Stable Oscillatory Mode Research Articles

Can Stochastic Resonance Explain Recurrence of Grand Minima?

Abstract The amplitude of the 11 yr solar cycle is well known to be subject to long-term modulation, including sustained periods of very low activity known as Grand Minima. Stable long-period cycles found in proxies of solar activity have given new momentum to the debate about a possible influence of the tiny planetary tidal forcing. Here, we study the solar cycle by means of a simple zero-dimensional dynamo model, which includes a delay caused by meridional circulation as well as a quenching of the α-effect at toroidal magnetic fields exceeding an upper threshold. Fitting this model to the sunspot record, we find a set of parameters close to the bifurcation point at which two stable oscillatory modes emerge. One mode is a limit cycle resembling normal solar activity including a characteristic kink in the decaying limb of the cycle. The other mode is a weak sub-threshold cycle that could be interpreted as Grand Minimum activity. Adding noise to the model, we show that it exhibits Stochastic Resonance, which means that a weak external modulation can toss the dynamo back and forth between these two modes, whereby the periodicities of the modulation get strongly amplified.

Oscillatory convection of a colloidal suspension in a horizontal cell

The finite difference numerical simulations performed are for convection of a non-uniformly heated colloidal suspension (Hyflon MFA) filling a horizontal cell of finite length. The cell has solid and impermeable boundaries and is heated from below. A linear temperature distribution is maintained at the side walls. Due to the negative effect of thermodiffusion (Soret) and gravity sedimentation, a heavy impurity is collected at the hot lower boundary, and convection transfers it inside the cell. The stable and transient oscillatory convection modes observed experimentally are analyzed. If the initial distribution of the concentration is uniform, then stationary convection occurs in the cell. As the concentration inhomogeneities accumulate, oscillatory perturbations begin to increase. A modulated traveling wave may form in the layer. Stable traveling waves are observed when the Rayleigh number exceeds a certain critical value, which depends on the cell length. There is a good agreement between the experimental data [1] and the results of numerical research. The spatial structure of the concentration field and the time evolution of the convective characteristics of the colloid suspension are determined. The behavior of the colloidal suspension in the regimes of modulated traveling waves and transient flows near the threshold of convection, including localized traveling waves and waves that change their direction, is simulated and elucidated. By analyzing the behavior of the world lines of the vortices and the concentration field, the mechanism of formation of defects in the form of vortex coalescence is clarified. In the process of defect formation, two vortices with the opposite direction of rotation are involved.

Controllable switching between stable modes in a small network of pulse-coupled chemical oscillators.

Switching between stable oscillatory modes in a network of four Belousov-Zhabotinsky oscillators coupled in a ring via unidirectional inhibitory pulsatile coupling with a time delay is analysed computationally and experimentally. There are five stable modes in this network: in-phase, anti-phase, walk, walk reverse, and three-cluster modes. Transitions between the modes are carried out by short external pulses applied to one or several oscillators. We consider three types of switching between the modes: (i) forced switching, when the phases of oscillators of an initial mode reset in such a way that they correspond to the phases of the final mode; internal pulses of the network play no role in this resetting; (ii) "specific" switching, when the phase of only one oscillator is changed by an external perturbation which induces a chain of phase changes in other oscillators due to internal coupling between oscillators; and (iii) multistep switching through intermediate modes, which can be either stable or unstable attractors. All these types of switching have been found in simulations and verified in laboratory experiments.

Dynamic flight stability of hovering mosquitoes

The flight of mosquitoes is unusual compared with many other insects, such as fruit-flies and honey bees: mosquitoes fly with their legs spread; they also have rather short stroke amplitude, hence use different aerodynamic mechanisms to produce lift. Could their flight-stability properties be different from those of other insects? Here, we first measured wing kinematics and morphological parameters of two hovering mosquitoes, and then use the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques of eigenvalue and eigenvector analysis to study their stability properties. We found that their natural-mode structure is the same as that of many other insects: for the longitudinal motion, one unstable oscillatory mode, one stable fast subsidence mode and one stable slow subsidence mode; for the lateral motion: an unstable divergence mode, a stable oscillatory mode and a stable subsidence mode. The different aerodynamic mechanisms of mosquitoes do not change the major aerodynamic derivatives. The spread legs of mosquitoes have great effect on the moments of inertia and make the eigenvalue of the stable lateral mode much smaller. However, the leg-spreading has only a small quantitative effect on the unstable eigenvalues: the magnitudes of the eigenvalues in the two unstable modes, or the growth rate of the disturbances, are reduced by approximately 11%, compared to those calculated without considering the spread legs.

