Abstract
In nonlinear systems, where explicit analytic solutions usually cannot be found, visualization is a powerful approach which can give insights into the dynamical behavior of models; it is also crucial for teaching this area of mathematics. In this paper, we present new software, Fireflies, which exploits the power of graphical processing unit (GPU) computing to produce spectacular interactive visualizations of arbitrary systems of ordinary differential equations. In contrast to typical phase portraits, Fireflies draws the current position of trajectories (projected onto 2D or 3D space) as single points of light, which move as the system is simulated. Due to the massively parallel nature of GPU hardware, Fireflies is able to simulate millions of trajectories in parallel (even on standard desktop computer hardware), producing “swarms” of particles that move around the screen in real-time according to the equations of the system. Particles that move forwards in time reveal stable attractors (e.g. fixed points and limit cycles), while the option of integrating another group of trajectories backwards in time can reveal unstable objects (repellers). Fireflies allows the user to change the parameters of the system as it is running, in order to see the effect that they have on the dynamics and to observe bifurcations. We demonstrate the capabilities of the software with three examples: a 2D “mean field” model of neuronal activity, the classical Lorenz system, and a 15D model of three interacting biologically realistic neurons.
Highlights
Many mathematical models are described by nonlinear ordinary differential equations (ODEs)
Bifurcation theory can be combined with numerical continuation in order to trace the boundaries in parameter space that separate different dynamical regimes, and several software packages exist for this purpose, such as AUTO [Doedel & Oldeman, 2012] and MATCONT [Dhooge et al, 2003]
When only the current position of each trajectory is shown, a very large number of particles must be used in order for the structure of the system to be visible — we find that several million are required for good results. Integrating this many equations quickly enough to display an interactive animation is not typically possible using the limited parallelism of traditional central processing units (CPU), but is an ideal problem for so-called “massively parallel” graphical processing units (GPUs)
Summary
Many mathematical models are described by nonlinear ordinary differential equations (ODEs). When only the current position of each trajectory is shown, a very large number of particles must be used in order for the structure of the system to be visible — we find that several million are required for good results Integrating this many equations quickly enough to display an interactive animation is not typically possible using the limited parallelism of traditional central processing units (CPU), but is an ideal problem for so-called “massively parallel” graphical processing units (GPUs). Fireflies consists of a user friendly graphical interface which can be used to produce simulations of N -dimensional systems of ODEs. Thanks to the power of GPU computing, these simulations can contain millions of particles while still running quickly enough to be interactive, allowing the user to change parameter values and immediately observe the effect of this on the particles’ motion. A collection of videos showing the systems described in this paper is available at https:// vimeo.com/album/3391568
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