## 学习目标：

• Dynamical system $(M,f)$: the space M of its possible states, and the law $f^t$ of their evolution in time.

• equilibria / fixed points

• Poincare map

• Explore the neighborhood by 线化 the flow； check the linear stability of its equilibria/ fixed points, their stability eigen-directions.

• Symmetry to simplify your problem

• Go global: train by partitioning the state space of 1-d maps. Label the regions by symbolic dynamics.

• Venture global distances across the system by continuing local tangent space into stable/ unstable manifolds. Their intersections partition the state space in a dynamically invariant way.

• Guided by this topological partition, compute a set of periodic orbits up to a given topological length.

## Introducation

• 长期行为很难或无法预测：即使是对混沌系统当前状态的非常精确的测量，也都无法指示系统的位置。必须再次测量系统以找出其位置。
• 对初始条件的敏感依赖（Poincare，Birkhoff甚至Turing指出的属性 ）：从非常接近的初始条件开始，混沌系统会迅速移动到不同的状态。
• 宽带频谱：混沌系统的输出听起来“嘈杂”。许多频率被激发。
• 错误的指数放大：在任何现实世界中，少量外部噪声都会迅速增长以控制系统。如果此噪声低于测量精度，则使实验者看不到或无法控制该噪声，则该系统似乎不可预测。微观的“热浴”被放大到人类规模。
• 局部不稳定性与全局稳定性：为了扩大小错误和噪声，行为必须是局部不稳定的：在短时间内，附近的状态会彼此远离。但是，为了使系统始终如一地产生稳定的行为，必须在很长一段时间内将一系列行为归为自身。这两个属性的张力导致结构非常优雅的混沌吸引子。

• 如果一个确定性系统具有局部不稳定性(positive Lyapunov exponent)globally mixing (positive entropy), 这个系统就成为是混沌的。

1. globally mixing (positive entropy)

## flows and maps

• A flow: the evolution rule $f^t$ can be used to map a region $M_i$ of the state space into the region $f^t(M_i)$.

闲话1：Manifolds 在微分几何的课程中有非常精彩的介绍。sagemath软件也有集成关于流行symbolic运算的功能，感兴趣可以了解。

闲话2The ring of fire - 这个着色器渲染出的，加的是随机噪声, 非物理。

ETD方法的思想就是利用数值近似求解该方程。

• ETD1：一阶精度， 认为F恒定：local truncation error= $h^2 F_t /2$

• ETD2：二阶精度，$F= Fn + \tau (F_n -F{n-1})/h + O(h^2)$, local truncation error= $5 h^3 F_{tt}/12$

• 再高阶的任意精度，见参考文献2

### 具体K-S代码如下：
• 结果展示：为40个周向模态的幅值随时间分布的演化图

除了最基本的主干部分，还一个添加一个velocity-field 计算函数：其功能是将所有周向模态叠加，转化回物理域。

还可以添加particles进行观察：

## basic flow dynamical model

• Heat equation
• Non-linear effects (Burger’s equation) 参考

Using the product rule, we can rewrite this as

• 另外一些简单的模型，基本差分格式，julia算法参考3

• N-S 不可压方程

涡量：

setp1 : we can derivate the vorticity equation:

那么，上面的第二项可以化为：

上述的第二项为vortex stretching term，在二维中为0:

setp2: The vorticity equation for two-dimensional incompressible flow becomes:

Step3: the kinematic relationship between streamfunction and vorticity is given by Possion equation:

The vorticity-streamfunction formulation has several advantages over solving N-S equation. It eliminates the pressure term from the momentum equation and hence, there is no odd-even coupling between the pressure and velocity.

### Two-step time-stepping alogorithm

• Start from an inition condition w at t=0
• Invert the Possion equation to determine $\psi^n$
• Timestep the vorticity equation forward to determine $\psi^{t+1}$ from $w^t$ and $\psi^t$
• Go back to step 3 and repeat to find $w^{t+2}$ from $w^{n+1}$ and $\psi^{n+1}$
1. G. Sivashinsky, “Nonlinear analysis of hydrodynamic instability in laminar flames I. Derivation of basic equations” Acta Astron. , 4 (1977) pp. 1177–1206
2. Exponential Time Differencing for Stiff Systems, JCP-2002, S.M. Cox & P.C. Matthews
3. CFD Julia: A Learning Module Structuring an Introductory Course on Computational Fluid Dynamics
4. Fluid dynamics with Oceananigans.jl | Week 13 | MIT 18.S191 Fall 2020 | Ali Ramadhan

## 草稿

Waleffe (1995, 1997) further developed these ideas into a ‘self- sustaining process theory’ that explains the quasi-cyclic roll-streak behavior in terms of the forced response of streaks to rolls, growth of streak instabilities, and nonlinear feedback from streak instabilities to rolls.

The preponderance of recurrent, coherent states in wall-bounded shear flows suggests
that their long-time dynamics lie on low-dimensional state-space attractors.

They improve on the POD models by capturing the linear stability of the
laminar flow and saddle-node bifurcations of non-trivial 3D equilibria consisting of rolls,
streaks, and streak undulations.

The work of Skufca et al. (2006), based on a Schmiegel
(1999) 9-variable model, offers an elegant dynamical systems picture, with the stable
manifold of a periodic orbit defining the basin boundary that separates the turbulent
and laminar attractors at Re < 402 and the stable set of a higher-dimensional chaotic
object defining the boundary at higher Re.

## invariant manifolds

We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Re and c

the calculation of exact invariant solutions of the fully-resolved Navier-Stokes
equations.

(a) that coherent structures are the physical images of the flow’s least unstable
invariant solutions, (b) that turbulent dynamics consists of a series of transitions between
these states, and (c) that intrinsic low-dimensionality in turbulence results from the low
number of unstable modes for each state (Waleffe (2002)).