Kuramoto Sivashinsky Research Articles

Regional Stabilization of the 1-D Kuramoto–Sivashinsky Equation via Modal Decomposition

In this letter, we suggest regional stabilization of the semilinear 1D KSE under nonlocal or boundary actuation. We employ modal decomposition and derive regional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$H^{1}$ </tex-math></inline-formula> stability conditions for the closed-loop system. Given a decay rate that defines the number of state modes in the controller, we provide LMIs for finding the the controller gain as well as a bound on the domain of attraction. In the case of boundary control, we suggest a dynamic extension with a novel internally stable dynamics. The latter allows to enlarge a bound on the domain of attraction. Numerical examples illustrate the efficiency of the method.

Rigorous validation of a Hopf bifurcation in the Kuramoto–Sivashinsky PDE

We use computer-assisted proof techniques to prove that a branch of non-trivial equilibrium solutions in the Kuramoto–Sivashinsky partial differential equation undergoes a Hopf bifurcation. Furthermore, we obtain an essentially constructive proof of the family of time-periodic solutions near the Hopf bifurcation. To this end, near the Hopf point we rewrite the time periodic problem for the Kuramoto–Sivashinsky equation in a desingularized formulation. We then apply a parametrized Newton–Kantorovich approach to validate a solution branch of time-periodic orbits. By construction, this solution branch includes the Hopf bifurcation point.

Analysis of a quasi-reversibility method for nonlinear parabolic equations with uncertainty data

In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed—i.e., the solution does not depend continuously on the data. To regularize the instable solution, we develop some new methods to construct some new regularized solution. We also investigate the convergence rate between the regularized solution and the solution of our equations. In particular, we establish results for several equations with constant coefficients and time dependent coefficients. The equations with constant coefficients include the heat equation, extended Fisher–Kolmogorov equation, Swift–Hohenberg equation, and many others. The equations with time dependent coefficients include Fisher-type logistic equations, the Huxley equation, and the Fitzhugh–Nagumo equation. The methods developed in this paper can also be applied to get approximate solutions to several other equations including the 1-D Kuramoto–Sivashinsky equation, 1-D modified Swift–Hohenberg equation, strongly damped wave equation, and 1-D Burgers equation with randomly perturbed operator. Some numerical examples are given which illustrate the effectiveness of our method.

On the Exponential Stability of a Nonlinear Kuramoto–Sivashinsky–Korteweg-de Vries Equation with Finite Memory

On the Exponential Stability of a Nonlinear Kuramoto–Sivashinsky–Korteweg-de Vries Equation with Finite Memory

The Solution of the General Kuramoto–Sivashinsky Equation Using the Compact Method in Conjunction with the ETD (1,3)-Padé Scheme

The Solution of the General Kuramoto–Sivashinsky Equation Using the Compact Method in Conjunction with the ETD (1,3)-Padé Scheme

Electrohydrodynamic instability of viscous liquid films: Electrostatically modified second-order Benney and Kuramoto–Sivashinsky models

Long wave evolution on viscous films flowing under gravity down an inclined plate in the presence of an electrostatic field acting normal to the plate is described using new nonlinear models. First, an asymptotic expansion is applied to derive a strongly nonlinear evolution equation for the interfacial position in the long-wave limit, which retains terms up to second order in a small film parameter (ε) and thus brings in dispersive effects, inclination-angle corrections for hydrostatic pressure, second-order contributions to inertia and electric stresses as well as the interaction of the last two. Two weakly nonlinear models (WNMs) for the disturbances with O(ε2) amplitude are then derived, which account for distinct effects of hydrostatic pressure, inertia and dispersion for shallow and steep angles. We compare the linear stability results based on the second-order electrified Benney equation (BE) and the WNMs with realistic parameter values. It is shown that the growth rates coincide, while the phase speed from the WNMs is a good approximation of that from the BE for small wavenumbers only. The comparison of exact dispersion relations from the Orr–Sommerfeld (OS) problem with the linear results of the second-order BE takes a step towards understanding the validity range of the BE with non-local terms that approximates the full electrohydrodynamic coupling system describing the inclined electrified film flow. The pertinent OS results demonstrate that a decrease in electrode distance can diminish the critical Reynolds number (Re). Furthermore, at subcritical Re with a slightly supercritical electric field, the finite wavelength of the least stable mode has been estimated using an energy balance, whose amplitude will be governed by two coupled Ginzburg–Landau (GL) equations, derived using a multiple-scale analysis by taking higher-order problems into account. The numerical solutions of the leading-order GL equation demonstrate that close to the bifurcation, it can fully govern the temporal behaviour of the system.

