High dimensional chaotic systems which behave like random walks in state space

TL;DR

This paper analyzes an n-dimensional chaotic system that exhibits random walk-like behavior on a hypersphere, revealing how error growth and predictability vary with control parameters and linking it to turbulence.

Contribution

It introduces a high-dimensional generalization of Thomas's attractor and demonstrates its behavior as a constrained random walk, connecting chaos theory with turbulence predictability.

Findings

Error growth is limited and depends on control parameters.

Chaotic behavior resembles a constrained random walk.

Link established between high-dimensional chaos and turbulence predictability.

Abstract

By analysing an n-dimensional generalisation of Thomas's cyclically symmetric attractor we find that this chaotic dynamical system behaves like a random walk constrained onto the surface of a hypersphere. The growth of error is limited, with qualitatively different behaviour depending on a control parameter. For moderate values of the control parameter, linear growth of error is seen. For low values of the control parameter, the error is limited by the random walk behaviour. Finally, we link this to the predictability of homogeneous isotropic turbulence, which we find here also behaves like a constrained random walk.

Equations

$$\cos (\theta )\sim \exp (-t/\tau ),$$

$$D=2E(1-\exp (-t/\tau )).$$

$$r_i(t)=\{\begin{array}{ll}+\sqrt{s/n}& \text{with probability}=1/2\\ -\sqrt{s/n}& \text{with probability}=1/2\end{array}.$$

$$D(t)=\sum _i^n(x_i(t)-x_i(0){)}^2,$$

$${\partial }_{t}x(t)$$

$${\partial }_{t}y(t)$$

$${\partial }_{t}z(t)$$

$${\partial }_t{x}_i(t)=\sin (x_{i+1}(t))-b{x}_i(t),$$

$$E_d(t)=\sum _i(x_i(t)-y_i(t){)}^2.$$

$$D(t)$$

$$E_d(t)$$

Full text

I Introduction

Understanding predictability allows us to measure the accuracy with which we can know the future state of a system given knowledge of the current state. Predictability is limited in chaotic systems, whereby initially close states become exponentially different with time Lorenz1963 . The difference between these two states is called the error. Even within systems that become exponentially different, there may be physical processes which allow a non-zero amount of predictability even after a long period of time. Real world systems often have large numbers of degrees of freedom and are chaotic, but also have non-zero predictability Lorenz1969 .

In this paper, we first look at a random walk constrained to the surface of an n-dimensional hypersphere. Under this constraint, the difference between an initial state and its evolved state after a given period of time can be predicted, with the theoretical prediction agreeing with simulation. We then analyse a specific high dimensional chaotic system. The difference between an initial state and its evolved state is consistent with a constrained random walk like behaviour.

Furthermore, when we analyse the predictability of this chaotic system, we find that the error does not grow exponentially for all times, but instead has a limited growth rate. The nature of this limited growth rate depends on the control parameter with qualitatively different behaviour depending on the parameter.

Finally, we link the findings from this specific system and recent results from simulations of hydrodynamic turbulence. The results are qualitatively similar, suggesting a similar physical origin. This further suggests that such random walk like behaviour may be a universal feature of systems constrained by their energy.

II Random walk on a unit hypersphere

We may consider a random walk on the surface of hypersphere, centred on the origin, with $n$ dimensions. If we take $\theta$ to be the angle between a point and its origin, then the evolution of a point on the surface of this hypersphere obeys Caillol2004

[TABLE]

where $\tau$ is some diffusion rate and $t$ the time. We take a point on the hypersphere to have a Euclidean norm $E$. By the cosine rule, this means that the difference, $D$, between a point and its origin grows as

[TABLE]

We test the prediction in Eqn. (2) numerically. We initialize an array of numbers, $x_{i}$, using a standard normal distribution, which is the special case of a Gaussian distribution with zero mean and unit variance. The hypersphere is of dimension $n$, with $i$ labelling the dimension and ranging from 1 to $n$. We then normalize this array such that $E=\sum_{i}x_{i}^{2}=1$.

At each step, $t$, a random walk, $r_{i}(t)$, is generated. This random walk assigns $+\sqrt{s/n}$ or $-\sqrt{s/n}$ to each dimension of the hypersphere according to a Bernoulli distribution, where $+$ and $-$ are equally likely. The distribution is updated at each step, with

[TABLE]

The normalisation by $\sqrt{s/n}$ ensures that the average step is of size $s$, because the Euclidean norm of $r_{i}$, $|r|$, has $|r|=s$. We then update $x_{i}(t+1)=x_{i}(t)+r_{i}(t)$. After this update, we renormalize $x_{i}(t+1)$ such that $E(t+1)=\sum_{i}(x_{i}(t+1))^{2}=1$.

The initial distribution is evolved for 1000 steps to further randomize the distribution. The initial state of the system is defined as the state of the system after these 1000 steps. It would be interesting to see how an arbitrary distribution would change in time under this set-up if it were not constrained. However, this is outside the scope of the current investigation.

