By Nisha Chandramoorthy, Adam A. Śliwiak, and Qiqi Wang
Large and abrupt changes mark the past, present, and future of our planet’s climate. These changes may be caused by small perturbations to certain control parameters, like those that describe Earth’s orbit. Such small parameter perturbations can mediate enormous long-term effects because they are amplified by the strongly nonlinear and often chaotic behavior of climate components. For example, the nonlinear interactions between the dynamics of vegetation and precipitation heavily influence the climate of North Africa’s Sahel-Sahara region.
The precipitation trajectories (time series) of this complex climate system indicate the presence of at least two regimes of dynamic stability: wet and dry Sahel [7, 8]. These dynamic climatic regimes can persist for decades at a time, with often catastrophic effects [2]. Small parameter perturbations—such as sea surface temperature variations that modify precipitation levels—can cause an abrupt shift between these regimes. Mathematically, these regime shifts are dramatic changes in the probability distributions of system states, which equivalently manifest as long-term changes in climate features (in this case, the vegetation). One can observe such changes to the ensemble distribution of states due to small parameter fluctuations even in one-dimensional chaotic maps. Before demonstrating this concept, we will briefly describe the ergodic theory of chaotic dynamical systems; in particular, we focus on a class of idealized chaotic systems called uniformly hyperbolic systems.
In a chaotic system, an infinitesimal perturbation to the initial condition produces a new trajectory that exponentially diverges from the original. But when the system is ergodic, both trajectories asymptotically evolve in the long term within the same region of phase space—known as an attractor—and share the same statistics (i.e., the same probability distribution of their time series) on the attractor. That is, long time averages are independent of the initial state for almost every initial state. As the averaging time tends toward infinity in uniformly hyperbolic systems, long time averages converge to expectations with respect to an ergodic, physical, and stationary probability distribution over phase space. This distribution—known as the Sinai–Ruelle–Bowen (SRB) distribution [4]—is evident (through time averages) along almost every orbit. There is a zero-volume set of initial conditions whose trajectories do not yield expectations with respect to the SRB measure. Since they are almost never observed (e.g., they are unstable periodic orbits of a chaotic system [3]), we refer to them as nonphysical orbits.
The SRB distribution—which is the distribution of physical orbits—can change once the system’s control parameters are perturbed. As such, long-term statistics or climate features may remain unaltered by changes in the state, but they can be modified by parameter perturbations (e.g., by anthropogenic forcings). This notion inspires a question that is similar to Ed Lorenz’s query in the 1960s: Can a butterfly in Brazil yield a desired change in the climate of Texas? This is not a question about the predictability of the weather; rather, we ask whether strategically devised small-scale interventions can produce large and predictable climate effects. Can small parameter perturbations remarkably change long-term or ensemble averages? If so, can we compute this change?
We can approach this question mathematically via the theory of linear response in chaotic systems [5]. This theory suggests that when linear response holds, the change in statistics is proportional to the small parameter change — up to first order in the parameter change. For uniformly hyperbolic systems, David Ruelle rigorously proved a formula for the proportionality constant (called linear response) that only depends on the unperturbed system’s statistics. But linear response may not be valid for systems that are not uniformly hyperbolic; even among uniformly hyperbolic systems, linear response theory does not preclude arbitrarily large magnitudes of the linear response derivative [6].
Figure 1. Effect of small parameter changes on the physical distribution of one-dimensional chaotic systems. 1a. Four different chaotic maps that we obtained using perturbations of the parameter \(s\). 1b. The associated stationary distribution of the unperturbed map at \(s = 0\) is uniform (shown in red), while that at \(s = 0.1\) (shown in blue) is remarkably different — even though the perturbation to the map in Figure 1a appears negligible. Figure courtesy of [1].
As a prototypical example, we consider the chaotic maps of the interval \([0,2]\) in Figure 1a. The reference map is located at parameter value \(s=0\); applying a small perturbation to \(s\) generates several different chaotic maps whose perturbations are tiny enough to be invisible. However, the physical probability distributions of the unperturbed map and one of the perturbed maps are remarkably different, as illustrated in Figure 1b. In this case, we can use linear response theory to construct perturbations that induce drastic changes in the map’s statistics.
While we have demonstrated thus far that linear response can be large, another property of uniformly hyperbolic systems exists that—at first glance—supports small statistical responses to perturbations. This property is called shadowing. For every orbit of a perturbed system that is generated by a small parameter change, we can find an orbit of the unperturbed system that closely follows the perturbed orbit for all time. Shadowing is used to justify the correctness of numerical simulations of chaotic partial or ordinary differential equations. Since the system is chaotic, any small errors in the computer-generated solution grow exponentially. We can therefore view a computer-generated numerical solution as a perturbed orbit of the true system. Because of the shadowing property, this numerical solution is close to some true solution. Yet are the numerical solutions physical? That is, do perturbed orbits have the same long-term distribution as a physical orbit of the unperturbed dynamics [1]?
