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

Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay

In this paper, the Hopf-zero bifurcation of the ring unidirectionally coupled Toda oscillators with delay was explored. First, the conditions of the occurrence of Hopf-zero bifurcation were obtained by analyzing the distribution of eigenvalues in correspondence to linearization. Second, the stability of Hopf-zero bifurcation periodic solutions was determined based on the discussion of the normal form of the system, and some numerical simulations were employed to illustrate the results of this study. Lastly, the normal form of the system on the center manifold was derived by using the center manifold theorem and normal form method

Mechanical Control of Sensory Hair-Bundle Function

Hair bundles detect sound in the auditory system, head position and rotation in the vestibular system, and fluid flow in the lateral-­‐‑line system. To do so, bundles respond to periodic, static, and hydrodynamic forces contingent upon the receptor organs in which they are situated. As the mechanosensory function of a hair bundle varies, so too do the mechanical properties of the bundle and its microenvironment. Hair bundles range in height from 1 μμm to 100 μμm and in stiffness from 100 μμN·∙m-­‐‑1 to 10,000 μμN·∙m-­‐‑1. They are composed of actin-­‐‑filled, hypertrophic microvilli—stereocilia—that number from fewer than 20 through more than 300 per bundle. In addition, bundles may or may not possess one true cilium, the kinocilium. Hair bundles differ in shape across organs and organisms: they may be isodiametric, fan-­‐‑shaped, or V-­‐‑shaped. Depending on the organ in which they occur, bundles may be free-­‐‑standing or they may be coupled to a tectorial membrane, otolithic membrane, cupula, or sallet. Because all hair bundles are comprised of similar molecular components, their distinct mechanosensory functions may instead be regulated by their mechanical loads. Dynamical-­‐‑systems analysis provides mathematical predictions of hair-­‐‑bundle behavior. One such model captures the effects of mechanical loading on bundle function in a state diagram. A mechanical-­‐‑load clamp permits exploration of this state diagram by robustly controlling the loads—constant force, load stiffness, virtual drag, and virtual mass—imposed on a hair bundle. Upon changes in these mechanical parameters, the bundle’s response characteristics alter. Subjected to particular control parameters, a bundle may oscillate spontaneously or remain quiescent. It may respond nonlinearly to periodic stimuli with high sensitivity, sharp frequency tuning, and easy entrainment; or it may respond linearly with low sensitivity, broad tuning, and reluctant entrainment. The bundle’s response to a force pulse may resemble that of an edge-­‐‑detection system or a low-­‐‑pass filter. Finally, a bundle from an amphibian vestibular organ can operate in a manner qualitatively similar to that from a mammalian auditory organ, implying an essential similarity between hair bundles. The bifurcation near which a bundle’s operating point resides controls its function: the state diagram provides a functional map of mechanosensory modalities. Auditory function is best tuned near a supercritical Hopf bifurcation, whereas vestibular function is captured by a subcritical Hopf bifurcation and a cusp bifurcation. Within the proposed region vestibular responsiveness, a hair bundle exhibits mechanical excitability analogous to the electrical excitability of neurons. This behavior implies for the first time a direct relationship between the mechanical behaviors of sensory organelles and the electrical behaviors of afferent neurons. Man-­‐‑made detectors function in limited capacities, each designed for a unique purpose. A single hair bundle, on the other hand, evolved to serve multiple purposes with the requirement of only two functional traits: adaptation and nonlinear channel gating. The remarkable conservation of these capabilities thus provides unique insight into the evolution of sensory systems

Lag Synchronization in Coupled Multistable van der Pol-Duffing Oscillators

We consider the system of externally excited identical van der Pol-Duffing oscillators unidirectionally coupled in a ring. When the coupling is introduced, each of the oscillator’s trajectories is on different attractor. We study the changes in the dynamics due to the increase in the coupling coefficient. Studying the phase of the oscillators, we calculate the parameter value for which we obtain the antiphase lag synchronization of the system and also the bifurcation values for which we observe qualitative changes in the dynamics of already synchronized system. We give evidence that lag synchronization is typical for coupled multistable systems

Anticipating Bifurcations for Identifying Dynamic Characteristics of Complex Systems

Complex systems are at risk of critical transitions when the system shifts abruptly from one state to another when a threshold is crossed. Recent studies have revealed that a variety of systems, ranging from systems examined by engineering, physics, and biology, to others related to climate sciences, medicine, social sciences, and ecology are susceptible to transitions leading to drastic re-organization or collapse. Such an unexpected transition is usually undesirable, because it is often difficult to restore a system to its pre-transition state once the transition occurs. It is exceedingly difficult to know if a system comes close to critical transitions because typically no easily noticeable changes can be observed unless the transition happens. Furthermore, accurate models are often not available, and predictions based on models of limited accuracy face difficulties. Hence, we are still ill-equipped to predict critical transitions, and there is an acute need for reliable methods to predict such catastrophic events. In this research, a data-driven, model-free approach is introduced to forecast critical points and post-critical dynamics of complex dynamical systems using measurements of the system response collected only in the pre-transition regime. Based on observations of the system response to perturbations only in the pre-transition regime, the method forecasts the bifurcation diagram and discovers the system’s stability after the transition. The forecasting approach is based on the phenomenon of critical slowing down, referring to the slowing down of a system's dynamics when approaching a tipping point. The rate of the system’s recovery from perturbations decreases when the system approaches the transition. Thus, the rate of recovery from perturbations can be used as an indicator, and is correlated to the distance to the transition. The method is employed to forecast critical transitions in several classes of complex systems including flutter instabilities in fluid-structural systems, collapse of natural populations in ecological systems, and the onset of traffic congestions in vehicular traffic flow systems. The theoretical and experimental results of this study address important challenges in forecasting safety and stability of complex systems. The capabilities of the methods proposed make them unique tools for analysis of complex systems in both computational and experimental studies.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/151735/1/aghadami_1.pd

Robustness Enhancement of Sensory Transduction by Hair Bundles

How do biological systems ensure robustness of function despite developmental and environmental variation? Our sense of hearing boasts exquisite sensitivity, precise frequency discrimination, and a broad dynamic range. Experiments and modeling imply, however, that the auditory system achieves this performance for only a narrow range of parameter values. Although the operation of some systems appears to require precise control over parameter values, I describe how the function of the ear might instead be made robust to parameter perturbation. The sensory hair cells of the cochlea are physiologically vulnerable: small changes in parameter values could compromise hair cells\u27 ability to detect stimuli. Most ears, however, remain highly sensitive despite differences in their physical properties. I propose that, rather than exerting tight control over parameters, the auditory system employs a homeostatic mechanism that increases the robustness of its operation to variation in parameter values. To slowly adjust the response to sinusoidal stimulation, the homeostatic mechanism feeds back to its adaptation process a rectified version of the hair bundle\u27s displacement. When homeostasis is enforced, the range of parameter values for which the sensitivity, tuning sharpness, and dynamic range exceed specified thresholds can increase by more than an order of magnitude. Certain characteristics of the hair cell\u27s behavior might provide a means to determine through experiment whether such a mechanism operates in the auditory system. This homeostatic strategy constitutes a general principle by which many biological systems might ensure robustness of function