To Jorge Sotomayor Principal foliations are tangent to the directions of maximal and minimal normal curvatures on surfaces embedded in three space. Jorge Sotomayor initiated the qualitative study of structural stability for these foliations. This paper answers a question he raised in a set of lecture notes coauthored with Ronaldo Garcia about the existence of principal foliations with dense lines of curvature on simply connected surfaces. It demonstrates that there is a devil’s staircase of such foliations in the family of surfaces defined by Sotomayor and Garcia.

Trade-offs between producing costly movements for gathering information (‘explore’) and using previously acquired information to achieve a goal (‘exploit’) arise in a wide variety of problems, including foraging, reinforcement learning and sensorimotor control. Determining the optimal balance between exploration and exploitation is computationally intractable, necessitating heuristic solutions. Here we show that the electric fish Eigenmannia virescens uses a salience-dependent mode-switching strategy to solve the explore–exploit conflict during a refuge-tracking task in which the same category of movement (fore-aft swimming) is used for both gathering information and achieving task goals. The fish produced distinctive non-Gaussian distributions of movement velocities characterized by sharp peaks for slower, task-oriented ‘exploit’ movements and broad shoulders for faster ‘explore’ movements. The measures of non-normality increased with increased sensory salience, corresponding to a decrease in the prevalence of fast explore movements. We found the same sensory salience-dependent mode-switching behaviour across ten phylogenetically diverse organisms, from amoebae to humans, performing tasks such as postural balance and target tracking. We propose a state-uncertainty-based mode-switching heuristic that reproduces the distinctive velocity distribution, rationalizes modulation by sensory salience and outperforms the classic persistent excitation approach while using less energy. This mode-switching heuristic provides insights into purposeful exploratory behaviours in organisms, as well as a framework for more efficient state estimation and control of robots.

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 .

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.

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.

The lines of curvature of a surface embedded in R-3 together with umbilic points comprise its principal foliations. Monge described the principal foliations of the triaxial ellipsoid in the eighteenth century, but few examples beyond these and surfaces of revolution have been characterized since. This paper analyzes the principal foliations of perturbations of the ellipsoid. The results are surprising. Notably, surfaces with dense lines of curvature are prevalent A geometric construction establishes an intimate relation of the principal foliation on perturbed ellipsoids with nonvanishing vector fields on the two dimensional torus T-2, rather than the two sphere S-2. The dynamics of nonvanishing vector fields on T-2 with a global cross-section has been extensively studied. In generic one-parameter families, vector fields with dense, quasiperiodic trajectories occur at positive measure sets of parameters. This paper establishes a comparable result proving that large sets of embeddings of S-2 in R-3 have dense lines of curvature.

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 El Ni\~no-Southern Oscillation (ENSO) is a mode of interannual variability in the coupled equatorial Pacific coupled atmosphere/ocean system. El Ni\~no describes a state in which sea surface temperatures in the eastern Pacific increase and upwelling of colder, deep waters diminishes. El Ni\~no events typically peak in boreal winter, but their strength varies irregularly on decadal time scales. There were exceptionally strong El Ni\~no events in 1982-83, 1997-98 and 2015-16 that affected weather on a global scale. Widely publicized forecasts in 2014 predicted that the 2015-16 event would occur a year earlier. Predicting the strength of El Ni\~no is a matter of practical concern due to its effects on hydroclimate and agriculture around the world. This paper discusses the frequency and regularity of strong El Ni\~no events in the context of chaotic dynamical systems. We discover a mechanism that limits their predictability in a conceptual "recharge oscillator" model of ENSO. Weak seasonal forcing or noise in this model can induce irregular switching between an oscillatory state that has strong El Ni\~no events and a chaotic state that lacks strong events, In this regime, the timing of strong El Ni\~no events on decadal time scales is unpredictable.

The El Niño-Southern Oscillation (ENSO) is a mode of interannual variability in the coupled equatorial ocean/atmosphere Pacific. El Niño describes a state in which sea surface temperatures in the eastern Pacific increase and upwelling of colder, deep waters diminishes. El Niño events typically peak in boreal winter, but their strength varies irregularly on decadal time scales. There were exceptionally strong El Niño events in 1982-83, 1997-98 and 2015-16 that affected weather on a global scale. Widely publicized forecasts in 2014 predicted that the 2015-16 event would occur a year earlier. Predicting the strength of El Niño is a matter of practical concern due to its effects on hydroclimate and agriculture around the world. This paper presents a new robust mechanism limiting the predictability of strong ENSO events: the existence of an irregular switching between an oscillatory state that has strong El Niño events and a chaotic state that lacks strong events, which can be induced by very weak seasonal forcing or noise.

