Original Ordinary Differential Equations Research Articles

Comparison of damping techniques in non-linear spring-mass systems for power transformers under short-circuit condition

PurposeThe simplified modeling of many physical processes results in a second-order ordinary differential equation (ODE) system. Often the damping of these resonating systems cannot be defined in the same simplified way as the other parameters due to the complexity of the physical effects. The purpose of this paper is to develop a mathematically stable approach for damping resonances in nonlinear ODE systems.Design/methodology/approachModifying the original ODE using the eigenvalues and eigenvectors of a linearized state leads to satisfying results.FindingsAn iterative approach is presented, how to modify the original ODE, to achieve a well-damped solution.Practical implicationsThe method can be applied for every physical resonating system, where the model complexity prevents the determination of the damping.Originality/valueThe iterative algorithm to modify the original ODE is novel. It can be used on different fields of the physics, where a second-order ODE is describing the problem, which has only measured or empirical damping.

Analytical theory for a droplet squeezing through a circular pore in creeping flows under constant pressures

We derived equations and closed-form solutions of transit time for a viscous droplet squeezing through a small circular pore with a finite length at microscale under constant pressures. Our analyses were motivated by the vital processes of biological cells squeezing through small pores in blood vessels and sinusoids and droplets squeezing through pores in microfluidics. First, we derived ordinary differential equations (ODEs) of a droplet squeezing through a circular pore by combining Sampson flow, Poiseuille flow, and Young–Laplace equations and took into account the lubrication layer between the droplet and the pore wall. Second, for droplets wetting the wall with small surface tension, we derived the closed-form solutions of transit time. For droplets with finite surface tension, we solved the original ODEs numerically to predict the transit time. After validations against experiments and finite element simulations, we studied the effects of pressure, viscosity, pore/droplet dimensions, and surface tension on the transit time. We found that the transit time is inversely linearly proportional to pressure when the surface tension is low compared to the critical surface tension for preventing the droplet to pass and becomes nonlinear when it approaches the critical tension. Remarkably, we showed that when a fixed percentage of surface tension to critical tension is applied, the transit time is always inversely linearly proportional to pressure, and the dependence of transit time on surface tension is nonmonotonic. Our results provided a quick way of quantitative calculations of transit time for designing droplet microfluidics and understanding cells passing through constrictions.

General Exact Schemes for Second-Order Linear Differential Equations Using the Concept of Local Green Functions

In this paper, we introduce a special system of linear equations with a symmetric, tridiagonal matrix, whose solution vector contains the values of the analytical solution of the original ordinary differential equation (ODE) in grid points. Further, we present the derivation of an exact scheme for an arbitrary mesh grid and prove that its application can completely avoid other errors in discretization and numerical methods. The presented method is constructed on the basis of special local green functions, whose special properties provide the possibility to invert the differential operator of the ODE. Thus, the newly obtained results provide a general, exact solution method for the second-order ODE, which is also effective for obtaining the arbitrary grid, Dirichlet, and/or Neumann boundary conditions. Both the results obtained and the short case study confirm that the use of the exact scheme is efficient and straightforward even for ODEs with discontinuity functions.

A Continuation Method for Large-Scale Modeling and Control: From ODEs to PDE, a Round Trip

In this article, we present a continuation method, which transforms spatially distributed ordinary differential equation (ODE) systems into a continuous partial differential equation (PDE). We show that this continuation can be performed for both linear and nonlinear systems, including multidimensional, space-varying, and time-varying systems. When applied to a large-scale network, the continuation provides a PDE describing the evolution of a continuous-state approximation that respects the spatial structure of the original ODE. Our method is illustrated by multiple examples, including transport equations, Kuramoto equations, and heat diffusion equations. As a main example, we perform the continuation of a Newtonian system of interacting particles and obtain the Euler equations for compressible fluids, thereby providing an original solution to Hilbert’s sixth problem. Finally, we leverage our derivation of Euler equations to solve a control problem multiagent systems, by designing a nonlinear control algorithm for robot formation based on its continuous approximation.

Two-parameter bifurcations analysis of a delayed high-temperature superconducting maglev model with guidance force.

