Autonomous Differential Equations Research Articles

On bifurcation of non-isolated singular points arising in a problem of two-phase fluid motion in a pipe

The dynamics of gas suspension in a pipe is studied. It is described by a system of three autonomous differential equations, whose equilibrium (singular) points are not isolated, but form several continuous curves. One of these curves belongs to the transonic plane. The dynamics in a neighborhood of this plane is interesting from the physical viewpoint. In this work, the dynamics in a neighborhood of the curves of singular points is investigated. The main tools of this work are the center manifold theorem and the reduction principle. A peculiarity of the system is that the curves of singular points intersect each other. Moreover, at the intersection point a bifurcation occurs. Results of the work are the description of the dynamics in a neighborhood of singular points, the proof that a curve of singular points belongs to a center manifold and the description of the bifurcation, which occurs at the intersection point of the curves of singular points.

Second-Order Modified Nonstandard Explicit Euler and Explicit Runge–Kutta Methods for n-Dimensional Autonomous Differential Equations

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems.

Gauss-Bonnet Cosmology: large-temperature behaviour and bounds from Gravitational Waves

We provide a transparent discussion of the high temperature asymptotic behaviour of Cosmology in a dilaton-Einstein-Gauss-Bonnet (dEGB) scenario of modified gravity with vanishing scalar potential. In particular, we show that it has a clear interpretation in terms of only three attractors (stable critical points) of a set of autonomous differential equations: w = -1/3, w = 1 and 1 < w < 7/3, where w ≡ p/ρ is the equation of state, defined as the ratio of the total pressure and the total energy density. All the possible different high-temperature evolution histories of the model are exhausted by only eight paths in the flow of the set of the autonomous differential equations. Our discussion clearly explains why five out of them are characterized by a swift transition of the system toward the attractor, while the remaining three show a more convoluted evolution, where the system follows a meta-stable equation of state at intermediate temperatures before eventually jumping to the real attractor at higher temperatures. Compared to standard Cosmology, the regions of the dEGB parameter space with w = -1/3 show a strong enhancement of the expected Gravitational Wave stochastic background produced by the primordial plasma of relativistic particles of the Standard Model. This is due to the very peculiar fact that dEGB allows to have an epoch when the energy density ρ rad of the relativistic plasma dominates the energy of the Universe while at the same time the rate of dilution with T of the total energy density is slower than what usually expected during radiation dominance. This allows to use the bound from Big Bang Nucleosynthesis (BBN) to put in dEGB a constraint T RH ≲ (108 – 109) GeV on the reheating temperature of the Universe T RH. Such BBN bound is complementary to late-time constraints from compact binary mergers.

Dynamic survival analysis: Modelling the hazard function via ordinary differential equations.

The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature on ODEs which, in particular, allows for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework. Although we focus on examples of Medical Statistics, the proposed framework is applicable in any context where the interest lies in estimating and interpreting the dynamics of the hazard function.

Limit cycles as stationary states of an extended harmonic balance ansatz

A limit cycle is a self-sustained, periodic, isolated motion appearing in autonomous differential equations. As the period of a limit cycle is unknown, finding it as a stationary state of a rotating ansatz is challenging. Correspondingly, its study commonly relies on numerical methodologies (e.g., brute-force time evolution, and variational shooting methods) or circumstantial evidence such as instabilities of fixed points. Alas, such approaches are (i) , as they rely on specific initial conditions, and (ii) do not provide analytical intuition about the physical origin of the limit cycles. Here, we (I) develop a multifrequency rotating ansatz with which we (II) find limit cycles as stationary-state solutions via a semianalytical homotopy continuation. We demonstrate our approach and its performance on the Van der Pol oscillator. Moving beyond this simple example, we show that our method captures all coexisting fixed-point attractors and limit cycles in a modified nonlinear Van der Pol oscillator. Our results facilitate the systematic mapping of out-of-equilibrium phase diagrams, with implications across multiple fields of the natural sciences. Published by the American Physical Society 2024

Koopman Spectral Linearization vs. Carleman Linearization: A Computational Comparison Study

