2,015 research outputs found
Modeling Mechanisms of Cell Secretion
No full textSecretion is a fundamental cellular process involving the regulated release of intracellular products from cells. Physiological functions such as neurotransmission, or the release of hormones and digestive enzymes, are all governed by cell secretion. Anomalies in the processes involved in secretion contribute to the development and progression of diseases such as diabetes and other hormonal disorders. To unravel the mechanisms that govern such diseases, it is essential to understand how hormones, growth factors and neurotransmitters are synthesized and processed, and how their signals are recognized, amplified and transmitted by intracellular signaling pathways in the target cells. Here, we discuss diverse aspects of the detailed mechanisms involved in secretion based on mathematical models. The models range from stochastic ones describing the trafficking of secretory vesicles to deterministic ones investigating the regulation of cellular processes that underlie hormonal secretion. In all cases, the models are closely related to experimental results and suggest theoretical predictions for the secretion mechanisms
Separating manifolds in slow-fast systems
No full textThere are many applications that lead to models involving different timescales. For example, this is particularly the case for models of neurons, which involve dynamics of ionic channels across the cell membrane. Due to the slow-fast nature of such models it is difficult to use numerical tools for the investigation of the global behaviour. This paper discusses the computation of global invariant manifolds for slow-fast systems. We explain how the different timescales cause the numerical difficulties and give suggestions on how to deal with these problems. We illustrate the techniques with the computation of separating manifolds in a Hodgkin-Huxley type model of a somatotroph cell; this is an endocrine cell in the anterior pituitary that secretes growth hormone. There are two co-existing attractors in this model and their basins of attraction are separated by global stable manifolds of equilibria or periodic orbits
The geometry of the solution set of nonlinear optimal control problems
No full textIn an optimal control problem one seeks a time-varying input to a dynamical systems in order to stabilize a given target trajectory, such that a particular cost function is minimized. That is, for any initial condition, one tries to find a control that drives the point to this target trajectory in the cheapest way. We consider the inverted pendulum on a moving cart as an ideal example to investigate the solution structure of a nonlinear optimal control problem. Since the dimension of the pendulum system is small, it is possible to use illustrations that enhance the understanding of the geometry of the solution set. We are interested in the value function, that is, the optimal cost associated with each initial condition, as well as the control input that achieves this optimum. We consider different representations of the value function by including both globally and locally optimal solutions. Via Pontryagin's Maximum Principle, we can relate the optimal control inputs to trajectories on the smooth stable manifold of a Hamiltonian system. By combining the results we can make some firm statments regarding the existence and smoothness of the solution set
Computing Invariant Manifolds via the Continuation of Orbit Segments
No full textPath following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits
Boundary crisis bifurcation in two parameters
No full textThe boundary crisis bifurcation is well known as a mechanism for destroying (or creating) a strange attractor by variation of one parameter: at the moment of the boundary crisis bifurcation the strange attractor touches its own basin of attraction. Here we follow this codimension-one bifurcation in two parameters. One might expect that this leads to a smooth curve in the two-parameter plane. Mathematically, a boundary crisis is effectively a homoclinic or heteroclinic tangency, the locus of which is a well-defined smooth curve in a two-parameter system. However, instead of a boundary crisis, the transition through this tangency curve may lead to a basin boundary metamorphosis or an interior crisis bifurcation, in which the attractor persists. This phenomenon is again well known: at the point where the type of transition changes, the boundary crisis switches to another branch of homoclinic or heteroclinic tangencies, associated with manifolds of a periodic point with a different period than before. The curve of boundary crisis bifurcations is not differentiable at such points. In this paper we show that the curve of boundary crisis bifurcations is, in fact, not even well defined as a piecewise-smooth curve. We illustrate that there are infinitely many gaps in much the same way as the one-parameter bifurcation diagram of the attractor contains infinitely many windows where the attractor is periodic and not strange or chaotic. Throughout, we use the Henon map to illustrate our findings
The sculpture Manifold: a band from a surface, a surface from a band
No full textThe steel sculpture Manifold consists of an 8 cm wide closed band of stainless steel that winds around in an intricate way, curving and coming very close to itself. It is based on a complicated mathematical surface, known as the Lorenz manifold, which has an important role in organising the chaotic dynamics of the well-known Lorenz equations. Namely, this surface consists of all points that, under the force field generated by the Lorenz equations, end up at the origin of the three-dimensional phase space. This is special because all other points go to the famous Lorenz butterfly attractor. The Lorenz manifold can be found and represented numerically by a set of smooth closed curves consisting of points that lie at the same geodesic distance (given by the length of the shortest path on the surface) from the origin. Any band between two such curves illustrates an aspect of the geometry of the surface. As is explained in this paper, the sculpture Manifold represents a choice of band that is motivated by aesthetic, practical and mathematical considerations. The goal was to create an element of dynamicism while only hinting at the underlying surface