A modified high-temperature superconducting maglev model is studied in this paper, mainly considering the influence of time delay on the dynamic properties of the system. For the original model without time delay, there are periodic equilibrium points. We investigate its stability and Hopf bifurcation and study the bifurcation properties by using the center manifold theorem and the normal form theory. For the delayed model, we mainly study the co-dimension two bifurcations (Bautin and Hopf-Hopf bifurcations) of the system. Specifically, we prove the existence of Bautin bifurcation and calculate the normal form of Hopf-Hopf bifurcation through the bifurcation theory of functional differential equations. Finally, we numerically simulate the abundant dynamic phenomena of the system. The two-parameter bifurcation diagram in the delayed model is given directly. Based on this, some nontrivial phenomena of the system, such as periodic coexistence and multistability, are well presented. Compared with the original ordinary differential equation system, the introduction of time delay makes the system appear chaotic behavior, and with the increase in delay, the variation law between displacement and velocity becomes more complex, which provides further insights into the dynamics of the high-temperature superconducting maglev model.

One-to-one internal resonance of a cable-beam structure subjected to a concentrated load

Cable-beam composite structure, such as a cable-stayed bridge in cantilever construction and a tower crane hoisting load, will sustain dynamic excitation. In order to investigate the dynamic behaviors at this stage, a cable-supported cantilever beam model subjected to a concentrated harmonic excitation at the cantilever end of the beam is established. The interactions including the second-order effect of the axial compressive load on the beam and the dynamic interaction between the beam and the cable due to cable's dynamic elongation are considered. A reduced two-degree-of-freedom system is obtained by using Galerkin discretization. Multiple time scales method is applied to solve the ordinary differential equations. Based on the derived modulation equations and the ordinary differential equations, frequency response, amplitude response, phase plane and time history curves are provided to explore the dynamic behaviors of the system. Three different lengths of the cable-stayed cantilever beam system are considered to study the changes in dynamic behaviors. In order to verify the validity in each case, Runge-Kutta method is used to directly solve the original ordinary differential equations and two results have a satisfying consistence. The analysis results show that frequency responses of both the beam and the cable exhibit hardening spring properties; the cable has more rich and complicated dynamic behaviors than the beam, for example the cable has more significant jump phenomenon.

Quantum homotopy perturbation method for nonlinear dissipative ordinary differential equations

While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving n-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state ϵ-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is O(gηT poly(log(nT/ϵ))), where η, g measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in n or ϵ.

A multiscale multicellular spatiotemporal model of local influenza infection and immune response

Respiratory viral infections pose a serious public health concern, from mild seasonal influenza to pandemics like those of SARS-CoV-2. Spatiotemporal dynamics of viral infection impact nearly all aspects of the progression of a viral infection, like the dependence of viral replication rates on the type of cell and pathogen, the strength of the immune response and localization of infection. Mathematical modeling is often used to describe respiratory viral infections and the immune response to them using ordinary differential equation (ODE) models. However, ODE models neglect spatially-resolved biophysical mechanisms like lesion shape and the details of viral transport, and so cannot model spatial effects of a viral infection and immune response. In this work, we develop a multiscale, multicellular spatiotemporal model of influenza infection and immune response by combining non-spatial ODE modeling and spatial, cell-based modeling. We employ cellularization, a recently developed method for generating spatial, cell-based, stochastic models from non-spatial ODE models, to generate much of our model from a calibrated ODE model that describes infection, death and recovery of susceptible cells and innate and adaptive responses during influenza infection, and develop models of cell migration and other mechanisms not explicitly described by the ODE model. We determine new model parameters to generate agreement between the spatial and original ODE models under certain conditions, where simulation replicas using our model serve as microconfigurations of the ODE model, and compare results between the models to investigate the nature of viral exposure and impact of heterogeneous infection on the time-evolution of the viral infection. We found that using spatially homogeneous initial exposure conditions consistently with those employed during calibration of the ODE model generates far less severe infection, and that local exposure to virus must be multiple orders of magnitude greater than a uniformly applied exposure to all available susceptible cells. This strongly suggests a prominent role of localization of exposure in influenza A infection. We propose that the particularities of the microenvironment to which a virus is introduced plays a dominant role in disease onset and progression, and that spatially resolved models like ours may be important to better understand and more reliably predict future health states based on susceptibility of potential lesion sites using spatially resolved patient data of the state of an infection. We can readily integrate the immune response components of our model into other modeling and simulation frameworks of viral infection dynamics that do detailed modeling of other mechanisms like viral internalization and intracellular viral replication dynamics, which are not explicitly represented in the ODE model. We can also combine our model with available experimental data and modeling of exposure scenarios and spatiotemporal aspects of mechanisms like mucociliary clearance that are only implicitly described by the ODE model, which would significantly improve the ability of our model to present spatially resolved predictions about the progression of influenza infection and immune response.

