Albert Luo

Southern Illinois University Edwardsville

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Infinity-large eigenvalues implies homoclinic orbits in the Lorenz systems

Albert Luo and Siyu Guo, Southern Illinois University EDwardsville

This paper presents how to determine infinite-many homoclinic orbits in the Lorenz system. Through this study, infinite-large eigenvalues implies homoclinic-orbits, and the corresponding unstable periodic motions induce the corresponding chaos. The bifurcation trees of (n1,1,n2)-periodic motion to chaos are presented as an example through discrete nodes and harmonic amplitudes. The homoclinic orbits are associated with unstable periodic motions on the bifurcation trees of the (n1,1,n2)-periodic motions to chaos. The homoclinic obits and periodic motions are illustrated from the bifurcation trees of the (n1,1,n2)-periodic motions to chaos. The numerical and analytical trajectories of unstable periodic motions were presented for comparison. If the numerical simulations did not have any computational errors, the numerical and analytical solutions of unstable periodic motions in the Lorenz system should be identical. Thus, one observed so-called strange attractors in the Lorenz system through numerical simulations, which are not real strange attractors.