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Tracers spread out along Lagrangian Coherent Structures in a turbulent flow experiment |
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The modern theory of dynamical systems seeks to explain and control nonlinear behavior through invariant manifolds. These manifolds are distinguished surfaces of solutions that have a decisive impact on nearby solutions. Invariant manifolds also interact with each other to form a global skeleton of overall system dynamics. Coherent structures, spiking behavior, chaotic motion, synchronization, turbulent mixing, aerodynamic separation, vortex formation, cell death and other physical phenomena can all be explained through appropriately defined invariant manifolds. This exciting approach, however, faces serious challenges in contemporary problems of applied science. Such problems are often high-dimensional, strongly nonlinear, time-dependent, multi-scale, non-smooth. They are also typically defined through spatially and temporally limited data sets, not equations. The focus of my research is to overcome these obstacles, and use nonlinear dynamical systems to solve complex real-life problems. This requires careful modeling, the analysis of numerical and experimental data, and the development of new mathematics. |
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Flow separation along a non-hyperbolic unstable manifold in a three-dimensional unsteady flow |
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Transition state geometry near a rank-N saddle point in a chemical reaction model |
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Inertial particle instabilities behind a cylinder in cross-flow |
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George Haller’s Nonlinear Dynamics Group |
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Department of Mechanical Engineering, McGill University |