Optimization-based convex relaxations for nonconvex parametric systems of ordinary differential equations

Novel convex and concave relaxations are proposed for the solutions of parametric ordinary differential equations (ODEs), to aid in furnishing bounding information for deterministic global dynamic optimization methods. These relaxations are constructed as the solutions of auxiliary ODE systems with embedded convex optimization problems, whose objective functions employ convex and concave relaxations of the original ODE right-hand side. Unlike established relaxation methods, any valid convex and concave relaxations for the original right-hand side are permitted in the approach, including the McCormick relaxations and the $$\alpha $$ BB relaxations. In general, tighter such relaxations will necessarily translate into tighter relaxations for the ODE solutions, thus providing tighter bounding information for an overarching global dynamic optimization method. Notably, if McCormick relaxations are employed in the new approach, then the obtained relaxations are guaranteed to be at least as tight as state-of-the-art ODE relaxations based on generalized McCormick relaxations. The new relaxations converge rapidly to the original system as the considered parametric subdomain shrinks. Several examples are presented for comparison with established ODE relaxations, based on a proof-of-concept implementation in MATLAB. In a global optimization case study, the new ODE relaxations are shown to lead to fewer branch-and-bound global optimization iterations than state-of-the-art relaxations.

A new hierarchical approach to multi-level model abstraction for simplifying ODE models of biological networks and a case study: The G1/S Checkpoint/DNA damage signalling pathways of mammalian cell cycle

Model reduction is an important topic in studies of biological systems. By reducing the complexity of large models through multi-level models while keeping the essence (biological meaning) of the model, model reduction can help answer many important questions about these systems. In this paper, we present a new reduction method based on hierarchical representation and a lumping approach. We used G1/S checkpoint pathway represented in Ordinary Differential Equations (ODE) in Iwamoto et al. (2011) as a case study to present this reduction method. The approach consists of two parts; the first part represents the biological network as a hierarchy (multiple levels) based on protein binding relations, which allowed us to model the biological network at different levels of abstraction. The second part applies different levels (level 1, 2 and 3) of lumping the species together depending on the level of the hierarchy, resulting in a reduced and transformed model for each level. The model at each level is a representation of the whole system and can address questions pertinent to that level. We develop and simulate reduced models for levels-1, 2 and 3 of lumping for the G1/S checkpoint pathway and evaluate the biological plausibility of the proposed method by comparing the results with the original ODE model of Iwamoto et al. (2011). The results for continuous dynamics of the G1/S checkpoint pathway with or without DNA-damage for reduced models of level- 1, 2 and 3 of lumping are in very good agreement and consistent with the original model results and with biological findings. Therefore, the reduced models (level-1, 2 and 3) can be used to study cell cycle progression in G1 and the effects of DNA damage on it. It is suitable for reducing complex ODE biological network models while retaining accurate continuous dynamics of the system. The 3 levels of the reduced models respectively achieved 20%, 26% and 31% reduction of the base model. Moreover, the reduced model is more efficient to run (30%, 44% and 52% time reduction for the three levels) and generate solutions than the original ODE model. Simplification of complex mathematical models is possible and the proposed reduction method has the potential to make an impact across many fields of biomedical research.

Theoretical analysis of dynamic behaviors of cable-stayed bridges excited by two harmonic forces

To better understand the dynamic behaviors of cable-stayed bridges, this study investigates the dynamic behaviors of a cable-stayed shallow arch subjected to two external harmonic excitations using the analytical approach. First, dimensionless planar vibration equations of the system are obtained by applying the Hamilton principle, and three ordinary differential equations of the arch and the two cables are obtained by using the Galerkin discretization method. Second, modulation equations involving the amplitude and phase of the dynamic response of the system are derived by applying the method of multiple scales. Third, three simultaneous resonance cases are considered. Finally, parametric study results are illustrated through frequency responses, amplitude responses, phase plane and bifurcation diagrams. Chaos phenomenon is also detected and presented. To validate the developed analytical solutions, numerical simulations are conducted by applying the Runge–Kutta method to integrate the original ordinary differential equations. The results demonstrate that acceptable consistency is reached in the results obtained from the analytical solutions and the Runge–Kutta method in the three simulated cases. The obtained results show that the system’s dynamic responses in the three simulated cases exhibit similarities in their frequency and amplitude responses, while some qualitative differences exist in the phase plane portraits (e.g., period-1, period-2, period-3 solutions) and their bifurcation diagrams.

