Simple Ordinary Differential Equation Research Articles

Construction and analysis of symmetric Sprott B multi-attractors with electric implementation

Abstract In chaos theory, a number of systems and models which apparently contain simple ordinary differential equations (ODEs) turn out to show a dynamic characterized by complicated behaviors and complex trajectories. One of such systems is the Sprott B model. We construct some set of multi-attractors based on the Sprott B model where additional parameters and operators are considered. After summarizing important preliminaries relevant to simple chaotic differential systems, the model is firstly solved analytically and numerically, then graphical simulations are provided. The later show coexistence and evolution of two chaotic attractors in a symmetrical representation. Lastly, similar results and expected outcomes are recovered via an electrical circuit implementation, realized using the Field Programmable Gate Array (FPGA) board, the Digital-to-Analog Converter (DAC) and the Rigol Oscilloscope. They also show progressing sets of coexisting multi-attractors.

Nonlinear Perturbations and Weak Shock Waves in Isentropic Atmospheres

Acoustic perturbations to stellar envelopes can lead to the formation of weak shock waves via nonlinear wave steepening. Close to the stellar surface, the weak shock wave increases in strength and can potentially lead to the expulsion of part of the stellar envelope. While accurate analytic solutions to the fluid equations exist in the limits of low-amplitude waves or strong shocks, connecting these phases generally requires simulations. We address this problem using the fact that the plane-parallel Euler equations, in the presence of a constant gravitational field, admit exact Riemann invariants when the flow is isentropic. We obtain exact solutions for acoustic perturbations and show that after they steepen into shock waves, Whitham’s approximation can be used to solve for the shock’s dynamics in the weak to moderately strong regimes, using a simple ordinary differential equation. Numerical simulations show that our analytic shock approximation is accurate up to moderate (∼few–15) Mach numbers, where the accuracy increases with the adiabatic index.

Truncated Taylor Solvers for Simple First-Order Ordinary Differential Equations

Certain exact solutions of first-order ordinary differential equations naturally become corresponding solvers for simple firstorder ordinary differential equations whose form is the equality of a twice continuously differentiable function of the dependent variable to the derivative with respect the independent variable of the dependent variable. When the twice continuously differentiable function of the dependent variable is replaced by its truncated Taylor expansions through second order about its initial value, the resulting first-order ordinary differential equations have exact solutions that naturally become corresponding solvers for those simple first-order ordinary differential equations

A Model of Linear Response with Progressive Decay Equally Suited for RiboGreen® Quantitation of RNA and Dissolution of Solid Oral Dosage forms

A new regression model is presented which offers flexibility, freedom from subjective determinations of linear range, and very wide applicability to measurement systems of industrial importance. This “progressive decay” model starts as a deceptively simple ordinary differential equation. We show here that its solution faithfully describes real but seemingly unconnected data from a plate-based assay for quantitation of RNA with RiboGreen® and dissolution data for a triple fixed-dose combination solid oral dosage form.

Short-time expansion of one-dimensional Fokker-Planck equationswith heterogeneous diffusion.

We formulate a short-time expansion for one-dimensional Fokker-Planck equationswith spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation(the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equationswhich can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Characterization of partial wetting by CMAS droplets using multiphase many-body dissipative particle dynamics and data-driven discovery based on PINNs

The molten sand that is a mixture of calcia, magnesia, alumina and silicate, known as CMAS, is characterized by its high viscosity, density and surface tension. The unique properties of CMAS make it a challenging material to deal with in high-temperature applications, requiring innovative solutions and materials to prevent its buildup and damage to critical equipment. Here, we use multiphase many-body dissipative particle dynamics simulations to study the wetting dynamics of highly viscous molten CMAS droplets. The simulations are performed in three dimensions, with varying initial droplet sizes and equilibrium contact angles. We propose a parametric ordinary differential equation (ODE) that captures the spreading radius behaviour of the CMAS droplets. The ODE parameters are then identified based on the physics-informed neural network (PINN) framework. Subsequently, the closed-form dependency of parameter values found by the PINN on the initial radii and contact angles are given using symbolic regression. Finally, we employ Bayesian PINNs (B-PINNs) to assess and quantify the uncertainty associated with the discovered parameters. In brief, this study provides insight into spreading dynamics of CMAS droplets by fusing simple parametric ODE modelling and state-of-the-art machine-learning techniques.

