Mathematics

The direct monitoring of a rotating detonation engine (RDE) combustion chamber has enabled the observation of combustion front dynamics that are composed of a number of co- and/or counter-rotating coherent traveling shock waves whose nonlinear mode-locking behavior exhibit bifurcations and instabilities which are not well understood. Computational fluid dynamics simulations are ubiquitous in characterizing the dynamics of RDE's reactive, compressible flow. Such simulations are prohibitively expensive when considering multiple engine geometries, different operating conditions, and the long-time dynamics of the mode-locking interactions. Reduced-order models (ROMs) provide a critically enabling simulation framework because they exploit low-rank structure in the data to minimize computational cost and allow for rapid parameterized studies and long-time simulations. However, ROMs are inherently limited by translational invariances manifest by the combustion waves present in RDEs. In this work, we leverage machine learning algorithms to discover moving coordinate frames into which the data is shifted, thus overcoming limitations imposed by the underlying translational invariance of the RDE and allowing for the application of traditional dimensionality reduction techniques. We explore a diverse suite of data-driven ROM strategies for characterizing the complex shock wave dynamics and interactions in the RDE. Specifically, we employ the dynamic mode decomposition and a deep Koopman embedding to give new modeling insights and understanding of combustion wave interactions in RDEs.

We establish exponential decay in Hölder norm of transfer operators applied to smooth observables of uniformly and nonuniformly expanding semiflows with exponential decay of correlations.

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are ϵ -approximate eigenfunctions of the Koopman evolution operator of the system, with a bound ϵ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, ϵ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are ϵ -approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in L 2 . Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.

Given ρ∈(0,1/3] , let μ be the Cantor measure satisfying μ= 1 2 μ f −1 0 + 1 2 μ f −1 1 , where f i (x)=ρx+i(1−ρ) for i=0,1 . The support of μ is a Cantor set C generated by the iterated function system { f 0 , f 1 } . Continuing the work of Feng et al.~(2000) on the pointwise lower and upper densities Θ s ∗ (μ,x)= lim inf r→0 μ(B(x,r)) (2r ) s , Θ ∗s (μ,x)= lim sup r→0 μ(B(x,r)) (2r ) s , where s=−log2/logρ is the Hausdorff dimension of C , we give a complete description of the sets D ∗ and D ∗ consisting of all possible values of the lower and upper densities, respectively. We show that both sets contain infinitely many isolated and infinitely many accumulation points, and they have the same Hausdorff dimension as the Cantor set C . Furthermore, we {compute the Hausdorff dimension of} the level sets of the lower and upper densities. Our proofs are based on recent progress on unique non-integer base expansions and ideas from open dynamical systems.

Inverse problems are ubiquitous because they formalize the integration of data with mathematical models. In many scientific applications the forward model is expensive to evaluate, and adjoint computations are difficult to employ; in this setting derivative-free methods which involve a small number of forward model evaluations are an attractive proposition. Ensemble Kalman based interacting particle systems (and variants such as consensus based and unscented Kalman approaches) have proven empirically successful in this context, but suffer from the fact that they cannot be systematically refined to return the true solution, except in the setting of linear forward models. In this paper, we propose a new derivative-free approach to Bayesian inversion, which may be employed for posterior sampling or for maximum a posteriori estimation, and may be systematically refined. The method relies on a fast/slow system of stochastic differential equations for the local approximation of the gradient of the log-likelihood appearing in a Langevin diffusion. Furthermore the method may be preconditioned by use of information from ensemble Kalman based methods (and variants), providing a methodology which leverages the documented advantages of those methods, whilst also being provably refineable. We define the methodology, highlighting its flexibility and many variants, provide a theoretical analysis of the proposed approach, and demonstrate its efficacy by means of numerical experiments.

A linear dynamical system is called k -positive if its dynamics maps the set of vectors with up to k?? sign variations to itself. For k=1 , this reduces to the important class of positive linear systems. Since stable positive linear time-invariant (LTI) systems always admit a diagonal quadratic Lyapunov function, i.e. they are diagonally stable, we may expect that this holds also for stable k -positive systems. We show that, in general, this is not the case both in the continuous-time (CT) and discrete-time (DT) case. We then focus on DT k -positive linear systems and introduce the new notion of DT k -diagonal stability. It is shown that this is a necessary condition for standard DT diagonal stability. We demonstrate an application of this new notion to the analysis of a class of DT nonlinear systems.

In this paper we introduce the notion of diameter diminishing to zero iterated function system, study its properties and provide alternative characterizations of it.

We consider Hölder continuous cocycles over an accessible partially hyperbolic system with values in the group of diffeomorphisms of a compact manifold M . We obtain several results for this setting. If a cocycle is bounded in C 1+γ , we show that it has a continuous invariant family of γ -Hölder Riemannian metrics on M . We establish continuity of a measurable conjugacy between two cocycles assuming bunching or existence of holonomies for both and pre-compactness in C 0 for one of them. We give conditions for existence of a continuous conjugacy between two cocycles in terms of their cycle weights. We also study the relation between the conjugacy and holonomies of the cocycles. Our results give arbitrarily small loss of regularity of the conjugacy along the fiber compared to that of the holonomies and of the cocycle.

In this paper, we prove the power-law in time upper bound for the diffusion of a 1D discrete nonlinear Anderson model. We remove completely the decaying condition restricted on the nonlinearity of Bourgain-Wang (Ann. of Math. Stud. 163: 21--42, 2007.). This gives a resolution to the problem of Bourgain (Illinois J. Math. 50: 183--188, 2006.) on diffusion bound for nonlinear disordered systems. The proof uses a novel ``norm'' based on tame property of the Hamiltonian.

Following the approach of [E1, M1, M2, S1, S2, SZJV] for reaction diffusion systems, we justify rigorously the Eckhaus stability criterion for stability of convective Turing patterns, as derived formally by complex Ginzburg-Landau approximation [SS, NW, WZ]. Notably, our analysis includes also higher-order, nonlocal, and even certain semilinear hyperbolic systems.