Matching of separatrix map and resonant dynamics, with application to global chaos onset between separatrices S. M. Soskin, 1,2, * R. Mannella, 3 and O. M. Yevtushenko 2,4 1 Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 03028 Kiev, Ukraine 2 Abdus Salam ICTP, 34100 Trieste, Italy 3 Dipartimento di Fisica, Università di Pisa, 56127 Pisa, Italy 4 Physics Department, Ludwig-Maximilians-Universität, München, D-80333 München, Germany Received 12 December 2006; revised manuscript received 10 October 2007; published 26 March 2008 We have developed a general method for the description of separatrix chaos, based on the analysis of the separatrix map dynamics. Matching it with the resonant Hamiltonian analysis, we show that, for a given amplitude of perturbation, the maximum width of the chaotic layer in energy may be much larger than it was assumed before. We use the above method to explain the drastic facilitation of global chaos onset in time- periodically perturbed Hamiltonian systems possessing two or more separatrices, previously discovered S. M. Soskin, O. M. Yevtushenko, and R. Mannella, Phys. Rev. Lett. 90, 174101 2003. The theory well agrees with simulations. We also discuss generalizations and applications. The method may be generalized for single- separatrix cases. The facilitation of global chaos onset may be relevant to a variety of systems, e.g., optical lattices, magnetic and semiconductor superlattices, meandering flows in the ocean, and spinning pendulums. Apart from dynamical transport, it may facilitate noise-induced transitions and the stochastic web formation. DOI: 10.1103/PhysRevE.77.036221 PACS numbers: 05.45.Ac, 05.45.Pq I. INTRODUCTION A weak perturbation of a Hamiltonian system causes the onset of chaotic layers around separatrices of the unperturbed system and/or separatrices surrounding nonlinear resonances generated by the perturbation 1–5. The system may be transported along the layer in a randomlike fashion and this chaotic transport plays an important role in many physical phenomena 3–5. If the perturbation is sufficiently weak, then the layers are thin and the chaos is called local 1–4. As the perturbation magnitude increases, the width of the layer grows and the layers corresponding to adjacent separa- trices reconnect at some, typically nonsmall, critical value of the perturbation. This conventionally marks the onset of glo- bal chaos 1–4, i.e., chaos in a large region of the phase space, with chaotic transport throughout the whole relevant energy range. The reconnection of the layers around separatrices of the resonances often correlates with the overlap in energy be- tween neighboring resonances calculated independently in the resonant approximation. The latter constitutes the heuris- tic Chirikov resonance-overlap criterion 1–4. But the Chir- ikov criterion may fail if the system is of the zero-dispersion ZD type 6, i.e., if the frequency of eigenoscillations pos- sesses a local maximum or minimum as a function of its energy cf. also studies of related maps 7,8 which are called nontwist, twistless, or nonmonotonic twist maps. In such systems, there are typically two resonances of one and the same order 9, and their overlap in energy does not result in the onset of global chaos 6–8. Even their overlap in phase space 10 results typically only in the reconnection of the thin chaotic layers associated with the resonances. As the amplitude of the time-periodic perturbation grows further, the layers may separate again 6–8. An example of the evo- lution of resonances in the plane of energy and slow angle is given in Fig. 1 the typical evolution of a real Poincaré sec- tion is shown, e.g., in 11. * Also at Physics Department, Lancaster University, UK. -3π -2π -π 0 π 2π 3π -3π -2π -π 0 π 2π 3π -3π -2π -π 0 π 2π 3π Slow angle Energy Energy Energy (a) (b) (c) FIG. 1. Typical evolution of thin chaotic layers in the plane of slow variables of a zero-dispersion system the perturbation magni- tude grows from the top to the bottom. PHYSICAL REVIEW E 77, 036221 2008 1539-3755/2008/773/03622129 ©2008 The American Physical Society 036221-1