Arnol′d tongues arising from a grazing-sliding bifurcation
No full textThe Neımark–Sacker bifurcation, or Hopf bifurcation for maps, is a well-known bifurcation for smooth dynamical systems. At this bifurcation a periodic orbit loses stability, and, except at certain “strong” resonances, an invariant torus is born. The dynamics on the torus is organized by Arnol'd tongues in parameter space; inside the Arnol'd tongues phase-locked periodic orbits exist that disappear in saddle-node bifurcations on the tongue boundaries, and outside the tongues the dynamics on the torus is quasi-periodic. In this paper we investigate whether a piecewise-smooth system with sliding regions may exhibit an equivalent of the Neımark–Sacker bifurcation. The vector field defining such a system changes from one region in phase space to the next, and the dividing (or switching) surface contains a sliding region if the vector fields on both sides point toward the switching surface. We consider the grazing-sliding bifurcation at which a periodic orbit becomes tangent to the sliding region and provide conditions under which it can be thought of as a Neımark– Sacker bifurcation. We find that the normal form of the Poincar´e map derived at the grazing-sliding bifurcation is, in fact, noninvertible. The resonances are again organized in Arnol'd tongues, but the associated periodic orbits typically bifurcate in border-collision bifurcations that can lead to more complicated dynamics than simple quasi-periodic motion. Interestingly, the Arnol'd tongues of piecewise-smooth systems look like strings of connected sausages, and the tongues close at double border-collision points. Since in most models of physical systems nonsmoothness is a simplifying approximation, we relate our results to regularized systems. As one expects, the phase-locked solutions deform into smooth orbits that, in a neighborhood of the Ne˘ımark–Sacker bifurcation, lie on a smooth torus. The deformation of the Arnol'd tongues is more complicated; in contrast to the standard scenario, we find several coexisting pairs of periodic orbits near the points where the Arnol'd tongues close in the piecewise-smooth system. Nevertheless, the unfolding near the double border-collision points is also predicted as a typical scenario for nondegenerate smooth systems
Numerical continuation of canard orbits in slow-fast dynamical systems
No full textA trajectory of a system with two clearly separated time scales generally consists of fast segments (or jumps) followed by slow segments where the trajectory follows an attracting part of a slow manifold. The switch back to fast dynamics typically occurs when the trajectory passes a fold with respect to a fast direction. A special role is played by trajectories known as canard orbits, which do not jump at a fold but, instead, follow a repelling slow manifold for some time. We concentrate here on the case of a slow-fast system with two slow and one fast variable, where canard orbits arise geometrically as intersection curves of two-dimensional attracting and repelling slow manifolds. Canard orbits are intimately related to the dynamics near special points known as folded singularities, which in turn have been shown to explain small-amplitude oscillations that can be found as part of so-called mixed-mode oscillations. In this paper we present a numerical method to detect and then follow branches of canard orbits in a system parameter. More specifically, we define well-posed two-point boundary value problems that represent orbit segments on the slow manifolds, and we continue their solution families with the package AUTO. In this way, we are able to deal effectively with the numerical challenge of strong attraction to and strong repulsion from the slow manifolds. Canard orbits are detected as the transverse intersection points of the curves along which attracting and repelling slow manifolds intersect a suitable section (near a folded node). These intersection points correspond to a unique pair of orbits segments, one on the attracting and one on the repelling slow manifold. After concatenation of the respective pairs of orbits segements, all detected canard orbits are represented as solutions of a single boundary value problem, which allows us to continue them in system parameters. We demonstrate with two examples -- the self-coupled FitzHugh-Nagumo system and a three-dimensional reduced Hodgkin-Huxley model -- that branches of canard orbits can be computed reliably. Furthermore, our computations illustrate that the continuation of canard orbits allows one to find and investigate new types of dynamics, such as the interaction between canard orbits and a saddle periodic orbit that is generated in a singular Hopf bifurcation
Visualizing the transition to chaos in the Lorenz system
No full textThe Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This paper addresses the role of the global stable and unstable manifolds in organising the dynamics. More precisely, for the standard system parameters, the origin has a two-dimensional stable manifold and the other two equilibria each have a two-dimensional unstable manifold. The intersections of these manifolds in the three-dimensional phase space form heteroclinic connections from the nontrivial equilibria to the origin. A parameter-dependent visualization of these manifolds clarifies the transition to chaos in the Lorenz syste
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