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The inescapable link between sensing and movement in organisms generates a behavioral conflict between producing costly movements for gathering information (“explore”)1–3versus using previously acquired information to achieve a goal (“exploit”).4Determining the optimal balance between explore and exploit is a computationally intractable problem,5–7necessitating the use of heuristics to understand how organisms resolve this conflict.8Here we show that the electric fishEigenmannia virescens2, 3, 9uses a mode-switching strategy to solve the explore–exploit conflict8during a refuge tracking task,10–13and this strategy is modulated by sensory salience. The fish produced distinctive non-Gaussian (i.e., non-normal) distributions of movement velocities characterized by sharp peaks for slower, task-oriented “exploit” movements and broad shoulders for faster, “explore” movements. Surprisingly, the measures of non-normality increased (rather than decreased) in relation to increased sensory salience. Reanalysis of published data for additional ten species revealed that phylogenetically diverse organisms, from amoebae to humans14–22: (i) produce this distinctive distribution of movement velocities, and (ii) exhibit the same counter-intuitive modulation of mode-switching in relation to sensory salience. We propose a parsimonious, state-uncertainty based mode-switching heuristic that reproduces the distinctive velocity distribution, and explains its relationship to sensory salience. This mode-switching strategy likely manifests in diverse biological mechanisms, from cellular processes for single-cell motility to network-level processes for the control of movement in organisms.

The lines of curvature of a surface embedded in $\R^3$ comprise its principal foliations. Principal foliations of surfaces embedded in $\R^3$ resemble phase portraits of two dimensional vector fields, but there are significant differences in their geometry because principal foliations are not orientable. The Poincar\'e-Bendixson Theorem precludes flows on the two sphere $S^2$ with recurrent trajectories larger than a periodic orbit, but there are convex surfaces whose principal foliations are closely related to non-vanishing vector fields on the torus $T^2$. This paper investigates families of such surfaces that have dense lines of curvature at a Cantor set $C$ of parameters. It introduces discrete one dimensional return maps of a cross-section whose trajectories are the intersections of a line of curvature with the cross-section. The main result proved here is that the return map of a generic surface has \emph{breaks}; i.e., jump discontinuities of its derivative. Khanin and Vul discovered a qualitative difference between one parameter families of smooth diffeomorphisms of the circle and those with breaks: smooth families have positive Lebesgue measure sets of parameters with irrational rotation number and dense trajectories while families of diffeomorphisms with a single break do not. This paper discusses whether Lebesgue almost all parameters yield closed lines of curvature in families of embedded surfaces.

Continuation methods are a well established tool for following equilibria and periodic orbits in dynamical systems as a parameter is varied. Properly formulated, they locate and classify bifurcations of these key components of phase portraits. Principal foliations of surfaces embedded in \begin{document}$ \mathbb{R}^3 $\end{document} resemble phase portraits of two dimensional vector fields, but they are not orientable. In the spirit of dynamical systems theory, Gutierrez and Sotomayor investigated qualitative geometric features that characterize structurally stable principal foliations and their bifurcations in one parameter families. This paper computes return maps and applies continuation methods to obtain new insight into bifurcations of principal foliations.Umbilics are the singularities of principal foliations and lines of curvature connecting umbilics are analogous to homoclinic and heteroclinic bifurcations of vector fields. Here, a continuation method tracks a periodic line of curvature in a family of surfaces that deforms an ellipsoid. One of the bifurcations of these periodic lines of curvature are connections between lemon umbilics. Differences between these bifurcations and analogous saddle connections in two dimensional vector fields are emphasized. A second case study tracks umbilics in a one parameter family of surfaces with continuation methods and locates their bifurcations using Taylor expansions in "Monge coordinates." Return maps that are generalized interval exchange maps of a circle are constructed for generic surfaces with no monstar umbilics.

Very strong El Ni\~no events occur sporadically every 10-20 years. The origin of this bursting behavior still remains elusive. Using a simplified 3-dimensional dynamical model of the tropical Pacific climate system, which captures the El Ni\~no-Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast-slow systems we show that decadal ENSO bursting behavior can be explained as a Mixed Mode Oscillation (MMO), which also predicts a critical threshold for rapid amplitude growth. It is hypothesized that the MMO dynamics of the low-dimensional climate model can be linked to a saddle-focus equilibrium point, which mimics a tropical Pacific Ocean state without ocean circulation., Comment: 18 pages, 7 figures

Running, walking, flying and swimming are all processes in which animals produce propulsion by executing rhythmic motions of their bodies. Dynamical stability of the locomotion is hardly automatic: millions of older people are injured by falling each year. Stability frequently requires sensory feedback. We investigate how organisms obtain the information they use in maintaining their stability. Assessing stability of a periodic orbit of a dynamical system requires information about the dynamics of the system off the orbit. For locomotion driven by a periodic orbit, perturbations that "kick" the trajectory off the orbit must occur in order to observe convergence rates toward the orbit. We propose that organisms generate excitations in order to set the gains for stabilizing feedback. We hypothesize further that these excitations are stochastic but have heavy-tailed, non-Gaussian probability distributions. Compared to Gaussian distributions, we argue that these are more effective for estimating stability characteristics of the orbit. Finally, we propose experiments to test the efficacy of these ideas.

The Koper model is a three-dimensional vector field that was developed to study complex electrochemical oscillations arising in a diffusion process. Koper and Gaspard described paradoxical dynamics in the model: they discovered complicated, chaotic behavior consistent with a homoclinic orbit of Shil'nikov type, but were unable to locate the orbit itself. The Koper model has since served as a prototype to study the emergence of mixed-mode oscillations (MMOs) in slow-fast systems, but only in this paper is the existence of these elusive homoclinic orbits established. They are found first in a larger family that has been used to study singular Hopf bifurcation in multiple time scale systems with two slow variables and one fast variable. A curve of parameters with homoclinic orbits in this larger family is obtained by continuation and shown to cross the submanifold of the Koper system. The strategy used to compute the homoclinic orbits is based upon systematic investigation of intersections of invariant manifolds in this system with multiple time scales. Both canards and folded nodes are multiple time scale phenomena encountered in the analysis. Suitably chosen cross-sections and return maps illustrate the complexity of the resulting MMOs and yield a modified geometric model from the one Shil'nikov used to study spiraling homoclinic bifurcations., Comment: 23 pages, 13 figures (17 subfigures). To appear in SIADS

This article is part of a series commemorating the 50th anniversary of Ed Lorenz's seminal paper 'Deterministic nonperiodic flow'. The system he studied has become the object of extensive mathematical investigations and serves as a paradigm of a low-dimensional chaotic dynamical system. Nonetheless, Lorenz maintained his focus on the relationship between dynamical models and empirical data. This paper reviews some of the difficulties encountered in fitting chaotic models to data and then pursues fits of models with stable periodic orbits to data in the presence of noise. Starting with Lorenz's model in a regime with a stable periodic orbit, it investigates how well properties of the deterministic system can be recovered from trajectories of stochastic perturbations. A numerical study suggests, surprisingly, that the stability properties of the periodic orbit cannot be fully recovered from a finite length trajectory as the magnitude of the stochastic perturbation converges to zero. A heuristic framework for the analysis is developed and shown empirically to yield good estimates for the return map of the periodic orbit and its eigenvalues as the length of a stochastic trajectory increases.