John Guckenheimer

John Guckenheimer, Christian Kuehn

Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm.

John Guckenheimer, Christian Kuehn

This paper investigates travelling wave solutions of the FitzHugh-Nagumo equation from the viewpoint of fast-slow dynamical systems. These solutions are homoclinic orbits of a three dimensional vector field depending upon system parameters of the FitzHugh-Nagumo model and the wave speed. Champneys et al. [A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd, When Shilnikov meets Hopf in excitable systems, SIAM Journal of Applied Dynamical Systems, 6(4), 2007] observed sharp turns in the curves of homoclinic bifurcations in a two dimensional parameter space. This paper demonstrates numerically that these turns are located close to the intersection of two curves in the parameter space that locate non-transversal intersections of invariant manifolds of the three dimensional vector field. The relevant invariant manifolds in phase space are visualized. A geometrical model inspired by the numerical studies displays the sharp turns of the homoclinic bifurcations curves and yields quantitative predictions about multi-pulse and homoclinic orbits and periodic orbits that have not been resolved in the FitzHugh-Nagumo model. Further observations address the existence of canard explosions and mixed-mode oscillations.

Juan M. Bello-Rivas, Anastasia Georgiou, John Guckenheimer et al.

We introduce a method to successively locate equilibria (steady states) of dynamical systems on Riemannian manifolds. The manifolds need not be characterized by an a priori known atlas or by the zeros of a smooth map. Instead, they can be defined by point-clouds and sampled as needed through an iterative process. If the manifold is an Euclidean space, our method follows isoclines, curves along which the direction of the vector field $X$ is constant. For a generic vector field $X$, isoclines are smooth curves and every equilibrium lies on isoclines. We generalize the definition of isoclines to Riemannian manifolds through the use of parallel transport: generalized isoclines are curves along which the directions of $X$ are parallel transports of each other. As in the Euclidean case, generalized isoclines of generic vector fields $X$ are smooth curves that connect equilibria of $X$. Our algorithm can be regarded as an extension of the method of Newton trajectories to the manifold setting when the manifold is unknown. This work is motivated by computational statistical mechanics, specifically high dimensional (stochastic) differential equations that model the dynamics of molecular systems. Often, these dynamics concentrate near low-dimensional manifolds and have transitions (saddle points with a single unstable direction) between metastable equilibria. We employ iteratively sampled data and isoclines to locate these saddle points. Coupling a black-box sampling scheme (e.g., Markov chain Monte Carlo) with manifold learning techniques (diffusion maps in the case presented here), we show that our method reliably locates equilibria of $X$.

Andrew Lamperski, Debojyoti Biswas, Eric S. Fortune et al.

Active sensing is traditionally defined as the expenditure of energy, typically in the form of movement, for obtaining information. Here, we propose that the combination of reliance on adaptive sensors, the linkage between movement and sensing, and task-level control inevitably gives rise to the emergence of active sensing movements. In this way, active sensing is not driven by sensory goals, such as minimizing uncertainty about the state, but rather is necessary for task-level control. This hypothesis, that active sensing subserves control, is supported by both empirical data from organisms and mathematical theory. Interestingly, active sensing behaviors often occur in discrete epochs, interspersed with goal-oriented behavior. This suggests that animals switch between two behavioral modes with distinct control policies, an `explore' mode in which animals produce dynamic movements to shape sensory feedback, and an `exploit' mode in which animals produce slower compensatory movements that are directly related to achieving task goals. This strategy for feedback control that relies on adaptive sensors, active sensing, and mode switching is not commonly used in engineered systems despite being ubiquitous in biology. Engineered systems comprising state-of-the-art sensors, actuators, and mechanical designs can outperform animals with respect to ``cost functions'' such as maximum force generation, precision, and speed. Nevertheless, animals routinely achieve robust, graceful behaviors that are currently unmatched by engineered systems, suggesting that current control systems are insufficient. These insights, expressed in the language of control theory, may be critical for improving robotic sensing and control.