A Parameter Identification Approach towards Hemodynamics based on Capnography

Research question Pulse detection decisions during cardiopulmonary resuscitation (CPR) are commonly augmented by end-tidal CO2-concentration (etCO2) levels and trend as etCO2 is a surrogate measure of systemic blood flow during (CPR)1. However, ventilatory and circulatory parameters influence the interpretability of etCO2 heavily. Recently developed methods of continuous cardiac output measurement based on capnography are not applicable during CPR due to the required controlled ventilation settings2. In this study, a parameter identification for a model of CO2 transport during resuscitation is used to estimate cardiac output based on capnography and tidal flow. Methodology After development of a compartmental ordinary differential equation (ODE) model to represent CO2 transport during CPR, parameters such as airway dead space and the level of systemic perfusion are estimated by a multishooting parameter identification approach3. To evaluate the method, tests are conducted on experimental data from a porcine model (approved by an ethics committee vote of the Austrian Federal Ministry of Education, Science and Research (GZ: 2021-0.895.386)). The estimated level of systemic perfusion is compared visually to invasively measured mean arterial pressure due to a lack of continuous cardiac output measurements as a ground truth. Results For exemplary data from a female pig (weight 50.4 kg), airway resistance was estimated to 20.8 hPa s /L, compliance: 25.2 mL/hPa, and dead space: 109 mL. Changes in cardiac output can be identified qualitatively as shown in Fig. 1. Conclusion A validated, simple ODE model for CO2 extraction during CPR could contribute quantifying the impact of ventilatory parameters on etCO2 and elucidate CO2 production, storage and transport during cardiac arrest. Assumptions on CO2 production during cardiac arrest and redistribution in the body, and the lacking cardiac output measurements complicate the analysis. Conflicts of Interest All authors have no conflict of interest to declare.

In situ study and assessment of the phosphorus-induced solute drag effect on the grain boundary mobility of austenite

The solute drag effect of phosphorus (P) in the single-phase austenite (γ-Fe) region was studied under isothermal annealing conditions at temperatures of 1050 °C, 1150 °C, 1250 °C and 1350 °C. High-temperature laser scanning confocal microscopy (HT-LSCM) was used to observe and quantify in situ grain growth on the surface of three different samples containing 0.026 wt.-% P, 0.044 wt.-% P and 0.102 wt.-% P. The dependence of the arithmetic mean grain sizes on time and temperature were modeled mathematically using a simple ordinary differential equation (ODE) according to normal grain growth (NGG) theory. As no other major effects, i.e., Zener-pinning by precipitates, occurred under the selected experimental conditions, grain growth interference was only considered by grain boundary (GB) segregation of P. Thus, the total grain boundary mobility (M) was directly determined depending on the P content and isothermal annealing temperature. The fitted GB mobility values enabled the determination of an average binding energy value between impurity P atoms and grain boundaries (E0 = -1.1 - -0.6 eV) in the system. Finally, the GB segregation of P in γ-Fe was derived from the observed grain growth kinetics. The results showed reasonable agreement with calculations using segregation enthalpies from the literature.

Multilevel optimization for policy design with agent-based epidemic models

Epidemiological modeling has a long history and is often used to forecast the course of infectious diseases or pandemics. These models come in different complexities, ranging from systems of simple ordinary differential equations (ODEs) to complex agent-based models (ABMs). The former allow a fast and straightforward optimization, but are limited in accuracy, detail, and parameterization, while the latter can resolve spreading processes in detail, but are extremely expensive to optimize. Epidemiological modeling can also be used to propose and design non-pharmaceutical interventions such as lockdowns. In general, their optimal design often leads to nonlinear optimization problems. We consider policy optimization in a prototypical situation modeled as both ODE and ABM, review numerical optimization approaches, and propose a heterogeneous multilevel approach based on combining a fine-resolution ABM and a coarse ODE model. Numerical experiments, in particular with respect to convergence speed, are given for illustrative examples.

A mathematical model of visceral leishmaniasis transmission and control: Impact of ITNs on VL prevention and elimination in the Indian subcontinent.

