Three-frequency quasiperiodicity, torus breakup, and multiple coexisting attractors in a higher dimensional dissipative dynamical system

The occurrence of quasiperiodic orbits in typical dissipative dynamical systems is of great importance. The equations for two driven coupled Van der Pol oscillators were integrated, and four Lyapunov exponents were calculated for every orbit. The results show that for small coupling and driving parameters, most orbits are three-frequency quasiperiodic. As the coupling and driving parameters are increased, three-frequency quasiperiodicity becomes less common, being first replaced by two-frequency quasiperiodicity, and then by periodic and chaotic motions. This example reinforces past work suggesting that three-frequency quasiperiodic attractors are common in typical dynamical systems.;Three-frequency quasiperiodic attractors lie on a 3-torus. The evidence presented here indicates that the torus is destroyed when the stable and unstable manifolds of an unstable orbit become tangent. Furthermore, no chaotic orbits lying on a torus were observed, suggesting that, in most cases, at least in the case of this system, orbits do not become chaotic before their tori are destroyed. To expedite the calculations, a method was developed, which can be used to determine if an orbit is on a torus, without actually displaying that orbit. The method was designed specifically for this system. The basic idea, however, could be used for studying attractors of other systems. Very few modifications of the method, if any, would be necessary when studying systems with the number of degrees of freedom equal to that of the Van der Pol system.;Several cases where multiple attractors coexist were studied. It was observed that the number of attractors tends to increase as one increases the value of the system's Jacobian. It was also observed that multiple coexisting attractors are often periodic. Several plots of periodic attractors' basins of attraction are shown, and the structures of their basin boundaries are discussed. Although those boundaries are not fractal, they are very finely intertwined even in the simplest cases with only two attractors. An example where periodic and chaotic attractors coexist is presented. In that case, orbits which asymptote to periodic attractors, have unusually long chaotic transients, suggesting the existence of fractal basin boundaries. This, in turn, suggests the existence of at least one chaotic basic set. One such set is isolated and studied.