Dynamics of the Stochastic Belousov-Zhabotinskii Chemical Reaction Model

In this paper, we discuss the dynamic behavior of the stochastic Belousov-Zhabotinskii chemical reaction model. First, the existence and uniqueness of the stochastic model’s positive solution is proved. Then we show the stochastic Belousov-Zhabotinskii system has ergodicity and a stationary distribution. Finally, we present some simulations to illustrate our theoretical results. We note that the unique equilibrium of the original ordinary differential equation model is globally asymptotically stable under appropriate conditions of the parameter value f, while the stochastic model is ergodic regardless of the value of f.

Two Stochastic Differential Equations for Modeling Oscillabolastic-Type Behavior

Stochastic models based on deterministic ones play an important role in the description of growth phenomena. In particular, models showing oscillatory behavior are suitable for modeling phenomena in several application areas, among which the field of biomedicine stands out. The oscillabolastic growth curve is an example of such oscillatory models. In this work, two stochastic models based on diffusion processes related to the oscillabolastic curve are proposed. Each of them is the solution of a stochastic differential equation obtained by modifying, in a different way, the original ordinary differential equation giving rise to the curve. After obtaining the distributions of the processes, the problem of estimating the parameters is analyzed by means of the maximum likelihood method. Due to the parametric structure of the processes, the resulting systems of equations are quite complex and require numerical methods for their resolution. The problem of obtaining initial solutions is addressed and a strategy is established for this purpose. Finally, a simulation study is carried out.

On finite-time stability of sub-homogeneous differential inclusions

Sub-homogeneity property is introduced and is related to a differential inclusion (DI). It is shown that a nonlinear ordinary differential equation (ODE), which may not admit a homogeneous approximation, can be transformed into a sub-homogeneous DI (which is a homogeneous extension of the original ODE). Using this homogeneous extension, one can directly recover finite-time stability property for some particular classes of nonlinear systems. In the last section, such a sub homogeneity property is used to design a nonlinear finite-time observer.

Predicting COVID-19 (Coronavirus Disease) Outbreak Dynamics Using SIR-based Models: Comparative Analysis of SIRD and Weibull-SIRD

The SIR type models are built by a set of ordinary differential equations (ODE), which are strongly initial value dependent. To fit multiple biological data with SIR type equations requires fitting coefficients of these equations by an initial guess and applying optimization methods. These coefficients are also extremely initial value-dependent. In the vast publication of these types, we hardly see, among simple to highly complicated SIR type methods, that these methods presented more than a maximum of two biological data sets. We propose a novel method that integrates an analytical solution of the infectious population using Weibull distribution function into any SIR type models. The Weibull-SIRD method has easily fitted 4 set of COVID-19 biological data simultaneously. It is demonstrated that the Weibull-SIRD method predictions for susceptible, infected, recovered, and deceased populations from COVID-19 in Kuwait and UAE are superior compared with SIRD original ODE model. The proposed method here opens doors for new deeper studying of biological dynamic systems with realistic biological data trends than providing some complicated, cumbersome mathematical methods with little insight into biological data's real physics.

The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O Δ E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and O Δ E). This aspect of the problem is the aim of further research.

Isochronous cosmological solutions of the Friedmann–Robertson–Walker model

In this paper, the periodic solutions of the equation of Friedmann–Robertson–Walker cosmology with a cosmological constant are investigated. Using variable transformation, the original second-order ordinary differential equation is converted to a planar dynamical system with cosmic time t. Numerical simulations indicate that period function T(h) of this dynamical system is monotonically increasing. However, a new planar dynamical system could be deduced by using conformal time variable [Formula: see text]. We prove that the new planar dynamical system has two isochronous centers under certain parameter conditions by using Picard–Fuchs equation. Explicitly, we find that there exist two families of periodic solutions with equal period for the new planar dynamical system which is derived from the Friedmann–Robertson–Walker model.