Visceral Leishmaniasis (VL) is a deadly, vector-borne, parasitic, neglected tropical disease, particularly prevalent on the Indian subcontinent. Sleeping under the long-lasting insecticide-treated nets (ITNs) was considered an effective VL prevention and control measures, until KalaNet, a large trial in Nepal and India, did not show enough supporting evidence. In this paper, we adapt a biologically accurate, yet relatively simple compartmental ordinary differential equations (ODE) model of VL transmission and explicitly model the use of ITNs and their role in VL prevention and elimination. We also include a game-theoretic analysis in order to determine an optimal use of ITNs from the individuals' perspective. In agreement with the previous more detailed and complex model, we show that the ITNs coverage amongst the susceptible population has to be unrealistically high (over 96%) in order for VL to be eliminated. However, we also show that if the whole population, including symptomatic and asymptomatic VL cases adopt about 90% ITN usage, then VL can be eliminated. Our model also suggests that ITN usage should be accompanied with other interventions such as vector control.

Infinite-memory classical wave-particle entities, attractor-driven active particles, and the diffusionless Lorenz equations.

A classical wave-particle entity (WPE) can materialize as a millimeter-sized droplet walking horizontally on the free surface of a vertically vibrating liquid bath. This WPE comprises a particle (droplet) that shapes its environment by locally exciting decaying standing waves, which, in turn, guides the particle motion. At high amplitude of bath vibrations, the particle-generated waves decay very slowly in time and the particle motion is influenced by the history of waves along its trajectory. In this high-memory regime, WPEs exhibit hydrodynamic quantum analogs where quantum-like statistics arise from underlying chaotic dynamics. Exploration of WPE dynamics in the very high-memory regime requires solving an integrodifferential equation of motion. By using an idealized one-dimensional WPE model where the particle generates sinusoidal waves, we show that in the limit of infinite memory, the system dynamics reduce to a 3D nonlinear system of ordinary differential equations (ODEs) known as the diffusionless Lorenz equations (DLEs). We use our algebraically simple ODE system to explore in detail, theoretically and numerically, the rich set of periodic and chaotic dynamical behaviors exhibited by the WPE in the parameter space. Specifically, we link the geometry and dynamics in the phase-space of the DLE system to the dynamical and statistical features of WPE motion, paving a way to understand hydrodynamic quantum analogs using phase-space attractors. Our system also provides an alternate interpretation of an attractor-driven particle, i.e., an active particle driven by internal state-space variables of the DLE system. Hence, our results might also provide new insights into modeling active particle locomotion.

Combining intransitive and higher-order effects in a coupled oscillator framework: A case study of an ant community.

A growing body of literature recognizes that pairwise species interactions are not necessarily an appropriate metaphorical molecule of community ecology. Two examples are intransitive competition and nonlinear higher-order effects. While these two processes have been discussed extensively, the explicit analysis of how the two of them behave when simultaneously part of the same dynamic system has not yet been explored theoretically. A concrete situation exists on coffee farms in Puerto Rico in which three ant species form an intransitive competitive triplet, and that triplet is strongly influenced, nonlinearly, by a fly parasitoid that modifies the competitive ability of one of the species. Using this arrangement as a template, we explore the dynamical consequences with a simple ordinary differential equation (ODE) model. Results are complicated and include alternative periodic and chaotic attractors. The qualitative structures of those complications, however, may be approximately retrieved from the basic natural history of the system.

The kind of silence: managing a reputation for voluntary disclosure in financial markets

We create a continuous-time setting in which to investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies) in the period between two consecutive mandatory disclosure dates: one with full and transparent disclosure termed candid, the other, termed sparing, under which values only above a dynamic threshold are disclosed. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and the optimal times for management to switch between the two. The model presented develops a number of insights, based on a very simple ordinary differential equation characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black–Scholes model and Poisson arrival of signals of firm value. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about). Significantly, with constant pay-for-performance parameters, at most one switching can occur.

