In mathematics, a system is said to be chaotic if it is extremely sensitive to change in initial conditions. In a joint project, http://arxiv.org/abs/1604.06951, Mathematical Sciences faculty members Elena Dimitrova and Lea Jenkins, Computer Science faculty member Brian Dean, and graduate students Sherli Koshy (Math Sciences) and Akshay Galande (Computer Science) developed a computational framework that helps researchers acquire better understanding of the causes of chaotic behavior in dynamical systems and connections between chaos and sustainability of biological species. The platform they built allows for efficient systematic characterization and visualization of factors leading to chaotic behavior.
In the 1963, Edward Lorenz developed a simplified model for atmospheric convection which exhibits chaotic behavior: the famous butterfly-shaped strange attractor in Figure 1. The model is a system of three ordinary differential equations now known as the Lorenz equations: 
Here x, y, and z make up the system state, t is time, and r , s , and b are the system parameters. The Lorenz equations also arise in simplified models for lasers, dynamos, thermosyphons, brushless DC motors, electric circuits, chemical reactions, and forward osmosis.
Figure 1: A sample solution in the Lorenz attractor when r = 28, s = 10, and b = 8/3.
Soon after, mathematical biologists began searching for similar behavior in biological systems and quickly found it in simple population models. The possibility of chaotic behavior in such models created quite a stir in the 1970s [6]. It suggested that the intuitive view that complicated population dynamics can arise only from complicated interactions and environmental fluctuations would had to be abandoned. However, further work on models of both laboratory and real-world insect populations led scientists to doubt that chaotic behavior was actually present in real population dynamics. While presently there is little data of sufficiently high quality and long enough duration to test the idea that simple population systems exhibit chaotic behavior, there is little doubt that more complex population models do. The first such discovery was made by Cushing et al. in 2001 [2], when they announced that a laboratory population of the flour beetle tribolium exhibits chaotic dynamics. A few years later, Becks et al. demonstrated existence of chaotic and non-chaotic states in an experimental study of a microbial system with a nutrient source [1] and were able to document transitions between chaotic and non-chaotic states after changing the strength of the nutrient source. Other studies followed, demonstrating that chaotic behavior is present in a variety of species populations [5, 9, 3, 7, 4].
Models of population dynamics give us means of better understanding chaotic dynamics in biological systems. However, it is hard to decide if a system is actually chaotic since it can exhibit a wide range of chaotic and non-chaotic behavior, related in some nontrivial way to certain combinations of system parameters and initial conditions, making it hard for scientists without enough mathematical sophistication to identify and analyze the presence of chaotic behavior in their systems of interest.
To address this difficulty, the authors provided a framework based on the well-known Metropolis-Hastings (MH) sampling algorithm that allows researchers to efficiently search over a design space including parameter values and initial conditions and discover possible connections between values of these constants and chaotic dynamics. They also developed an interactive visualization platform to help the user understand the structure of the parameter space which produces chaotic behavior — a challenging task, due to the potentially complex, high-dimensional nature of this set. The platform is based on parallel coordinates, a popular means of visualizing high-dimensional data.
Figure 2: Chaotic samples visualized using parallel coordinates. These results were obtained using one of the systems described in the work by Sprott [8]. Note the wide range of values for and , while the range for is smaller.
Figure 2 shows an example of chaotic sample points generated from the equations of Case N in Table I of [8] and visualized using parallel coordinates.
The general system in [8] is given as
The varied parameters are , , , , and ; the remainder of the parameters were chosen to be zero. Each of the selected varied parameters is mapped to an individual coordinate axis, arranged in parallel from left to right, with each sample point drawn as a “poly line” that intersects each coordinate axis at the appropriate location. All five parameters were constrained to lie in the interval [–5, 5] during initial sampling, although by dynamically dragging the upper and lower endpoint markers on each axis, the user can further restrict the display so it only shows samples generated within a smaller sub-rectangle. Here, the upper bound on the parameter was decreased to –2.4, filtering out some of the 500 initial sample points initially present.
Figure 3: The result (from left to right) of dragging down the upper limit on the axis, observing a corresponding increase in the mean of the axis due to anticorrelation between the two.
Parallel coordinate plots allow us to understand a number of useful properties by visual inspection and interactive manipulation. For example, in Figure 2 we see that in order to achieve chaos with restricted to such a low range, the parameters and need to be anticorrelated: that is, there needs to be a negative relationship in which increases as decreases. This is visually apparent from the “X” pattern between the and axes, and we can also see it by dynamically dragging the upper bound on downward, watching the marker for the mean value on the axis move upward in lock step, as shown in Figure 3. Interestingly, if one raises the lower bound on , forcing this parameter to take large values, then the pattern between and becomes one of mostly straight horizontal lines, indicating correlation rather than anticorrelation.
By restricting several coordinates at a time, the user can filter an initially large number of sample points down to only a few. For example, if we restrict the range to half of each of five axes, this will on average show only of all sample points. In order to populate the filtered space with sufficiently many samples to understand its geometric structure, it may be necessary to re-launch a new round of MH sampling within this restricted space, requiring tight coupling between the user interface and the “back end” parallel MH sampler.
The ability to search the parameter space of a system and visualize the results can reveal chaotic behavior in systems not previously known to have chaotic regime and reveal the existence of parameters and initial conditions not previously known to yield chaotic behavior in studied systems. Furthermore, it allows for studying system behavior that is virtually impossible to observe in laboratory environment, thus making it useful to experimentalists.
We are currently studying the connection between chaotic dynamics and long-term survival of populations in ecological context. We focus on perhaps the most widely accepted criterion for long-term survival, permanence. Roughly speaking, a system is permanent if no orbit comes too close to the boundary. Our preliminary results demonstrate that the high responsiveness of chaotic systems allows for easier transition towards the desirable state of permanence, identifying chaos as a beneficial property of a system.
References
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[6] R.M. May. Simple mathematical models with very complicated dynamics. Nature, 261:459–567, 1978.
[7] F. Molz and B. Faybishenko. Increasing evidence for chaotic dynamics in the soil-plant-atmosphere system: A motivation for future research. Procedia Environmental Sciences, 19:681–690, 2013.
[8] J.C. Sprott. Some simple chaotic flows. Phys. Rev. E, 50(2):647–650, 1994.
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