scholarly journals A Mathematical Model of Demand-Supply Dynamics with Collectability and Saturation Factors

2017 ◽  
Vol 27 (01) ◽  
pp. 1750016 ◽  
Author(s):  
Y. Charles Li ◽  
Hong Yang
Keyword(s):  
Mathematical Model ◽  
Market Equilibrium ◽  
Basins Of Attraction ◽  
Periodic Attractor ◽  
Danger Zone ◽  
Small Perturbations ◽  
Demand And Supply ◽  
Flash Crash ◽  
Period Map ◽  
Irrational Exuberance

We introduce a mathematical model on the dynamics of demand and supply incorporating collectability and saturation factors. Our analysis shows that when the fluctuation of the determinants of demand and supply is strong enough, there is chaos in the demand-supply dynamics. Our numerical simulation shows that such a chaos is not an attractor (i.e. dynamics is not approaching the chaos), instead a periodic attractor (of period-3 under the Poincaré period map) exists near the chaos, and coexists with another periodic attractor (of period-1 under the Poincaré period map) near the market equilibrium. Outside the basins of attraction of the two periodic attractors, the dynamics approaches infinity indicating market irrational exuberance or flash crash. The period-3 attractor represents the product’s market cycle of growth and recession, while period-1 attractor near the market equilibrium represents the regular fluctuation of the product’s market. Thus our model captures more market phenomena besides Marshall’s market equilibrium. When the fluctuation of the determinants of demand and supply is strong enough, a three leaf danger zone exists where the basins of attraction of all attractors intertwine and fractal basin boundaries are formed. Small perturbations in the danger zone can lead to very different attractors. That is, small perturbations in the danger zone can cause the market to experience oscillation near market equilibrium, large growth and recession cycle, and irrational exuberance or flash crash.

Control of Current Reversal and Separation of Particles in Inertia Ratchets

2003 ◽  
Vol 790 ◽  
Author(s):  
F. Family ◽  
H.A. Larrondo ◽  
C.M. Arizmendi
Keyword(s):  
Driving Force ◽  
Control Parameter ◽  
Basins Of Attraction ◽  
Fractal Nature ◽  
Mean Velocity ◽  
Small Perturbations ◽  
Control Techniques ◽  
The Mean ◽  
Current Reversal ◽  
External Driving

ABSTRACTWe have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of both the amplitude and the frequency of an external driving force. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. Control processes employing small perturbations on the frequency and the amplitude of the external force may be designed in view of the intermixed fractal nature of the domains of attraction of the mean velocity attractors. The range where each control parameter is capable of reversing the current is determined. Quenched noise has significant effect on the basins of attraction. In particular, with increasing disorder the direction of a packet of particles can be reversed, leading to disappearance or weakening of the negative velocity attractor. The influence of the mass of the particles is also considered in order to design control techniques capable of separating particles of different masses.

Coupling between phosphate and calcium homeostasis: a mathematical model

2017 ◽  
Vol 313 (6) ◽  
pp. F1181-F1199 ◽  
Author(s):  
David Granjon ◽  
Olivier Bonny ◽  
Aurélie Edwards
Keyword(s):  
Mathematical Model ◽  
Vitamin D ◽  
Parathyroid Hormone ◽  
Production Rate ◽  
Fibroblast Growth Factor 23 ◽  
Feedback Loops ◽  
Fibroblast Growth ◽  
Small Perturbations ◽  
Intracellular Pool ◽  
Negative Feedback Loops

We developed a mathematical model of calcium (Ca) and phosphate (PO4) homeostasis in the rat to elucidate the hormonal mechanisms that underlie the regulation of Ca and PO4balance. The model represents the exchanges of Ca and PO4between the intestine, plasma, kidneys, bone, and the intracellular compartment, and the formation of Ca-PO4-fetuin-A complexes. It accounts for the regulation of these fluxes by parathyroid hormone (PTH), vitamin D3, fibroblast growth factor 23, and Ca2+-sensing receptors. Our results suggest that the Ca and PO4homeostatic systems are robust enough to handle small perturbations in the production rate of either PTH or vitamin D3. The model predicts that large perturbations in PTH or vitamin D3synthesis have a greater impact on the plasma concentration of Ca2+([Ca2+]p) than on that of PO4([PO4]p); due to negative feedback loops, [PO4]pdoes not consistently increase when the production rate of PTH or vitamin D3is decreased. Our results also suggest that, following a large PO4infusion, the rapidly exchangeable pool in bone acts as a fast, transient storage PO4compartment (on the order of minutes), whereas the intracellular pool is able to store greater amounts of PO4over several hours. Moreover, a large PO4infusion rapidly lowers [Ca2+]powing to the formation of CaPO4complexes. A large Ca infusion, however, has a small impact on [PO4]p, since a significant fraction of Ca binds to albumin. This mathematical model is the first to include all major regulatory factors of Ca and PO4homeostasis.