Thermohaline circulation in the North Atlantic and its simulation with a box model

Features of the North Atlantic thermohaline circulation response to periodic, stochastic, and instantaneous forcing are studied using a four-box model. The present-day circulation is shown to be characterized by a stable quasi-periodic oscillatory mode that manifests itself as the Atlantic Multidecadal Oscillation. The thermohaline catastrophe is unlikely in the modern climate epoch.

Hovering and forward flight of the hawkmoth Manduca sexta: trim search and 6-DOF dynamic stability characterization

We show that the forward flight speed affects the stability characteristics of the longitudinal and lateral dynamics of a flying hawkmoth; dynamic modal structures of both the planes of motion are altered due to variations in the stability derivatives. The forward flight speed ue is changed from 0.00 to 1.00 m s−1 with an increment of 0.25 m s−1. (The equivalent advance ratio is 0.00 to 0.38; the advance ratio is the ratio of the forward flight speed to the average wing tip speed.) As the flight speed increases, for the longitudinal dynamics, an unstable oscillatory mode becomes more unstable. Also, we show that the up/down (wb) dynamics become more significant at a faster flight speed due to the prominent increase in the stability derivative Zu (up/down force due to the forward/backward velocity). For the lateral dynamics, the decrease in the stability derivative Lv (roll moment due to side slip velocity) at a faster flight speed affects a slightly damped stable oscillatory mode, causing it to become more stable; however, the thalf (the time taken to reach half the amplitude) of this slightly damped stable oscillatory mode remains relatively long (∼12T at ue = 1 m s−1; T is wingbeat period) compared to the other modes of motion, meaning that this mode represents the most vulnerable dynamics among the lateral dynamics at all flight speeds. To obtain the stability derivatives, trim conditions for linearization are numerically searched to find the exact trim trajectory and wing kinematics using an algorithm that uses the gradient information of a control effectiveness matrix and fully coupled six-degrees of freedom nonlinear multibody equations of motion. With this algorithm, trim conditions that consider the coupling between the dynamics and aerodynamics can be obtained. The body and wing morphology, and the wing kinematics used in this study are based on actual measurement data from the relevant literature. The aerodynamic model of the flapping wings of a hawkmoth is based on the blade element theory, and the necessary aerodynamic coefficients, including the lift, drag and wing pitching moment, are experimentally obtained from the results of previous work by the authors.

Lateral Flight Stability of Two Hovering Model Insects

The longitudinal disturbance motion of different insects at hovering flight has the same modal structure. Here, we consider the case of lateral motion. The lateral dynamic flight stability of two model insects, hoverfly and honeybee, at hovering flight is studied. The method of computational fluid dynamics is applied to compute the stability derivatives. The techniques of eigenvalue and eigenvector analysis are used to solve the equations of motion. Results show that the lateral disturbance motion of the hoverfly has three natural modes of motion: an unstable divergence mode, a stable oscillatory mode and a stable subsidence mode, and the flight is unstable; while the honeybee has a different modal structure (a stable slow subsidence mode, a stable fast subsidence mode, and a nearly neutrally stable oscillatory mode) and the flight is nearly neutrally stable. The change in modal structure between the two insects is due to their roll-moment/side-velocity derivative having different sign, and the sign difference is because that the hoverfly has a relatively small, but the honeybee has a relatively large, distance between the wing roots and the center of mass. Thus, unlike the case of longitudinal motion, for lateral motion, some insects have different modal structures and stability properties from others.

Attractor in Circular Structure of Oscillatory Generalized Neural Elements

A perspective model of a neuron cell — the generalized neural element (GNE) is studied in this article. The model has an universal character. It combines properties of a neuron-oscillator and a neuron-detector. In this structure cyclic sequential pulse generation elements of the ring are studied. A nonlinear mapping for mismatches between pulses of neighboring elements is constructed. We prove the existence of a fixed point of this mapping (threshold value of mismatches) and its stability in a small neighborhood of the fixed point. In doing so the existence of a stable oscillatory mode of neural activity (attractor) of a certain type is proved. The parameters of the attractor (threshold values of mismatches) can be controlled in advance, due to the choice of synaptic weights of the links in the ring.