Dynamic transitions and bifurcations of 1D reaction–diffusion equations: The self‐adjoint case

This paper deals with the classification of transition phenomena in the most basic dissipative system possible, namely, the 1D reaction–diffusion equation. The emphasis is on the relation between the linear and nonlinear terms and the effect of the boundaries which influence the first transitions. We consider the cases where the linear part is self‐adjoint with second‐order and fourth‐order derivatives which is the case which most often arises in applications. We assume that the nonlinear term depends on the unknown function and its first derivative which is basically the semilinear case for the second‐order reaction–diffusion system. As for the boundary conditions, we consider the typical Dirichlet, Neumann, and periodic boundary settings. In all the cases, the equations admit a trivial steady state which loses stability at a critical parameter. We aim to classify all possible transitions and bifurcations that take place. Our analysis shows that these systems display all three types of transitions: continuous, jump and mixed. Moreover they exhibit transcritical, supercritical bifurcations with bifurcated states such as finitely many equilibria, circle of equilibria, and slowly rotating limit cycle. Many applications found in the literature are basically corollaries of our main results. We apply our results to classify the first transitions of the Chaffee–Infante equation, the Fisher‐KPP equation, the Kuramoto–Sivashinsky equation, and the Swift–Hohenberg equation.

Knowledge-based learning of nonlinear dynamics and chaos.

Extracting predictive models from nonlinear systems is a central task in scientific machine learning. One key problem is the reconciliation between modern data-driven approaches and first principles. Despite rapid advances in machine learning techniques, embedding domain knowledge into data-driven models remains a challenge. In this work, we present a universal learning framework for extracting predictive models from nonlinear systems based on observations. Our framework can readily incorporate first principle knowledge because it naturally models nonlinear systems as continuous-time systems. This both improves the extracted models' extrapolation power and reduces the amount of data needed for training. In addition, our framework has the advantages of robustness to observational noise and applicability to irregularly sampled data. We demonstrate the effectiveness of our scheme by learning predictive models for a wide variety of systems including a stiff Van der Pol oscillator, the Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz system, different types of domain knowledge are incorporated to demonstrate the strength of knowledge embedding in data-driven system identification.

An improvised extrapolated collocation algorithm for solving Kuramoto–Sivashinsky equation

In the present work, the numerical solution of the Kuramoto–Sivashinsky (KS) equation is obtained using an improvised quintic B‐spline extrapolated collocation technique. This equation helps in the study of wave production in dissipative medium, investigates the hydrodynamic uncertainty in laminar flames, describes the turbulence of diverse physical processes, and so on. In this work, splines are used due to their higher smoothness property and the sparse nature of the matrices corresponding to the B‐splines. The improvised B‐splines are formed by forcing the quintic B‐spline interpolant to satisfy the interpolatory and some special end conditions. These basis functions are used in the collocation method for space integration and a weighted finite difference scheme is used for temporal domain integration. The stability of the technique is analyzed using the von Neumann scheme, and it is found to be unconditionally stable. The proposed method is found to be sixth‐order convergent in space and second‐order convergent in the time direction. The theoretical order of convergence is matching perfectly with the numerical one. The efficiency of this scheme is demonstrated by applying it to a number of examples. The L2, L∞, and global relative errors are calculated and compared with the previous work, especially with those where quintic B‐splines are used as basis functions. Also, the behavior of some KS equations is discussed for which the exact solution is not available. The aim of the paper is to show that such an improvised technique can be implemented to solve partial differential equations like the KS equation.

The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method

In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.

Evolutional deep neural network.

The notion of an evolutional deep neural network (EDNN) is introduced for the solution of partial differential equations (PDE). The parameters of the network are trained to represent the initial state of the system only and are subsequently updated dynamically, without any further training, to provide an accurate prediction of the evolution of the PDE system. In this framework, the network parameters are treated as functions with respect to the appropriate coordinate and are numerically updated using the governing equations. By marching the neural network weights in the parameter space, EDNN can predict state-space trajectories that are indefinitely long, which is difficult for other neural network approaches. Boundary conditions of the PDEs are treated as hard constraints, are embedded into the neural network, and are therefore exactly satisfied throughout the entire solution trajectory. Several applications including the heat equation, the advection equation, the Burgers equation, the Kuramoto Sivashinsky equation, and the Navier-Stokes equations are solved to demonstrate the versatility and accuracy of EDNN. The application of EDNN to the incompressible Navier-Stokes equations embeds the divergence-free constraint into the network design so that the projection of the momentum equation to solenoidal space is implicitly achieved. The numerical results verify the accuracy of EDNN solutions relative to analytical and benchmark numerical solutions, both for the transient dynamics and statistics of the system.