After this process, we take the initial state of the system, evolve it, and measure $D$, where

[TABLE]

and where $t=0$ is defined as the step corresponding to the initial state.

In Figure. 1 we show the result of a random walk constrained onto the surface of a unit hypersphere in $n$ = 100,000 dimensions. The step size, $s$, is 0.0005. The solid black line is the prediction, whereas the squares are the simulation results. As can be seen from the figure, the predicted and measured results show very good agreement. We find that the analogous quantity to the diffusion constant, $\tau$, has $1/\tau=s/2$.

III Application to an example chaotic system

In this section, we try to relate the above results regarding random walks on hyperspheres to many dimensional chaotic systems. A three-dimensional dissipative cyclically symmetric dynamical system was proposed by Thomas Thomas1999

[TABLE]

where $\partial_{t}$ is the derivative with respect to time, $t$. The control parameter $b$ represents a linear damping in the system, which if low enough allows chaotic dynamics.

Thomas’s cyclically symmetric attractor (TCSA) can be expanded to $n$ dimensions as

[TABLE]

where $i$ ranges from 1 to $n$, and $\partial x_{n}(t)=\sin(x_{1}(t))-bx_{n}(t)$. This system has been thoroughly examined and, at $b=0$, shows behaviour like a random walk, with independent and identically distributed steps Sprott2007 ; Chlouverakis2007 . In this system, energy, $E$, is defined $E(t)=\sum_{i}(x_{i}(t))^{2}$. Such a system is interesting to compare to the random walk constrained on a hypersphere because, for $b>0$, $E$ varies only slightly.

The simulation results gathered here come from solving TCSA in $n=10,000$ dimensions using Euler’s method. Using prior tests, it was found that, for high dimensions, an increase in dimension did not change the dynamics or results appreciably, but did allow for smoother statistics. Also, for $b\lesssim 0.01$, $E\sim 1/b$, but for higher values of $b$, $E$ was higher than this. As such, the field is initialized with a Gaussian distribution with zero mean and standard deviation $1/\sqrt{2b}$. This initialisation allows results which quickly reach a steady state of energy. Even so, the simulation is evolved for 50 time units to remove any artefacts from the initialisation.

After the initial evolution, both the difference, $D$, and error, $E_{d}$, are measured. Difference $D$ is defined as in Eqn. (4). To measure the evolution of the error, we create field $y_{i}(0)=x_{i}(0)+\delta_{i}$, where $\delta_{i}$ is a small white noise perturbation. Field $y_{i}(t)$ is then evolved according to Eqn. (6), with $x$ replaced by $y$. The error is defined

[TABLE]

Figure 2 shows $D$ as a function of time for TCSA in $n$ = 10,000 dimensions with $b=0.1$. For low $t$ there is an anomalous diffusion like behaviour where $D\sim t^{2}$, as is seen in the inset of the figure. However, for high $t$, $D$ behaves similar to the prediction for a random walk on a hypersphere given in Eqn. (2), but with a slight offset of time. At small times, the behaviour is dominated by the chaotic dynamics which produces the anomalous diffusion, but at large times, the system behaves like a constrained random walk.

Figure 3 shows $E_{d}$ as a function of time for TCSA in $n=10,000$ dimensions with $b=0.1$. This is the same system as shown in Fig. 2. For small time, error grows exponentially as expected in a chaotic system. However, for large times there is a limit on growth such that $E_{d}$ starts to grow linearly in time for an appreciable duration. Because the system has finite energy, saturation occurs at $2E$.

Surprisingly, for lower values of $b$, qualitatively different behaviour is found for $E_{d}$. Figure 4 shows $E_{d}$ as a function of time for TCSA in $n=10,000$ dimensions with $b=0.001$. Whilst there is still an initial exponential behaviour, with a similar Lyapunov exponent to the case of $b=0.1$, the long term behaviour is very different. Instead of being linear, the long term behaviour of $E_{d}(t)$ looks like that of $D(t)$.

The growth of $D$ for $b=0.001$ is slightly different than for $b=0.1$. For $b=0.1$, the $t^{2}$ growth transitions smoothly into the late $2E(1-\exp(-t/\tau))$ growth. However, for $b=0.001$, there is a kink in the graph such that the $t^{2}$ part can have a higher gradient than the $2E(1-\exp(-t/\tau))$ part.

The limiting behaviour of $E_{d}$ for low $b$ is presumably due to the following physical effect. We may consider an initial point in state space, after time $t$, the evolved state is distance $d$ from the initial state. If we have two points which are initially close to the initial state, then after time $t$, they should be at most distance $2d$ apart. As such, if $D$ has a certain time dependence, we should expect $E_{d}$ to be bounded in time by a similar time dependence. Because $E_{d}$ grows qualitatively differently for low and high $b$, the high $b$ behaviour of $E_{d}(t)$ must be limited by a different effect.