scholarly journals Dynamics of viscoelastic element of flow channel

2019 ◽  
Vol 21 (4) ◽  
pp. 488-506
Author(s):  
Nina I. Eremeeva ◽  
Petr A. Velmisov
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Computer Simulation ◽  
Numerical Method ◽  
Linear Formulation ◽  
Vector Equation ◽  
Small Perturbations ◽  
Subsonic Flows ◽  
Small Oscillations ◽  
Viscoelastic Element

We consider the plane problem of aerohydroelasticity on small oscillations arising during bilateral flow around a viscoelastic element located on the rectilinear wall of an infinite channel. A mathematical model describing the problem in a linear formulation and corresponding to small perturbations of homogeneous subsonic flows and small deflections of a viscoelastic element is formulated. Using the methods of the theory of functions of a complex variable, the solution of the problem is reduced to the study of the integro-differential equation with partial derivatives with respect to the deflection function of the element. To solve this equation, a numerical method based on the application of the Bubnov-Galerkin method is proposed, followed by the reduction of the resulting system of integro-differential equations to the Volterra vector equation of the second kind. On the basis of the developed numerical method the computer simulation of the dynamics of the deformable element is carried out.

Nonlinear Dynamics of the Parametric Pendulum with a View on Wave Energy Harvesting Applications

2021 ◽  
Author(s):  
FrancoE Dotti ◽  
JuanN Virla
Keyword(s):  
Nonlinear Dynamics ◽  
Mathematical Model ◽  
Power Generation ◽  
Wave Energy ◽  
Basins Of Attraction ◽  
Sea Waves ◽  
Bifurcation Diagrams ◽  
Parameter Spaces ◽  
Parametric Pendulum ◽  
Parameter Ranges

Abstract In this article, nonlinear dynamics tools are employed to quantify the ability of pendulum harvesters to recover energy from the sea waves. The versatility of pendulum harvesters is highlighted, as it is shown that devices can be scaled to produce a usable energy from 6 W to 10 kW. Several aspects of the pendulum's dynamics having a key influence on power generation are discussed by means of bifurcation diagrams, parameter spaces and basins of attraction. Parameter ranges that minimize the need for a control action are identified, and an explanation is provided on why tilting the pendulum's plane of rotation improves power generation. A practical mathematical model of the parametric pendulum is formulated for such purpose. This model incorporates the possibility of accounting an arbitrary number of concentric masses, while allowing a simple and direct correlation between dimensionless approaches and the myriad possible physical configurations of the system.

scholarly journals You Need Three Butterflies to Cause a Hurricane

2020 ◽  
Vol 67 (1) ◽  
pp. 139-155
Author(s):  
Gianluca Piero Maria Virgilio
Keyword(s):  
Mathematical Model ◽  
Flash Crash ◽  
Audit Trail ◽  
Model Based ◽  
Suggestive Evidence ◽  
Two Factors ◽  
High Volatility ◽  
Stop Loss ◽  
The Impact ◽  
Abnormal Events

The aim of this study is verifying the impact of high volatility, scarce liquidity and stop-loss orders on abnormal events like the May 6, 2010 Flash Crash. The paper assumes those three factors to be the main drivers, proposes a mathematical model based upon them and analyses audit trail data to verify whether those factors actually were at the origin of that event. It uses the concept of 'run', an uninterrupted sequence of trades all occurring in the same direction and compares volatility, liquidity and occurrence of stop-loss orders over the analysis period. The results found provide suggestive evidence that a combination of the three factors contributed to the crash. Each of them, taken individually, does not usually lead to extreme behaviours. Even two factors together may not disturb the orderly functioning of the markets but the combination of volatility, scarce liquidity and stop-loss orders may lead to a crisis.

Multistability in a Simplified Underwater Supercavity System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750121 ◽  
Author(s):  
Yipin Lv ◽  
Tianhong Xiong ◽  
Wenjun Yi
Keyword(s):  
Feedback Control ◽  
Deflection Angle ◽  
Initial Conditions ◽  
Major Change ◽  
Basins Of Attraction ◽  
Cavitation Number ◽  
Stable Equilibrium Point ◽  
Periodic Attractor ◽  
Control Gain ◽  
The Stability

Supercavity can increase the velocity of underwater vehicles greatly, however the launching state of vehicle and systematic parameters often lead to unstable motion. To solve the problem, the effect of parameters and initial conditions on the stability of vehicles is studied. With two variable parameters, namely cavitation number and feedback control gain of fin deflection angle, a simple dynamic model of supercavity system is studied. The multistability is verified through simulation. Robustness of the system is also analyzed based on its basins of attraction. There are various coexisting attractors in a relatively large region of parameter space of the supercavity system, namely coexistence of a stable equilibrium point and a periodic attractor, coexistence of various periodic attractors, coexistence of a periodic attractor with a chaotic attractor and so on, which explain the effect of parameters and initial values on stability of vehicles qualitatively. In addition, without major change in cavitation number, there is a negative correlation between the robustness of the vehicle and feedback control gain of fin deflection angle. The robustness can be improved through optimization of parameters.

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