Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
@article{Tobasco2017OptimalBA, title={Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems}, author={Ian Tobasco and David Goluskin and Charles R. Doering}, journal={Physics Letters A}, year={2017}, volume={382}, pages={382-386} }
43 Citations
Bounding Extreme Events in Nonlinear Dynamics Using Convex Optimization
- Mathematics, Computer ScienceSIAM J. Appl. Dyn. Syst.
- 2020
A convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs) is studied and it is shown that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events.
Optimal time averages in non-autonomous nonlinear dynamical systems
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- 2020
The auxiliary function method allows computation of extremal long-time averages of functions of dynamical variables in autonomous nonlinear ordinary differential equations via convex optimization.…
Convex computation of maximal Lyapunov exponents
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We describe an approach for finding upper bounds on an ODE dynamical system’s maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is…
Convex Computation of Extremal Invariant Measures of Nonlinear Dynamical Systems and Markov Processes
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We propose a convex-optimization-based framework for computation of invariant measures of polynomial dynamical systems and Markov processes, in discrete and continuous time. The set of all invariant…
Bounding extrema over global attractors using polynomial optimisation
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In particular, nonnegativity of certain polynomial expressions is enforced by requiring them to be representable as sums of squares, leading to a convex optimisation problem that can be recast as a semidefinite program and solved computationally.
Bounding Averages Rigorously Using Semidefinite Programming: Mean Moments of the Lorenz System
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Methods for proving bounds on infinite-time averages in differential dynamical systems rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions.
Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations
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The aim is to present a minimax optimization formula that yields optimal bounds for time and/or ensemble averages for the twoand threedimensional Navier-Stokes equations.
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Borders are presented for bounding infinite-time averages in dynamical systems governed by nonlinear PDEs, which constitute the first positive evidence for this conjecture up to finite L, and they offer some guidance for analytical proofs.
Optimal minimax bounds for time and ensemble averages of dissipative infinite-dimensional systems with applications to the incompressible Navier-Stokes equations
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The aim is to present a minimax optimization formula that yields optimal bounds for time and ensemble averages of dissipative infinite-dimensional systems, including the two- and three-dimensional Navier-Stokes equations.
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Methods for proving bounds on infinite-time averages in differential dynamical systems rely on the construction of nonnegative polynomials with certain properties, similarly to the way nonlinear stability can be proved using Lyapunov functions.
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