To investigate the different behaviour for different $b$, $b$ is varied from 0.001 to 0.22. We measure the maximum gradient for the growth of $E_{d}$ and $D$, which are labelled $m_{E}$ and $m_{D}$ respectively. An average is taken of 10 runs each value of $b$.

Figure 5 shows $m_{E}$ (triangles) and $m_{D}$ (circles) normalized by energy $E$ for varying $b$. From the figure we see that, going from low to high $b$, $m_{E}$ and $m_{D}$ increase proportionally to each other. Physically, an increase in $m_{D}$ allows $m_{E}$ to also increase because the bound on the growth of $m_{E}$ is increased. The values for $m_{E}$ are slightly higher than $m_{D}$, but this is possible since the bound on $E_{d}$ is $2D$. For $b$ = 0.001, the ratio $m_{E}/m_{D}=1.35\pm 0.04$.

However, around $b\approx 0.05$, an increase in $m_{D}$ is no longer accompanied by an increase in $m_{E}$. In this region, a different effect must be limiting the growth of $E_{d}$, Around $b=0.1$, the rate of $E_{d}$ growth is independent of $b$, and $m_{E}$ is roughly proportional to $E$, because $E$ is almost inversely dependent on $b$ in this region.

IV Comparison to turbulence

We compare the qualitative behaviours just described in the n-dimensional TCSA with other high dimensional chaotic systems to see if there are analogous behaviours. One such physical system is that of turbulent hydrodynamics. Turbulent flows are characterized by their Reynolds number, Re = $uL/\nu$, where $u$ is the typical velocity, $L$ a typical length scale, and $\nu$ the kinematic viscosity. A fluid field $\bm{u}$, evolving according to the Navier-Stokes equation, with high enough Re becomes turbulent, which means that the flow develops highly complex spatial and temporal structure and becomes chaotic.

Many recent exciting directions in understanding turbulence have come from the perspective of dynamical systems theory. These include relaminarisation of turbulence in pipe flow eckhardt2007turbulence and directed percolation models for understanding turbulent lifetimes lemoult2016directed . In this section we try to use the insights gained in the above section and apply them to the case of turbulence. We compare the behaviour of $D$ and $E_{d}$ between the n-dimensional TCSA and hydrodynamic turbulence. In hydrodynamics the difference, $D$, and error, $E_{d}$, are defined

[TABLE]

which are integrals over all space.

Figure 6 shows the result of measuring $D$ for a simulation of homogeneous isotropic turbulence with Re = 490. The details of this simulation are the same as those used for the main results in Berera2018 . As can be seen from the figure, the growth of $D$ is well approximated as following the prediction in Eqn. (2). The departure from prediction at later time is due to the fact that the assumption of constant energy is not as good in turbulence as it is for the n-dimensional TCSA. As well, similar to the n-dimensional TCSA, at small time there is an anomalous diffusion type growth where $D\sim t^{2}$, which can be seen in the inset. As such, we can say that the behaviour of hydrodynamic turbulence is consistent with that of a random walk on a high dimensional hypersphere.

V Conclusion and discussion

For three-dimensional turbulence with moderate dissipation, it has recently been seen that $E_{d}$ grows linearly in time with a rate proportional to the dissipation Berera2018 ; Boffetta2017 . This is analogous to the behaviour of the n-dimensional TCSA found in this paper at moderate $b$. However, it has also been found that for very low dissipation there is a qualitatively different type of behaviour for hydrodynamic turbulence. In two-dimensional flows, which allow for lower dissipation than three-dimensional ones, the evolution of $E_{d}(t)$ is more similar to that of $D(t)$ in Fig. 6. An example of this can be seen in Boffetta2001 . This suggests that the same bounding of error growth by $D$ that has been found for the n-dimensional TCSA here, also may hold in other high dimensional chaotic systems. Thus, in turbulence, we see both type of bounding effect seen here for the n-dimensional TCSA, linear and random walk like, as well as a dependence on a dissipation/damping parameter.

The fact that there are analogous behaviours for both $D$ and $E_{d}$ between the n-dimensional TCSA and hydrodynamic turbulence suggests that this may represent some more universal behaviour of high dimensional chaotic systems which are bounded by some energy constraint. We note finally that $D$ and $E_{d}$ are important quantities from the point of studies of predictability. They each give an idea of the way in which bounds may be made on future prediction given the degree of current knowledge of a system. There are many systems that are bounded by finite energy, but are otherwise chaotic, and the limit of their predictability should be that of a random walk on the surface of a hypersphere.

Acknowledgements.

This work has used resources from ARCHER archer via the Director’s Time budget. R.D.J.G.H is supported by the U.K. Engineering and Physical Sciences Research Council (EP/M506515/1). The author thanks Arjun Berera for helpful discussion.