
Rare extreme events: Complex irregular behavior is a characteristic of chaotic systems, which is usually visualized through the time series of an observable. Many natural and engineering systems exhibit a second level of complexity typified by intermittent bursts in the time series of certain observables. Examples include rogue waves in the ocean, unusual weather patterns, spikes in neural networks and intermittent energy dissipation in turbulent fluid flows. Prediction of rogue waves: Farazmand and Sapsis, JCP 2017 Precursor for rare, extreme events in highdimensional systems: Farazmand and Sapsis, PRE 2016 Intermittent energy dissipation in turbulent flows: Farazmand, JFM 2016 
Invariant solutions in highdimensional systems: Valuable information about the global behavior of chaotic systems can be achieved through their simple invariant solutions (i.e. equilibria, traveling waves and periodic orbits). In high dimensional systems, the unstable invariant solutions are difficult to compute. We developed an adjointbased method for efficient computation of these solutions. Invariant solutions of NavierStokes and intermittency: Farazmand, JFM 2016 
Lagrangian Coherent Structures:
Chaotic dynamical systems often exhibit islands of regular behavior (elliptic islands) interrupting the surrounding chaotic sea that is dominated by hyperbolicity. A fascinating example of such coherent behavior is vortex tubes in turbulent fluid flow. While elliptic and hyperbolic structures are well understood in steady and timeperiodic systems, their treatment in fully unsteady flows remains a topic of current research.
We have developed methods (based on variational principles) for quantifying and computing these structures.
Polar rotationbased detection of vortices:
Farazmand and Haller, Physica D 2016
Variational theory of shearless LCS:
Farazmand, Blazevski and Haller, Physica D 2014
Computation of LCS:
Farazmand and Haller, Chaos 2013
Farazmand and Haller, Chaos 2012
Lagrangian Coherent Structures: Chaotic dynamical systems often exhibit islands of regular behavior (elliptic islands) interrupting the surrounding chaotic sea that is dominated by hyperbolicity. A fascinating example of such coherent behavior is vortex tubes in turbulent fluid flow. While elliptic and hyperbolic structures are well understood in steady and timeperiodic systems, their treatment in fully unsteady flows remains a topic of current research. We have developed methods (based on variational principles) for quantifying and computing these structures. Polar rotationbased detection of vortices: Farazmand and Haller, Physica D 2016 Variational theory of shearless LCS: Farazmand, Blazevski and Haller, Physica D 2014 Computation of LCS: Farazmand and Haller, Chaos 2013 Farazmand and Haller, Chaos 2012 