Jinqiao Duan, Vincent Ervin
In this paper we study the effects of a ``nonlocal'' term on the global dynamics of the Kuramoto-Sivashinsky equation. We show that the equation possesses a ``family of maximal attractors'' parameterized by the mean value of the initial data. The dimension of the attractor is estimated as a function of the coefficient of the nonlocal term and the width of the periodic domain.
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The nonlocal Kuramoto-Sivashinsky equation arises in the modeling of the flow of a thin film of viscous liquid falling down an inclined plane, subject to an applied electric field. In this paper, the authors show that, as the coefficient of the nonlocal integral term goes to zero, the solution trajectories and the maximal attractor of the nonlocal Kuramoto-Sivashinsky equation converge to those of the usual Kuramoto-Sivashinsky equation.
We present numerical results and computer assisted proofs of the existence of periodic orbits for the Kuramoto-Sivashinky equation. These two results are based on writing down the existence of periodic orbits as zeros of functionals. This leads to the use of Newton's algorithm for the numerical computation of the solutions and, with some a posteriori analysis in combination with rigorous interval arithmetic, to the rigorous verification of the existence of solutions. This is a particular case of the methodology developed in [19] for several types of orbits. An independent implementation, cover…
The Kuramoto-Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto-Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan-Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto-Sivashinsky PDE, and the transition to its 1D turbulence.
The Kuramoto-Sivashinsky equation was introduced as a simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating in its own right. One reason is that this equation is a simple model of Galilean-invariant chaos with an arrow of time. Starting from random initial conditions, manifestly time-asymmetric stripe-like patterns emerge. As we move forward in time, it appears that these stripes are born and merge, but do not die or split. We pose a precise conjecture to this effect, which requires a precise definition of 'stripes'.
In this article we study the solution of the Kuramoto-Sivashinsky equation (for surface erosion or surface growth) on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data.
Symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line II
该研究通过计算机辅助的新严格积分算法，证明了特定参数与边界条件下直线上Kuramoto-Sivashinsky偏微分方程存在无穷多同宿和异宿轨道，为该类耗散PDE的符号动力学研究提供了支撑。
Symbolic dynamics for Kuramoto-Sivashinsky PDE on the line --- a computer-assisted proof
本文研究了具有奇数和周期边界条件的Kuramoto-Sivashinsky PDE在直线上的符号动力学，通过计算机辅助证明了存在可数无穷多个具有任意大周期的周期轨道。
Pitchfork bifurcation and heteroclinic connections in the Kuramoto--Sivashinsky PDE
本文提出分析耗散PDE叉式分岔动力学的方法，并对实线上的Kuramoto–Sivashinsky方程给出非对称不动点分支及异宿连接的计算机辅助证明。
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 b…
We investigate the bifurcation structure of the Kuramoto-Sivashinsky equation with homogeneous Dirichlet boundary conditions. Using hidden symmetry principles, based on an extended problem with periodic boundary conditions and $\Otwo$ symmetry, we show that the zero solution exhibits two kinds of pitchfork bifurcations: one that breaks the reflection symmetry of the system with Dirichlet boundary conditions and one that breaks a shift-reflect symmetry of the extended system. Using Lyapunov-Schmidt reduction, we show both to be supercritical. We extend the primary branches by means of numerical…
Rigorous Numerics for Partial Differential Equations: the Kuramoto-Sivashinsky equation
本文提出基于自洽先验界的拓扑方法，严格证明Galerkin投影，将耗散PDE约化为低维ODE并用Conley指标分析，最终计算机辅助证明Kuramoto-Sivashinsky方程不动点的存在性。
In this paper, a partial proof of a conjecture raised by Galaktionov and Svirshchevskii concerning existence and global uniqueness of an asymptotically stable periodic orbit in a fourth-order piecewise linear ordinary differential equation is presented. The fourth-order equation comes from the study of traveling wave patterns in a signed Kuramoto-Sivashinsky equation with absorption. The proof is twofold. First, the problem of solving for the periodic orbit is transformed into a zero finding problem in four dimensional space, which is solved with a computer-assisted proof based on Newton's met…
This paper presents a finite difference method combined with the Crank--Nicolson scheme of the Kuramoto--Sivashinsky equation defined on an expanding circle (\cite{KUY}), and the existence, uniqueness, and second-order error estimate of the scheme. The equation can be obtained as a perturbation equation from the circle solution to an interfacial equation and can provide guidelines for understanding the wavenumber selection of solutions to the interfacial equation. Our proposed numerical scheme can help with such a mathematical analysis.
