Hopf Bifurcation and Chaos in a Single Inertial Neuron Model with Time Delay

A delayed differential equation modelling a single neuron with inertial term is considered in this paper. Hopf bifurcation is studied by using the normal form theory of retarded functional differential equations. When adopting a nonmonotonic activation function, chaotic behavior is observed. Phase plots, waveform plots, and power spectra are presented to confirm the chaoticity.Comment: 12 pages, 7 figure

Complex oscillations in the delayed Fitzhugh-Nagumo equation

Motivated by the dynamics of neuronal responses, we analyze the dynamics of the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canard-induced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

In this paper, the Hopf-pitchfork bifurcation of coupled van der Pol with delay is studied. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly, the normal form is gotten by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly, bifurcation diagrams and phase portraits are given through analyzing the unfolding structure. Finally, numerical simulations are used to support theoretical analysis

Bogdanov–Takens and triple zero bifurcations in general differential systems with m delays

This paper mainly concerns the derivation of the normal forms of the Bogdanov–Takens (BT) and triple zero bifurcations for differential systems with m discrete delays. The feasible algorithms to determine the existence of the corresponding bifurcations of the system at the origin are given. By using center manifold reduction and normal form theory, the coefficient formulas of normal forms are derived and some examples are presented to illustrate our main results

Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators

A chimera state is a spatio-temporal pattern in a network of identical coupled oscillators in which synchronous and asynchronous oscillation coexist. This state of broken symmetry, which usually coexists with a stable spatially symmetric state, has intrigued the nonlinear dynamics community since its discovery in the early 2000s. Recent experiments have led to increasing interest in the origin and dynamics of these states. Here we review the history of research on chimera states and highlight major advances in understanding their behaviour.Comment: 26 pages, 3 figure

Two adaptation processes in auditory hair cells together can provide an active amplifier

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves calcium ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce `self-tuned critical oscillations', i.e. maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the calcium dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations.Comment: 13 pages, 6 figures,REVTeX 4, To appear in Biophysical Journa

The Kuramoto model: A simple paradigm for synchronization phenomena

Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included

Neuroinspired control strategies with applications to flapping flight

This dissertation is centered on a theoretical, simulation, and experimental study of control strategies which are inspired by biological systems. Biological systems, along with sufficiently complicated engineered systems, often have many interacting degrees of freedom and need to excite large-displacement oscillations in order to locomote. Combining these factors can make high-level control design difficult. This thesis revolves around three different levels of abstraction, providing tools for analysis and design. First, we consider central pattern generators (CPGs) to control flapping-flight dynamics. The key idea here is dimensional reduction - we want to convert complicated interactions of many degrees of freedom into a handful of parameters which have intuitive connections to the overall system behavior, leaving the control designer unconcerned with the details of particular motions. A rigorous mathematical and control theoretic framework to design complex three-dimensional wing motions is presented based on phase synchronization of nonlinear oscillators. In particular, we show that flapping-flying dynamics without a tail or traditional aerodynamic control surfaces can be effectively controlled by a reduced set of central pattern generator parameters that generate phase-synchronized or symmetry-breaking oscillatory motions of two main wings. Furthermore, by using a Hopf bifurcation, we show that tailless aircraft (inspired by bats) alternating between flapping and gliding can be effectively stabilized by smooth wing motions driven by the central pattern generator network. Results of numerical simulation with a full six-degree-of-freedom flight dynamic model validate the effectiveness of the proposed neurobiologically inspired control approach. Further, we present experimental micro aerial vehicle (MAV) research with low-frequency flapping and articulated wing gliding. The importance of phase difference control via an abstract mathematical model of central pattern generators is confirmed with a robotic bat on a 3-DOF pendulum platform. An aerodynamic model for the robotic bat based on the complex wing kinematics is presented. Closed loop experiments show that control dimension reduction is achievable - unstable longitudinal modes are stabilized and controlled using only two control parameters. A transition of flight modes, from flapping to gliding and vice-versa, is demonstrated within the CPG control scheme. The second major thrust is inspired by this idea that mode switching is useful. Many bats and birds adopt a mixed strategy of flapping and gliding to provide agility when necessary and to increase overall efficiency. This work explores dwell time constraints on switched systems with multiple, possibly disparate invariant limit sets. We show that, under suitable conditions, trajectories globally converge to a superset of the limit sets and then remain in a second, larger superset. We show the effectiveness of the dwell-time conditions by using examples of nonlinear switching limit cycles from our work on flapping flight. This level of abstraction has been found to be useful in many ways, but it also produces its own challenges. For example, we discuss death of oscillation which can occur for many limit-cycle controllers and the difficulty in incorporating fast, high-displacement reflex feedback. This leads us to our third major thrust - considering biologically realistic neuron circuits instead of a limit cycle abstraction. Biological neuron circuits are incredibly diverse in practice, giving us a convincing rationale that they can aid us in our quest for flexibility. Nevertheless, that flexibility provides its own challenges. It is not currently known how most biological neuron circuits work, and little work exists that connects the principles of a neuron circuit to the principles of control theory. We begin the process of trying to bridge this gap by considering the simplest of classical controllers, PD control. We propose a simple two-neuron, two-synapse circuit based on the concept that synapses provide attenuation and a delay. We present a simulation-based method of analysis, including a smoothing algorithm, a steady-state response curve, and a system identification procedure for capturing differentiation. There will never be One True Control Method that will solve all problems. Nature's solution to a diversity of systems and situations is equally diverse. This will inspire many strategies and require a multitude of analysis tools. This thesis is my contribution of a few

Codimension two and three bifurcations of a predator–prey system with group defense and prey refuge

A predator–prey system with nonmonotonic functional response and prey refuge is considered. We mainly obtain that the system has the bifurcations of cusp-type codimension two and three, these illustrate that the dynamic behaviors of the model with prey refuge will become more complicated than the system with no refuge

Spontaneous oscillations of elastic filaments induced by molecular motors.

It is known from the wave-like motion of microtubules in motility assays that the piconewton forces that motors produce can be sufficient to bend the filaments. In cellular phenomena such as cytosplasmic streaming, molecular motors translocate along cytoskeletal filaments, carrying cargo which entrains fluid. When large numbers of such forced filaments interact through the surrounding fluid, as in particular stages of oocyte development in Drosophila melanogaster, complex dynamics are observed, but the detailed mechanics underlying them has remained unclear. Motivated by these observations, we study here perhaps the simplest model for these phenomena: an elastic filament, pinned at one end, acted on by a molecular motor treated as a point force. Because the force acts tangential to the filament, no matter what its shape, this 'follower-force' problem is intrinsically non-variational, and thereby differs fundamentally from Euler buckling, where the force has a fixed direction, and which, in the low-Reynolds-number regime, ultimately leads to a stationary, energy-minimizing shape. Through a combination of linear stability theory, analytical study of a solvable simplified 'two-link' model and numerical studies of the full elastohydrodynamic equations of motion, we elucidate the Hopf bifurcation that occurs with increasing forcing of a filament, leading to flapping motion analogous to the high-Reynolds-number oscillations of a garden hose with a free end