Flapping Flight Stability in Hover: A Comparison of Various Aerodynamic Models

The interest in flapping wing MAVs has been rising progressively in the past years. The complex aerodynamic mechanisms responsible for high manoeuvrability and energy efficient lift production have been modelled by multiple techniques. Here we present a comparison of several models from the perspective of flight stability in hover. We estimated the stability derivatives by aerodynamic models of various levels of complexity, ranging from analytical expressions derived from a quasi steady model to CFD results that were taken as a reference. The stability of the complete 6DOF linearized system, split into longitudinal and lateral part, was evaluated in terms of eigenvalues and eigenvectors. While the pole locations and modes of longitudinal system were consistent for all the models (two stable subsidence modes, one unstable oscillatory mode), the lateral dynamics showed that quasi steady models with only translational force are insufficient and revealed the necessity of including the rotational lift to obtain pole locations that are in accordance with the CFD study (one slow divergence mode, one fast subsidence mode and one stable oscillatory mode).

Dipteran insect flight dynamics. Part 1 Longitudinal motion about hover

This paper presents a reduced-order model of longitudinal hovering flight dynamics for dipteran insects. The quasi-steady wing aerodynamics model is extended by including perturbation states from equilibrium and paired with rigid body equations of motion to create a nonlinear simulation of a Drosophila-like insect. Frequency-based system identification tools are used to identify the transfer functions from biologically inspired control inputs to rigid body states. Stability derivatives and a state space linear system describing the dynamics are also identified. The vehicle control requirements are quantified with respect to traditional human pilot handling qualities specification. The heave dynamics are found to be decoupled from the pitch/fore/aft dynamics. The haltere-on system revealed a stabilized system with a slow (heave) and fast subsidence mode, and a stable oscillatory mode. The haltere-off (bare airframe) system revealed a slow (heave) and fast subsidence mode and an unstable oscillatory mode, a modal structure in agreement with CFD studies. The analysis indicates that passive aerodynamic mechanisms contribute to stability, which may help explain how insects are able to achieve stable locomotion on a very small computational budget.

Dynamic flight stability of a hovering model insect: lateral motion

The lateral dynamic flight stability of a hovering model insect (dronefly) was studied using the method of computational fluid dynamics to compute the stability derivatives and the techniques of eigenvalue and eigenvector analysis for solving the equations of motion. The main results are as following. (i) Three natural modes of motion were identified: one unstable slow divergence mode (mode 1), one stable slow oscillatory mode (mode 2), and one stable fast subsidence mode (mode 3). Modes 1 and 2 mainly consist of a rotation about the horizontal longitudinal axis (x-axis) and a side translation; mode 3 mainly consists of a rotation about the x-axis and a rotation about the vertical axis. (ii) Approximate analytical expressions of the eigenvalues are derived, which give physical insight into the genesis of the natural modes of motion. (iii) For the unstable divergence mode, t d, the time for initial disturbances to double, is about 9 times the wingbeat period (the longitudinal motion of the model insect was shown to be also unstable and t d of the longitudinal unstable mode is about 14 times the wingbeat period). Thus, although the flight is not dynamically stable, the instability does not grow very fast and the insect has enough time to control its wing motion to suppress the disturbances.

Mechanism of long transient oscillations in cyclic coupled systems

We consider unidirectionally coupled ring networks of neurons with inhibitory connections. It is known that when a number of inhibitory neurons is even, the system never has stable oscillatory modes. However, we observed oscillatory modes in such a case. In this paper, we clarify the oscillation mechanism in a state space. We investigate the relationship between these oscillatory modes and the unstable periodic solutions (UPSs) generated by the Hopf bifurcations. As a result we find that these transient oscillatory modes are generated by trajectories closing to the UPS and staying around it for a long time. We also confirm this phenomenon in a simple electrical circuit using inverting operational amplifiers.

