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Heteroclinic channels are ensembles of trajectories that can connect saddle equilibrium points in phase space. Dynamical systems and their associated phase spaces can be used to describe natural phenomena in mathematical terms; heteroclinic channels, and the cycles that they produce, are features in phase space that can be designed to occupy specific locations in that space. Heteroclinic channels move trajectories from one equilibrium point to another. More formally, a heteroclinic channel is a region in phase space in which nearby trajectories are drawn closer and closer to one unique limiting trajectory, the heteroclinic orbit. Equilibria connected by heteroclinic trajectories form heteroclinic cycles and cycles can be connected to form heteroclinic networks. Heteroclinic cycles and networks naturally appear in a number of applications, such as fluid dynamics, population dynamics, and neural dynamics. In addition, dynamical systems are often used as methods for robotic control. In particular, for robotic control, the equilibrium points can correspond to robotic states, and the heteroclinic channels can provide smooth methods for switching from state to state.
References
- Kirk, V; Silber, M (1994-11-01). "A competition between heteroclinic cycles". Nonlinearity. IOP Publishing. 7 (6): 1605–1621. Bibcode:1994Nonli...7.1605K. doi:10.1088/0951-7715/7/6/005. ISSN 0951-7715. S2CID 250772340.
- Rabinovich, M.; Volkovskii, A.; Lecanda, P.; Huerta, R.; Abarbanel, H. D. I.; Laurent, G. (2001-07-20). "Dynamical Encoding by Networks of Competing Neuron Groups: Winnerless Competition" (PDF). Physical Review Letters. American Physical Society (APS). 87 (6): 068102. Bibcode:2001PhRvL..87f8102R. doi:10.1103/physrevlett.87.068102. ISSN 0031-9007. PMID 11497865.
- Rabinovich, Mikhail I.; Afraimovich, Valentin S.; Varona, Pablo (2010). "Heteroclinic binding". Dynamical Systems. Informa UK Limited. 25 (3): 433–442. doi:10.1080/14689367.2010.515396. ISSN 1468-9367. S2CID 11334340.
- Ashwin, Peter; Lavric, Aureliu (2010). "A low-dimensional model of binocular rivalry using winnerless competition". Physica D: Nonlinear Phenomena. Elsevier BV. 239 (9): 529–536. Bibcode:2010PhyD..239..529A. doi:10.1016/j.physd.2009.06.018. hdl: 10871/14537 . ISSN 0167-2789.
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