Very strong El Nino events occur sporadically every 10-20 yr. The origin of this bursting behavior still remains elusive. Using a simplified three-dimensional dynamical model of the tropical Pacific climate system, which captures El Nino-Southern Oscillation (ENSO) combined with recently developed mathematical tools for fast-slow systems, the authors 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.

Owing to aerodynamic instabilities, stable flapping flight requires ever-present fast corrective actions. Here, we investigate how flies control perturbations along their body roll angle, which is unstable and their most sensitive degree of freedom. We glue a magnet to each fly and apply a short magnetic pulse that rolls it in mid-air. Fast video shows flies correct perturbations up to 100° within 30 ± 7 ms by applying a stroke-amplitude asymmetry that is well described by a linear proportional–integral controller. For more aggressive perturbations, we show evidence for nonlinear and hierarchical control mechanisms. Flies respond to roll perturbations within 5 ms, making this correction reflex one of the fastest in the animal kingdom.

Invariant manifolds are key objects in describing how trajectories partition the phase spaces of a dynamical system. Examples include stable, unstable, and center manifolds of equilibria and periodic orbits, quasiperiodic invariant tori, and slow manifolds of systems with multiple timescales. Changes in these objects and their intersections with variation of system parameters give rise to global bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families of dynamical systems also play a prominent role in dynamical systems theory. Much progress has been made in developing theory and computational methods for invariant manifolds during the past 25 years. This article highlights some of these achievements and remaining open problems.

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 Shilnikov 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 singularities 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 Shilnikov used to study spiraling homoclinic bifurcations.

Running is an essential mode of human locomotion, during which ballistic aerial phases alternate with phases when a single foot contacts the ground. The spring-loaded inverted pendulum (SLIP) provides a starting point for modelling running, and generates ground reaction forces that resemble those of the centre of mass (CoM) of a human runner. Here, we show that while SLIP reproduces within-step kinematics of the CoM in three dimensions, it fails to reproduce stability and predict future motions. We construct SLIP control models using data-driven Floquet analysis, and show how these models may be used to obtain predictive models of human running with six additional states comprising the position and velocity of the swing-leg ankle. Our methods are general, and may be applied to any rhythmic physical system. We provide an approach for identifying an event-driven linear controller that approximates an observed stabilization strategy, and for producing a reduced-state model which closely recovers the observed dynamics.

This paper investigates the stability of mosquito flight. We employ a reduced model that represents the mosquito as a rigid body with two rigid wings. Wing motions derived from analysis of high speed movies are used as inputs to a dynamical model of the mosquito body. Our simulations use a variational integrator with a quasi-steady approximation for aerodynamic forces. We study uniform flight, consisting of periodic wing motions in a body frame that drive a net translation of the body each period. These trajectories are periodic orbits in a suitable translating reference frame. We analyze the stability of body motions in this frame by computing a linearized return map of the periodic orbit along with its eigenvalues and singular values. Long time stability corresponds to eigenvalues all smaller than one in magnitude and depends upon model parameters. For our mosquito geometry, we find that hovering flight is unstable but locate parameter ranges in which forward flight is stable. We also investigate the effect of varying the location of the joint between wings and body on stability of hovering flight. All cases are close to marginal stability. In contrast to eigenvalues, singular values measure short term growth of perturbations. Throughout the parameter ranges we investigated, there is a large singular value whose right singular vector locates a perturbation direction with growth on timescales comparable with the timescale of the insect's sensory feedback systems. Pseudospectra show that, even in cases when passive stability is achieved, there exist nearby systems that are unstable.

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.

Due to aerodynamic instabilities, stabilizing flapping flight requires ever-present fast corrective actions. Here we investigate how flies control body roll angle, their most susceptible degree of freedom. We glue a magnet to each fly, apply a short magnetic pulse that rolls it in mid-air, and film the corrective maneuver. Flies correct perturbations of up to 100^∘ within 30±7ms by applying a stroke-amplitude asymmetry that is well described by a linear PI controller. The response latency is ∼5ms, making the roll correction reflex one of the fastest in the animal kingdom.

The word chaos has become firmly embedded in the literature on dynamical systems. Indeed, James Gleick’s book, Chaos Theory [17], established that term as a description of the entire subject in the public mind. Nonetheless, there is no authoritative technical meaning of “chaos” in dynamical systems. Li and Yorke first used the word in the title of their paper “Period three implies chaos” [31], but it does not appear in the text.