Our research is primarily in applications of contemporary developments in dynamical systems and nonlinear dynamics to problems of physical interest in fluid and plasma physics. The particular problems we are working on include:
Nonlinear stability of fluid flows in geophysical settings. We have developed a method for separating modes of an inviscid flow with significantly different time scales while retaining the Hamiltonian structure of the dynamics. This has been applied to the description of quasi-geostrophic flows in the ocean and atmosphere. This method also promises the ability to analyze the time scales in which nonlinear instabilities develop in physical systems. We are now applying this to the description of other flows with separated time scales such as waves on interfaces and wave motions internal to stratified fluids.
Using notions of nonlinear dynamics, we are investigating the manner in which friction is exhibited by parts of a Hamiltonian system which develop on separated time scales. Characterizing the flow of information from one subsystem to another by the Liapunov exponents natural to chaotic motions and by correlation functions in phase space, we are able to determine on what time scales a system may look dissipative and possess strange attractors even though it is part of an overall Hamiltonian system.
We are also beginning to investigate the spatial and temporal scales associated with nonlinear instabilities in a continuum system such as a fluid or plasma. This is an entirely new direction in the study on nonlinear instability and promises to have direct implications on experiments done here at UCSD and elsewhere which seek characteristics of these instabilities.
Dynamical estimation of neuron and network properties II: Path integral Monte Carlo methods. Kostuk M, Toth BA, Meliza CD, Margoliash D, Abarbanel HD. Biol Cybern. 2012 Mar;106(3):155-67. doi: 10.1007/s00422-012-0487-5. Epub 2012 Apr 13. PMID: 22526358
Dynamic state and parameter estimation applied to neuromorphic systems.Neftci EO, Toth B, Indiveri G, Abarbanel HD. Neural Comput. 2012 Jul;24(7):1669-94. doi: 10.1162/NECO_a_00293. Epub 2012 Mar 19. PMID: 22428591
Dynamical estimation of neuron and network properties I: variational methods. Toth BA, Kostuk M, Meliza CD, Margoliash D, Abarbanel HD. Biol Cybern. 2011 Oct;105(3-4):217-37. doi: 10.1007/s00422-011-0459-1. Epub 2011 Oct 11. PMID: 21986979
Parameter and state estimation of experimental chaotic systems using synchronization. Quinn JC, Bryant PH, Creveling DR, Klein SR, Abarbanel HD. Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jul;80(1 Pt 2):016201. Epub 2009 Jul 6. PMID: 19658786
Brain stem feedback in a computational model of birdsong sequencing. Gibb L, Gentner TQ, Abarbanel HD. J Neurophysiol. 2009 Sep;102(3):1763-78. doi: 10.1152/jn.91154.2008. Epub 2009 Jun 24. PMID: 19553477
Inhibition and recurrent excitation in a computational model of sparse bursting in song nucleus HVC. Gibb L, Gentner TQ, Abarbanel HD. J Neurophysiol. 2009 Sep;102(3):1748-62. doi: 10.1152/jn.00670.2007. Epub 2009 Jun 10. PMID: 19515949