Numerical Investigation of Magnetohydrodynamic Flow of Reiner– Philippoff Nanofluid with Gyrotactic Microorganism Using Porous Medium

Nanoparticles facilitate the enrichment of heat transmission, which is crucial in many industrial and technical phenomena. The suspension of nanoparticles with microbes is another intriguing study area that is pertinent to biotechnology, health sciences, and medicinal applications. In the dispersion of nanoparticles, the conventional non-Newtonian fluid Reiner-Philippoff flows across a stretching sheet, which is examined in this article using numerical analysis. This study investigates the numerical investigation of Arrhenius reaction, heat radiation, and vicious variation variations on a Reiner-Philippoff nanofluid of MHD flow through a stretched sheet. Thus, for the current nanofluid, nanoparticles and bio-convection are highly crucial. The set of nonlinear differential equations is translated into Ordinary Differential Equations (ODEs) utilizing the requisite translation of similarities. These collected simple ODE are solved using the MATLAB computational tool bvp4c method. The graphical results for the velocity, concentration, motile microorganisms, and temperature profile are defined using the thermophoresis parameter and the Brownian motion respectively. Consider a tube containing gyrotactic microbes and a regular flow of nanofluid which is electrically conducted through a porous stretched sheet surface. This nonlinear differential problem is solved by a hybrid numerical solution method using fourth-order Runge-Kutta with shooting technique. The optimization method also performs well in terms of predicting outcomes accurately. As a result, the research applies the Bayesian Regularization Method (BRM) to improve the accuracy of the prediction results. Physical constraints are plotted against temperature, velocity, concentration, and microorganism profile trends and they are briefly described.

One-parameter Darboux-deformed Fibonacci numbers

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (noninteger) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov–Lewis invariants for these sequences are also discussed.

Optimizing differential equations to fit data and predict outcomes.

Many scientific problems focus on observed patterns of change or on how to design a system to achieve particular dynamics. Those problems often require fitting differential equation models to target trajectories. Fitting such models can be difficult because each evaluation of the fit must calculate the distance between the model and target patterns at numerous points along a trajectory. The gradient of the fit with respect to the model parameters can be challenging to compute. Recent technical advances in automatic differentiation through numerical differential equation solvers potentially change the fitting process into a relatively easy problem, opening up new possibilities to study dynamics. However, application of the new tools to real data may fail to achieve a good fit. This article illustrates how to overcome a variety of common challenges, using the classic ecological data for oscillations in hare and lynx populations. Models include simple ordinary differential equations (ODEs) and neural ordinary differential equations (NODEs), which use artificial neural networks to estimate the derivatives of differential equation systems. Comparing the fits obtained with ODEs versus NODEs, representing small and large parameter spaces, and changing the number of variable dimensions provide insight into the geometry of the observed and model trajectories. To analyze the quality of the models for predicting future observations, a Bayesian-inspired preconditioned stochastic gradient Langevin dynamics (pSGLD) calculation of the posterior distribution of predicted model trajectories clarifies the tendency for various models to underfit or overfit the data. Coupling fitted differential equation systems with pSGLD sampling provides a powerful way to study the properties of optimization surfaces, raising an analogy with mutation-selection dynamics on fitness landscapes.

Foreword to the special issue on new developments in structural stability.

Foreword to the special issue on new developments in structural stability.

Energy computation and multistability in a class of second-order chaotic systems with simple nonlinearities: numerical, experimental and analytical results

The present work is aimed at the design, implementation and analysis of generic new second-order non-autonomous chaotic systems with simple nonlinear functions that have algebraically simple representations in mathematical form. These systems involve simple second-order non-autonomous ordinary differential equations with different simple nonlinearities like . The present study deals with numerical simulation, experimental analog circuit simulation results to demonstrate the ordered and chaotic phenomena being exhibited by these systems. Of most interest the striking phenomenon of multistability is uncovered for certain nonlinearities. Coexisting attractors are harnessed through the offset boosting approach. Also, the Hamilton energy is established through the Helmholtz theorem. It is found that both methods can be exploited as a non-bifurcation approach in characterizing multistability in the model. Further, analytical solutions developed for systems with certain nonlinearities are presented and the chaotic behavior of the systems studied using the solutions validating the numerical and experimental results are reported.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

New Solvable System of 2 First-Order Nonlinearly-Coupled Ordinary Differential Equations

In this short communication we introduce a rather simple autonomous system of 2 nonlinearly-coupled first-order Ordinary Differential Equations (ODEs), whose initial-values problem is explicitly solvable by algebraic operations. Its ODEs feature 2 right-hand sides which are the ratios of 2 homogeneous polynomials of first degree divided by the same homogeneous polynomial of second degree. The model features only 4 arbitrary parameters. We also report its isochronous variant featuring 4 nonlinearly-coupled first-order ODEs in 4 dependent variables, featuring 9 arbitrary parameters.