Complex dynamics in the stretch-twist-fold flow. (English) Zbl 1204.37083
Summary: The system associated with fluid particle motions of the stretch-twist-fold (STF) flow has displayed rich and attractive dynamic properties. Detailed research on the system has been done in this work. By using a high-dimensional generalization of the Melnikov method, the explicit parametric conditions for the existence of periodic solutions in the system can be determined. Then, by using the new-KAM-like theorems for perturbations of a three-dimensional generalized Hamiltonian system, the criteria for the existence of invariant tori in the STF flow have been obtained. In addition, one new first integral is found. On the basis of it, nonexistence of chaos in the system at \(\alpha =0\) is rigorously proved. Nonexistence of homoclinic orbits is also proved in the system if some conditions hold. And an interesting phenomenon is found, where the unit circle in the \((y,z)\)-plane is filled with heteroclinic orbits of the system at \(\alpha =0\). The system with \(\alpha =0\) is also successfully reduced to a generalized Hamiltonian system, and further transformed to slowly varying oscillators.
MSC:
| 37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
| 76R99 | Diffusion and convection |
References:
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