# 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

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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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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.…

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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.

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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

- Computer Science
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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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