Global Solutions of the Two-Dimensional Kuramoto–Sivashinsky Equation with a Linearly Growing Mode in Each Direction

We consider the Kuramoto–Sivashinsky equation in two space dimensions. We establish the first proof of global existence of solutions in the presence of a linearly growing mode in both spatial directions for sufficiently small data. We develop a new method to this end, categorizing wavenumbers as low (linearly growing modes), intermediate (linearly decaying modes that serve as energy sinks for the low modes), and high (strongly linearly decaying modes). The low and intermediate modes are controlled by means of a Lyapunov function, while the high modes are controlled with operator estimates in function spaces based on the Wiener algebra.

Deep learning of conjugate mappings

Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. This work proposes a method for obtaining explicit Poincaré mappings by using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. Indeed, the enforcement of topological conjugacies is the critical neural network regularization for learning the coordinate and dynamics pairing. We provide expository applications of the method to low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on infinite-dimensional systems, such as the Kuramoto–Sivashinsky equation.

Learning a reduced basis of dynamical systems using an autoencoder.

Machine learning models have emerged as powerful tools in physics and engineering. In this work, we use an autoencoder with latent space penalization to discover approximate finite-dimensional manifolds of two canonical partial differential equations. We test this method on the Kuramoto-Sivashinsky (K-S), Korteweg-de Vries (KdV), and damped KdV equations. We show that the resulting optimal latent space of the K-S equation is consistent with the dimension of the inertial manifold. We then uncover a nonlinear basis representing the manifold of the latent space for the K-S equation. The results for the KdV equation show that it is more difficult to recover a reduced latent space, which is consistent with the truly infinite-dimensional dynamics of the KdV equation. In the case of the damped KdV equation, we find that the number of active dimensions decreases with increasing damping coefficient.

Efficient computation of coordinate-free models of flame fronts

We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit. doi:10.1017/S1446181121000079

Temporal Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

Let U=U(t,x) for (t,x)∈R+×Rd and ∂xU=∂xU(t,x) for (t,x)∈R+×R be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels and symmetry analysis, yielding exact, dimension-dependent, and temporal moduli of non-differentiability for U(·,x) and ∂xU(·,x). It has been confirmed that almost all sample paths of U(·,x) and ∂xU(·,x), in time, are nowhere differentiable.

Spatial Moduli of Non-Differentiability for Linearized Kuramoto–Sivashinsky SPDEs and Their Gradient

We investigate spatial moduli of non-differentiability for the fourth-order linearized Kuramoto–Sivashinsky (L-KS) SPDEs and their gradient, driven by the space-time white noise in one-to-three dimensional spaces. We use the underlying explicit kernels and symmetry analysis, yielding spatial moduli of non-differentiability for L-KS SPDEs and their gradient. This work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. Moreover, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of L-KS SPDEs and their gradient.

Lyapunov based on-line model reduction and control of semilinear dissipative distributed parameter systems with minimum feedback information

We focus on the Lyapunov-based output feedback control problem for a class of distributed parameter systems with spatiotemporal dynamics described by input-affine linear and semilinear dissipative partial differential equations (DPDEs). The control problem is addressed via model order reduction. Galerkin projection is applied to discretize the DPDE and derive low-dimensional reduced order models (ROMs). The empirical basis functions needed for this discretization are recursively computed using adaptive proper orthogonal decomposition (APOD). To update the basis functions during process operation, APOD needs measurements of the system state’s complete profile (called snapshots) at revision times. This paper’s main objective is to minimize the demand for snapshots from the spatially distributed sensors by the control structure while maintaining closed-loop stability and performance. A control Lyapunov function is defined, and its value is monitored as the system evolves. Only when the value violates a closed-loop stability threshold, snapshots are requested for a brief period by APOD after which the ROM is updated, and the controller is reconfigured. The proposed approach is applied to stabilize the Kuramoto–Sivashinsky equation.

Large deviations for stochastic Kuramoto–Sivashinsky equation with multiplicative noise

The Kuramoto–Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation.

Predicting high-dimensional heterogeneous time series employing generalized local states

We generalize the concept of local states (LS) for the prediction of high-dimensional, potentially mixed chaotic systems. The construction of generalized local states (GLS) relies on defining distances between time series on the basis of their (non-)linear correlations. We demonstrate the prediction capabilities of our approach based on the reservoir computing (RC) paradigm using the Kuramoto-Sivashinsky (KS), the Lorenz-96 (L96) and a combination of both systems. In the mixed system a separation of the time series belonging to the two different systems is made possible with GLS. More importantly, prediction remains possible with GLS, where the LS approach must naturally fail. Applications for the prediction of very heterogeneous time series with GLSs are briefly outlined.