chaotic oscillating chaotic phenomena chaotic solution chaotic transient motion coexisting attractors concept control parameter cross-well chaotic attractor damping coefficient displacement driving frequency dynamical systems equation of motion forcing amplitude forcing parameter fractal structure global bifurcation global homoclinic bifurcation heteroclinic bifurcation hilltop saddle initial conditions intersect irregular Large Orbit Lyapunov exponents mathematical Melnikov criterion nondimensional nonlinear dynamics nonlinear oscillations nonresonant attractor Sn number of points obtained oscillating attractors oscillating chaotic attractor parameters F pendulum period doubling period-doubling bifurcations periodic attractor periodic force periodic solution phase plane phase portrait Poincare map potential energy principal resonance Regions of existence resonant attractor S1OR saddle Dn saddle-node bifurcation shown in Figure single-well stable and unstable steady-state oscillations strange attractor time-history transient chaos Ueda unpredictable unstable manifolds unstable solution velocity