Ordinary differential equations with point interactions: An inverse problem

Given a linear ordinary differential equation (ODE) on $\RE$ and a set of interface conditions at a finite set of points $I \subset \RE$, we consider the problem of determining another differential equation whose {\it global} solutions satisfy the original ODE on $\RE \backslash I $, and the interface conditions at $I $. Using an extension of the product of distributions with non-intersecting singular supports presented in [L. H\"ormander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983], we determine an {\it intrinsic} solution of this problem, i.e. a new ODE, satisfying the required conditions, and strictly defined within the space of Schwartz distributions. Using the same formalism, we determine a singular perturbation formulation for the $n$-th order derivative operator with interface conditions.

Real-time performance of DAE and ODE based estimators evaluated on a diesel engine

Computation and sampling time requirements for real-time implementation of observers is studied. A common procedure for state estimation and observer design is to have a system model in continuous time that is converted to sampled time with Euler forward method and then the observer is designed and implemented in sampled time in the real time system. When considering state estimation in real time control systems for production there are often limited computational resources. This becomes especially apparent when designing observers for stiff systems since the discretized implementation requires small step lengths to ensure stability. One way to reduce the computational burden, is to reduce the model stiffness by approximating the fast dynamics with instantaneous relations, transforming an ordinary differential equations (ODE) model into a differential algebraic equation (DAE) model. Performance and sampling frequency limitations for extended Kalman filter (EKF)’s based on both the original ODE model and the reduced DAE model are here analyzed and compared for an industrial system. Furthermore, the effect of using backward Euler instead of forward Euler when discretizing the continuous time model is also analyzed. The ideas are evaluated using measurement data from a diesel engine. The engine is equipped with throttle, exhaust gas recirculation (EGR), and variable geometry turbines (VGT) and the stiff model dynamics arise as a consequence of the throttle between two control volumes in the air intake system. The process of simplifying and modifying the stiff ODE model to a DAE model is also discussed. The analysis of the computational effort shows that even though the ODE, for each time-update, is less computationally demanding than the resulting DAE, an EKF based on the DAE model achieves better estimation performance than one based on the ODE with less computational effort. The main gain with the DAE based EKF is that it allows increased step lengths without degrading the estimation performance compared to the ODE based EKF.

A model of emerging intelligence in Universe

AbstractThe paper proposes a scenario of origin and emerging of intelligent life in Universe based upon the mathematical discovery of a new class of dynamical systems described by ordinary differential equation (ODE) coupled with their Liouville equation. These systems called self-controlled since the role of actuators is played by the probability produced by the Liouville equation. Following the Madelung equation that belongs to this class, non-Newtonian and quantum-like properties such as randomness, entanglement and probability interference typical for quantum systems have been described. At the same time, these systems expose properties of livings: decomposition into motor and mental dynamics, the capability of self-identification and self-awareness, as well as self-supervision. But the most surprising discovery is the existence of a special sub-class, in which the dynamical systems can violate the second law of thermodynamics, and that makes them different from both Newtonian and quantum physics. This sub-class should be associated with intelligent livings due to capability to move from disorder to order without external help. Based upon the mathematical discovery described above, one can assume that there are good chances that similar dynamical systems representing intelligent livings exist in real physical world. This provides a reason for a ‘rehabilitation’ of the Maxwell demon and put it into physics of intelligent systems. Indeed, the Maxwell demon is implemented by the feedback from the Liouville equation to the original ODE, while this feedback is capable to rearrange the probability distribution against the second law of thermodynamics. In addition to that, the same feedback removes the entropy paradox by explaining high order in our surrounding by ‘intelligent life support’. Two-steps transition: from the Newtonian physics to the linear model of life, and from the latter to the model of intelligent life are analysed. The first transition is triggered by the Hadamard instability of the Newtonian physics with respect to small random disturbances in linear terms of the Liouville feedback. The second transition is triggered by instability of linear model of life with respect to small random disturbances of non-linear terms of Liouville feedback. This transition could be implemented by such physical phenomena as shock waves or negative diffusion in probability space. Both transitions can be associated with catastrophe theory, in which sudden shifts in behaviour arises from small changes in parameters of the model. In view of the proposed model, possible competition between artificial and human intelligence are discussed.