Computational methods for sensitivity analysis are invaluable tools for scientists and engineers investigating a wide range of physical phenomena. However, many of these methods fail when applied to chaotic systems, such as the Kuramoto-Sivashinsky (K-S) equation, which models a number of different chaotic systems found in nature. The following paper discusses the application of a new sensitivity analysis method developed by the authors to a modified K-S equation. We find that least squares shadowing sensitivity analysis computes accurate gradients for solutions corresponding to a wide range o…
Melnikov Analysis of Deterministic and Stochastic Manifold Splitting in the Kuramoto--Sivashinsky Equation
本文基于Melnikov框架研究Kuramoto-Sivashinsky方程中确定性和随机曼-fold分裂，通过分析稳定与不稳定曼-fold的分裂，揭示了耗散偏微分方程中时空混沌的几何机制。
We consider the Kuramoto-Sivashinsky (KS) equation in finite domains of the form $[-L,L]^d$. Our main result provides refined Gevrey estimates for the solutions of the one dimensional differentiated KS, which in turn imply effective new estimates for higher Sobolev norms of the solutions in terms of powers of $L$. We illustrate our method on a simpler model, namely the regularized Burger's equation. We also show local well-posedness for the two dimensional KS equation and provide an explicit criteria for (eventual) blow-up in terms of its $L^2$ norm. The common underlying idea in both results …
Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation
本文基于中心流形理论，通过整体离散化方法对Kuramoto–Sivashinsky方程进行粗网格上的精确建模，并验证了该方法在稳态和动力学再现上的优异性能。
Concurrent multi-parameter learning demonstrated on the Kuramoto-Sivashinsky equation
本研究提出一种基于nudging数据同化方案的并行多参数估计算法，可在状态部分观测的耗散偏微分方程系统中同时估计多个标量参数，经一维Kuramoto-Sivashinsky方程测试验证了有效性，且易推广到其他系统。
Degree-of-freedom and optimization-dynamic effects on the observability of Kuramoto-Sivashinsky systems
通过将可观测性与耗散动力系统的嵌入理论联系起来，建立了变分状态估计应用于Kuramoto-Sivashinsky方程的可观测性准则，并分析了优化动力学限制，提出了结合非凸牛顿更新和伪投影步骤的鲁棒重构策略。
A master-slave coupling scheme for synchronization and parameter estimation in the generalized Kuramoto-Sivashinsky equation
本文针对广义 Kuramoto-Sivashinsky 方程，提出一种基于主从同步的在线参数估计方法，利用数据驱动从系统持续调整参数直至完全同步，并通过数值模拟证明其计算快速且对初始化误差、观测噪声和空间分辨率变化具有鲁棒性。
Amortized Generative Modeling of Invariant Measures across A Family of Kuramoto-Sivashinsky PDEs
针对Kuramoto-Sivashinsky方程族，训练单一条件扩散模型以覆盖不同超粘性下的不变测度，发现弱混沌低维性导致困难，限制训练于强混沌成员可提升约40%性能并匹配单参数模型。
Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming
本文提出了一种通过半正定规划计算非线性PDE动力系统无限时间平均的边界方法，通过构造辅助泛函来约束能量平均，并应用于Kuramoto-Sivashinsky方程，得到不同截断阶数下的能量上界。
Emergent behaviors of the singular continuum Kuramoto model and its graph limit
本文研究了奇异连续库莫托模型（SCKM）及其图极限的涌现动力学，探讨了奇异性的关键作用，证明了相同自然频率下有限时间内完全相位同步，非相同自然频率下出现实用同步，并建立了有限系统规模到SCKM的图极限。
Phase Oscillator Networks with Nonlocal Higher-Order Interactions: Twisted States, Stability and Bifurcations
本文研究了环状非局部耦合无穷粒子极限下Kuramoto模型的扭转态与分岔，发现高阶非成对相互作用能稳定仅靠成对耦合时不稳定的扭转态。
We prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive Lévy noise. The main result is applicable to various types of SPDE such as stochastic Burgers type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$α$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard type equations, stochastic Kuramoto-Sivashinsky type equations, stochastic porous media equations and stochastic $p$-Laplace equations.
Embedding techniques allow the approximations of finite dimensional attractors and manifolds of infinite dimensional dynamical systems via subdivision and continuation methods. These approximations give a topological one-to-one image of the original set. In order to additionally reveal their geometry we use diffusion mapst o find intrinsic coordinates. We illustrate our results on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky equation, as well as for the attractor of the Mackey-Glass delay differential equation.
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