Negative energy waves and MHD stability of rotating plasmas

Eigenmode analysis of ideal magnetohydrodynamic (MHD) systems with flows is performed. It is shown that the energy of stable oscillatory modes (waves) can be both positive and negative. Negative energy waves always correspond to non-symmetric modes which are nonuniform along the direction of the flow. Coupling of negative and positive energy waves is shown to be a universal mechanism of non-symmetric MHD instabilities in flowing media. To study the stability of non-symmetric modes, a new variational approach is developed based on Lyapunov theory. This approach provides a sufficient and (under some assumptions) necessary stability condition. Specific examples are given to illustrate the developed approach.

A Numerical Analysis of Dynamic Flight Stability of Hawkmoth Hovering

A numerical analysis of dynamic flight stability of a hovering hawkmoth is presented. A computational fluid dynamic (CFD) method is used to simulate the unsteady flow about a realistic hawkmoth model and to compute the aerodynamic derivatives of the aerodynamic forces and pitching moment in response with a series of small disturbances. With these parameters, the techniques of eigenvalue and eigenvector analysis is employed to investigate dynamic flight stability of the hawkmoth hovering. In the longitudinal disturbance motion, three natural modes are identified of a stable oscillatory mode, a stable fast subsidence mode and a stable slow subsidence mode, which indicate that the hawkmoth hovering flight is stable. In short, a hovering hawkmoth, if the body motion is dynamically stable and hence the disturbance dies out fast, might not need to make any adjustment with wing motions and could return to the equilibrium state ‘automatically’.

Energy of eigenmodes in magnetohydrodynamic flows of ideal fluids

Energy of eigenmodes in magnetohydrodynamic (MHD) flows of ideal fluids is studied analytically. It is shown that the energy of unstable modes is zero, while the energy of stable oscillatory modes (waves) can assume both positive and negative values. Negative energy waves always correspond to eigenmodes with a finite component of the wave-vector along the flow. Coupling of negative and positive energy waves is shown to be a universal mechanism of MHD instabilities in flowing media. As an example, the energy of eigenmodes of magnetorotational instability is calculated.

Elastic stability and the onset of plastic flow in accelerated solid plates

For accelerated incompressible, ideal elastoplastic plates of finite thickness with a preformed sinusoidal perturbation at the interface, we investigate the stability behavior encompassing neutral and most-unstable modes, stable oscillatory modes, and the onset of plastic flow. We show that the largest perturbation wavelength that can maximize the growth rate corresponds to a finite thickness plate. For elastically stable configurations, stress gradients that arise as a result of the interfacial disturbance can lead to the formation of counter-rotating particle displacement trajectories that tessellate the extent of the plate. By computing the spatiotemporal evolution of the stress tensor, we are able to construct the boundaries that demarcate the transition from elastically stable oscillatory modes to the onset of plastic flow based on the von Mises yield criterion. Earlier estimates of these boundaries for thick plates differ qualitatively and quantitatively from the present results, in which the common simplifying assumptions of thin-plate theory were not invoked. We show that multimodal solutions are necessary to accurately represent the actual oscillatory behavior of the stress tensor, which in general is not time periodic, that thin-plate bimodal solutions are unable to capture.

1134 昆虫羽ばたき飛行の動的安定性に関する研究(J04-1 生物の運動機能/バイオミメティクスとバイオメカニクス/バイオロボティクスとバイオメカトロニクス(1),J04 生物の運動機能/バイオミメティクスとバイオメカニクス/バイオロボティクスとバイオメカトロニクス)

A numerical study of dynamic flight stability of a hovering hawkmoth is presented. The method of computational fluid dynamics (CFD) is used to compute the aerodynamic derivatives of the aerodynamic forces and pitching moment in response with a series of small disturbances and the techniques of eigenvalue and eigenvector analysis is used for solving the equations of motion. In the longitudinal disturbance motion, three natural modes are identified of a stable oscillatory mode, a stable fast subsidence mode and a stable slow subsidence mode, which indicate that the hawkmoth hovering flight is stable. In short, a hovering hawkmoth, if the motion is dynamically stable and the disturbance die out fast, might not need to make any adjustment with wing motion and will return to the equilibrium state 'automatically'.

Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats

Single-nephron proximal tubule pressure in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the corresponding complex power spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero thick ascending limb (TAL) NaCl permeability, was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis. These techniques revealed four parameter regions, consistent with TGF gain and delays in SHR, where multiple stable model solutions are possible: 1) a region having one stable, time-independent steady-state solution; 2) a region having one stable oscillatory solution only, of frequency f1; 3) a region having one stable oscillatory solution only, of frequency f2, which is approximately equal to 2f1; and 4) a region having two possible stable oscillatory solutions, of frequencies f1 and f2. In addition, we conducted simulations in which TAL volume was assumed to vary as a function of time and simulations in which two or three nephrons were assumed to have coupled TGF systems. Four potential sources of spectral complexity in SHR were identified: 1) bifurcations that permit switching between different stable oscillatory modes, leading to multiple spectral peaks and their respective harmonic peaks; 2) sustained lability in delay parameters, leading to broadening of peaks and of their harmonics; 3) episodic, but abrupt, lability in delay parameters, leading to multiple peaks and their harmonics; and 4) coupling of small numbers of nephrons, leading to multiple peaks and their harmonics. We conclude that the TGF system in SHR may exhibit multistability and that the complex power spectra of the irregular TGF fluctuations in this strain may be explained by switching between multiple dynamic modes, temporal variation in TGF parameters, and nephron coupling.

Dynamic flight stability in the desert locust Schistocerca gregaria.

Here we provide the first formal quantitative analysis of dynamic stability in a flying animal. By measuring the longitudinal static stability derivatives and mass distribution of desert locusts Schistocerca gregaria, we find that their static stability and static control responses are insufficient to provide asymptotic longitudinal dynamic stability unless they are sensitive to pitch attitude (measured with respect to an inertial or earth-fixed frame) as well as aerodynamic incidence (measured relative to the direction of flight). We find no evidence for a 'constant-lift reaction', previously supposed to keep lift production constant over a range of body angles, and show that such a reaction would be inconsequential because locusts can potentially correct for pitch disturbances within a single wingbeat. The static stability derivatives identify three natural longitudinal modes of motion: one stable subsidence mode, one unstable divergence mode, and one stable oscillatory mode (which is present with or without pitch attitude control). The latter is identified with the short period mode of aircraft, and shown to consist of rapid pitch oscillations with negligible changes in forward speed. The frequency of the short period mode (approx. 10 Hz) is only half the wingbeat frequency (approx. 22 Hz), so the mode would become coupled with the flapping cycle without adequate damping. Pitch rate damping is shown to be highly effective for this purpose - especially at the small scales associated with insect flight - and may be essential in stabilising locust flight. Although having a short period mode frequency close to the wingbeat frequency risks coupling, it is essential for control inputs made at the level of a single wingbeat to be effective. This is identified as a general constraint on flight control in flying animals.

Dynamics of biochemical oscillators in a large number of interacting cells

6. Summary We investigated here minimal models for coupled metabolic oscillators in cell suspensions. They are based on a feedback-activation mechanism, proposed for the explanation of the glycolytic oscillations. Despite the simplicity of the models we found a wide variety of complex dynamical phenomena. Depending on the kinetic parameters interacting cells may oscillate synchronous or asynchronous. It was shown, that there are different possibilities for asynchronous oscillations. (a) Depending on the type of coupling regular asynchronous behaviour may occur near to the boundary of stability. This type of behaviour was only possible, if the coupling metabolite belongs to the pool of products of the autocatalytic reaction. (b) Leaving the near neighbourhood of the points of Hopf-bifurcations in both models nonregular asynchronous behaviour may arise by secondary symmetry breaking bifurcations, either from the branch of synchronous oscillations or from the branch of regular asynchronous oscillations. In model II the coupling opens the possibility of multiple steady states. The whole population may be within different stable steady states or different oscillatory modes. The results of our study may be helpful for the interpretation of experimental data in this field. The mechanism of interaction, which was investigated here, has been proposed for many biological systems. Especially for yeast cell suspensions the question of the coupling intermediates is studied intensively. In recent time acetaldehyde was reported to be the synchronizer of the oscillations [5], but this substance was shown to desynchronize the oscillations too [4]. Without discussing the problem, whether or not this substance mediates the coupling, we want to emphasize that any metabolite may play the role of a synchronizer as well as that of a desynchronizer. Moreover we have demonstrated that the regular asynchronous oscillations may lead to the phenomenon of hidden oscillations, where it may be difficult to observe cellular oscillations.