The perturbed orbit and shadowing orbit (which satisfies the unperturbed dynamics) have similar probability distributions in phase space because they are close to each other. However, this does not mean that a parameter perturbation cannot remarkably alter the statistics. Shadowing orbits can belong to the zero-volume set of initial conditions, which result in completely different statistics when compared to expectations with respect to the true system’s SRB measure. In other words, shadowing orbits do not need to be physical orbits of the system. As a result, the shadowing property does not guarantee that numerical simulations of chaos are physical.
Figure 2. Shadowing orbits can have very different distributions compared to the typical distribution of the system. 2a. Small perturbations of the parameter \(s\) produce slight variations of the tent map. 2b. The blue lines depict the histogram of a shadowing orbit (i.e., its probability distribution) that corresponds to a physical orbit at \(s = 0.1\), while the dotted black line represents a physical orbit at \(s = 0\) with a uniform probability distribution. Figure courtesy of [1].
Again, we use a one-dimensional chaotic system to see that shadowing orbits can almost surely be nonphysical. In Figure 2a, we plot perturbations of the same chaotic map from Figure 1a that we obtained by changing the parameter \(s\). The map at \(s=0\)—which is considered the unperturbed system—is shown in blue. In Figure 2b, the probability distribution of an orbit of the unperturbed system—in this case, a shadowing orbit that closely follows a physical orbit of the perturbed system at \(s=0.1\)—is shown in blue. A physical orbit of the unperturbed system, which has a uniform probability distribution, is marked by the dotted black line. This example shows that shadowing orbits can be nonphysical; they can have different probability distributions (i.e., different long-term statistics) when compared to the system’s physical orbits.
To provide broader context, computational linear response studies that utilize detailed chaotic climate models remain computationally prohibitive. Moreover, quantifying uncertainties in stochastic parameterizations—approximate modeling of turbulent oceanic and atmospheric fluid mechanics, for example—also remains an active challenge. Meanwhile, very low-dimensional chaotic models can produce unique insights into the behaviors of the extremely complex dynamical processes that make up Earth’s climate. Here we described two lessons from one-dimensional models that provide an improved understanding of the climate system and other large-scale chaotic dynamics: (i) Small perturbations can cause enormous changes in the long-term behavior of chaotic systems, and (ii) numerical simulations of chaos may not reproduce the true long-term behavior of the underlying physics.
Qiqi Wang presented this research during a minisymposium at the 2021 SIAM Conference on Computational Science and Engineering (CSE21), which took place virtually last year.
References
[1] Chandramoorthy, N., & Wang, Q. (2021). On the probability of finding nonphysical solutions through shadowing. J. Comput. Phys., 440, 110389.
[2] Foley, J., Coe, M., Scheffer, M., & Wang, G. (2003). Regime shifts in the Sahara and Sahel: Interactions between ecological and climatic systems in northern Africa. Ecosyst., 6, 524-532.
[3] Grebogi, C., Ott, E., & Yorke, J.A. (1988). Unstable periodic orbits and the dimensions of multifractal chaotic attractors. Phys. Rev. A, 37(5), 1711-1724.
[4] Ruelle, D. (1976). A measure associated with Axiom-A attractors. Am. J. Math., 98(3), 619-654.
[5] Ruelle, D. (1998). General linear response formula in statistical mechanics, and the fluctuation-dissipation theorem far from equilibrium. Phys. Lett. A, 245(3-4), 220-224.
[6] Śliwiak, A.A., Chandramoorthy, N., & Wang, Q. (2021). Computational assessment of smooth and rough parameter dependence of statistics in chaotic dynamical systems. Commun. Nonlin. Sci. Numer. Simul., 101, 105906.
[7] Wang, G., & Eltahir, E.A.B. (2000). Biosphere–atmosphere interactions over West Africa. II: Multiple climate equilibria. Q. J. R. Meteorol. Soc., 126(565), 1261-1280.
[8] Wang, G., & Eltahir, E.A.B. (2000). Ecosystem dynamics and the Sahel drought. Geophys. Res. Lett., 27(6), 795-798.
Nisha Chandramoorthy is a postdoctoral researcher at the Massachusetts Institute of Technology (MIT). She is interested in taking a dynamical systems approach to computational and mathematical problems related to the climate and other physical systems, as well as machine learning theory. Adam A. Śliwiak is a Ph.D. candidate in computational science and engineering at MIT, where he currently works on sensitivity analysis of chaotic dynamical systems. He is primarily interested in developing practical algorithms for prediction, control, and uncertainty quantification of complex physical systems. Qiqi Wang is a tenured associate professor in the MIT Department of Aeronautics and Astronautics. His research includes engineering design of chaotic dynamical systems, unsteady aerodynamics, and turbulence; numerical methods for exascale computation; and design optimization of uncertainty. He co-founded FlexCompute, which offers a fast and easy computational fluid dynamics service.