Periodic orbits, invariant tori and chaotic behavior in certain nonequilibrium quadratic three-dimensional differential systems
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In (Jafari et al, Phys Lett A 377(9):699-702, 2013) the authors gave the expressions of seventeen classes of quadratic differential systems defined in R3, depending on one real parameter a, which present chaotic behavior even without having any equilibrium point, for suitable choices of the parameter a > 0. In that paper, such systems are denoted by NE1 to NE17. As these systems have no equilibrium points, a natural question arises: how chaotic motion is generated in their nonequilibrium phase spaces? In this note we combine analytical and numerical results in order to study the integrability and dynamics of systems NE1, NE6, NE8 and NE9 among those listed in Jafari et al. (2013). We show that they exhibit a quite similar dynamical behavior and, consequently, the mechanisms for birth of chaos in these systems are similar. In this way, we intend to give at least a partial answer to the above question and contribute to better understand the complicated dynamics of the considered systems, in particular concerning the existence of periodic orbits and invariant tori and the emergence of chaotic behavior. The periodic orbits are studied using the Averaging Theory while the invariant tori are proved to exist via KAM Theorem. The chaotic dynamics arises from the broken of some of these invariant tori.