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Global transversality and chaos in two-dimensional nonlinear dynamical systems

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Albert Luo, Department of Mechanical and Industrial Engineering, University of Southern Illinois
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The global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function (L-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The first integral quantity increment ( i.e., L-function) is presented to observe the periodicity of flows. In additions, the global tangency of periodic flows in such an oscillator is measured by the G-function and -function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincare mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincare mapping points are presented to observe the complexity of chaos. The first integral quantity increment ( i.e., L-function) of chaotic flows at the Poincare mapping points is non-zero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.