Homework 16

This homework is not required. It serves as bonus to help you improve your final grade.
The template code and data can be download here .
For this homework, you can submit up to 3 times.

Q16.1 \(\quad\) Invariant subspace in Kuramoto-Sivashinsky system

The one dimensional Kuramoto-Sivashinsky equation \[ u_t + \frac{1}{2}(u^2)_x + u_{xx} + u_{xxxx} = 0\,,\quad x \in [0, L] \] describes the velocity of flame front in a circular shape. Expand \(u(t,x)\) in Fourier basis: \( u(x_n,t) = \sum^{\infty}_{n=-\infty} \hat{u}_k e^{iq_k x_n} \,,\) where \(q_k = 2\pi k/L\), We obtain the dynamics in terms of Fourier modes: \[ \dot{\hat{u}}_k = (q_k^2 - q_k^4) \hat{u}_k - \frac{iq_k}{2} \sum^{\infty}_{m=-\infty} \hat{u}_m \hat{u}_{k-m} \] Note, Kuramoto-Sivashinsky equation is anti-symmetric under reflection: \( u(x) \to -u(-x) \). Thus, pure imaginary modes form invariant subspace of this system. In this homework, we work in this subspace: \(\hat{u}_k = i a_k\). The flow becomes \[ \dot{a}_k = (q_k^2 - q_k^4) a_k + \frac{q_k}{2} \sum^{N}_{m=-N} a_m a_{k-m} \] Here, we truncate the Fourier modes, only keeping the lowest \(2N+1\) modes. Due to Galilean invariance \(u(x,t) \to u(x-ct, t)+c\), we can set the mean velocity \(\int_0^L u dx \) to zero without loss of generality, which is \(a_0 = 0\). This can be confirmed by \(\dot{a}_0 = 0\). Also, real field \(u(x,t)\) means \(\hat{u}^*_k = \hat{u}_{-k}\), so \(a_{-k} = -a_k\). Taking all these facts into consideration, we end up with a state space: \((a_1, a_2, \cdots, a_N)\). In this homework, set \( N = 16 \).

From everyday experience, we know that flame front can flutter every so often, so the system is actually chaotic. Here, we study how the domain size \(L\) affects the dynamics in the invariant subspace. In the template code, please fill out all required functions and set "case=1" to validate your implementation. In this case, you are supposed to get 2 figures. One is the state space projected onto the first 3 Fourier modes. The other is the color-mapped configuration space. They may be (a) and (b), or (c) or (d) at bottom of this page. It depends because the system is chaotic for the domain sizes used in case = 1. Also confirm that the velocity printed out is correct. Please consult chapter 30 section 30.7 to get the energy formula. Set domain size \(L = 36.23\), integrate the system for a long time. What is the average energy ? ( 2 significant digits should be accurate. )

Q16.2 \(\quad\) Period-doubling bifurcation

Period-doubling is a common source of bifurcation. A slight change in parameter space may give birth to a sudden transition in the state space from a single stable state to double states oscillation. Dynamics in the anti-symmetric subspace in Kuramoto-Sivashinsky system exhibits this phenomenon. In order to observe this bifurcation, let us work on a Poincare section \(a_1 = 0\). Start from a random state point, and evolve the system forward. Discard the transient evolution, and record the energy whenever this ergodic trajectory reaches the Poincare section. Please finish case 2 in the template code. Try different domain sizes \(L \in [36.20, 36.40]\), and plot the dependence of energy on \(L\). You will get a Feigenbaum bifurcation tree as shown in figure (e) below.

The Poincare section direction I used is from negative \(a_1\) to positive \(a_1\). If you get an upside down figure compared to figure (e), please change the direction of your Poincare section. Also, you have noticed that there is a sharp cliff in the bifurcation tree. That is related to the discrete symmetry \(a_{2k} \to a_{2k}\), \(a_{2k+1} \to -a_{2k+1}\). For more details, see reference [1] at the bottom.

Period can not only get doubled but also tripled as shown in (e). At \(L\sim 36.25\), the first doubling occurs. As \(L\) is increased, each branch goes trough doubling and this process repeats until it becomes chaotic. At a specific \(L\), system reaches a period-3 cycle and then after some chaotic region, reaches a period-5 cycle... Look at figure (e), there is a window containing a stable period-3 cycle. At what \(L\) this window starts ? ( Note, please keep at least 2 decimal digits accurate. )

Q16.3 \(\quad\) Equilibria in anti-symmetric space

Every time we try to figure out the geometry of a nonlinear flow, we first try to find some equilibria ( or relative equilibria ) which cooperate to shape the structure of this flow. It turns out that there are dozens of equilibria for Kuramoto-Sivashinsky system in the anti-symmetric invariant subspace with relatively large \(L\). They are not equally important. For example, let \(L=36.33\). Figures (f) and (g) below display the same stable equilibrium, but figure (g) also shows the main attractor ( green curve ). You see that the stable manifold of this equilibrium is far away from the main attractor. In figure (f), trajectory is spiraling inward the equilibrium. However, figure (h) shows an unstable equilibrium and its unstable manifold, which lands on the main attractor. So you see, this unstable equilibrium plays a more important role than that stable equilibrium in shaping the geometry in the state space.

Both equilibria are given to you in file "eq.npz". Please finish case 3 to visualize these two equilibria. Note, please set \(L=36.33\) in this experiment. What is the real part of the leading expanding stability exponent of this unstable equilibrium ? ( Please keep at least 2 significant digits accurate and do not use scientific notation. )

Q16.4 \(\quad\) Return map and cycle expansion

We are equipped with all basic tools: integrator, stability matrix, important equilibrium, etc. Then we can construct the one dimensional return map as we did in two-mode system. At the unstable equilibrium, use the unstable eigenvectors to construct a Poincare section and choose a state point close to this equilibrium in the expanding direction as initial condition and generate an ergodic trajectory. Store the Poincare intersection points and order them by their curvilinear distance to the equilibrium. Therefore, we get the 1-d return map.

Then we figure out the pruning rule of this map by using kneading theory, after which the symbolic dynamics is obtained. Then start from the points where the map intersects the diagonal line, we retrieve and find all the periodic orbits up to a certain topological length. Finally, with this set of periodic orbits, we turn to cycle expansion of dynamical zeta function or spectral determinant to predict some physical properties inside this system.

I just sketched the basic procedure of applying What ChaosBook teaches us towards a real world problem. We are trying to convince you that high dimensional real world problem can be reduced to a one dimensional map and cycle expansion is effective to predict the long time behavior of flow in this high dimensional space. This homework comes from the work done in reference [1] and [2], which have the cycle expansion result.

Due to limited time, we do not expect your cycle expansion result, so this problem will not be graded. However, we strongly recommend finishing the remaining work and compare your result with that in references [1] and [2].

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[1] Spatiotemporal chaos in terms of unstable recurrent patterns, F Christiansen, P Cvitanovic and V Putkaradze, arXiv:chao-dyn/9606016
[2] Unstable recurrent patterns in Kuramoto-Sivashinksy dynamics, Yueheng Lan, Predrag Cvitanovic, arXiv:0804.2474 [nlin.PS]