Nonlinearity presents a significant challenge in developing quantum algorithms involving differential equations, prompting the exploration of various linearization techniques, including the well-known Carleman Linearization. Instead, this paper introduces the Koopman Spectral Linearization method tailored for nonlinear autonomous ordinary differential equations. This innovative linearization approach harnesses the interpolation methods and the Koopman Operator Theory to yield a lifted linear system. It promises to serve as an alternative approach that can be employed in scenarios where Carleman Linearization is traditionally applied. Numerical experiments demonstrate the effectiveness of this linearization approach for several commonly used nonlinear ordinary differential equations.

A Study of Movable Singularities in Non-Algebraic First-Order Autonomous Ordinary Differential Equations

We studied the movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in the form of y′=I(y(t)) and y′=I1(y(t))+I2(y(t))+⋯+In(y(t)), aiming to prove that all movable singularities of all complex solutions of these equations are at most algebraic branch points. This study explores the use of the constructing triangle method to analyze complex solutions of autonomous non-algebraic first-order ordinary differential equations. For complex solutions in the form of y=w+iv, we treat the constructing triangle method as a way to construct a right-angled triangle in the complex plane, with the lengths of the adjacent sides being w and v. We use the definitions of the trigonometric functions sin and cos (the ratio of the adjacent side to the hypotenuse) to represent the trigonometric functions of complex solutions y=w+iv. Since the movable singularities of the inverse functions of trigonometric functions are easy to analyze, the properties of the movable singularities of the complex solutions are then easy to deal with.

Coefficientwise Hankel-total positivity of the row-generating polynomials for the output matrices of certain production matrices

Total positivity of matrices is deeply studied and plays an important role in various branches of mathematics. The aim of this paper is to study the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of generalized m-Jacobi-Rogers triangles and their applications.Using the theory of production matrices, we present the criteria for coefficientwise Hankel-total positivity of the row-generating polynomials of the output matrices of certain production matrices. In particular, we gain a criterion for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the generalized m-Jacobi-Rogers triangle. This immediately implies that the corresponding generalized m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex of higher order in all the indeterminates. In consequence, for m=1, we immediately obtain some results on Hankel-total positivity for the Catalan-Stieltjes matrices. In particular, we in a unified manner apply our results to some combinatorial triangles or polynomials including the generalized Jacobi Stirling triangle, a generalized elliptic polynomial, a refined Stirling cycle polynomial and a refined Eulerian polynomial. For the general m, combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence of exponential Rirodan arrays. In addition, we also derive some results for coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes-type continued fractions. Finally, we apply our criteria to: (1) rook polynomials and signless Laguerre polynomials (confirming a conjecture of Sokal on coefficientwise Hankel-total positivity of rook polynomials), (2) labeled trees and forests (proving some conjectures of Sokal on total positivity and Hankel-total positivity), (3) rth-order Eulerian polynomials (giving a new proof for the coefficientwise Hankel-total positivity of rth-order Eulerian polynomials, which in particular implies the conjecture of Sokal on the coefficientwise Hankel-total positivity of reversed 2th-order Eulerian polynomials), (4) multivariate Ward polynomials, labeled series-parallel networks and nondegenerate fanout-free functions, (5) an array from the Lambert function and a generalization of Lah numbers and associated triangles, and so on.

All meromorphic solutions of the autonomous Schwarzian differential equations

AbstractThis paper considers a specific autonomous Schwarzian differential equation given by and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case.

НЕЙРОННА МЕРЕЖА ТИПУ АВТОКОДУВАЛЬНИК ДЛЯ ВКЛАДЕННЯ ОДНОВИМІРНИХ ЧАСОВИХ РЯДІВ

The problem of time series embedding is a universal one. It is the main prerequisite when it comes to modeling of dynamical processes using systems of autonomous ordinary differential equations (ODEs) because they have hard requirements for the dimensionality of the problem. One-dimensional ODE can only exhibit 3 types of behavior while two-dimensional ODE can exhibit 9. This is why it is important to increase the dimensionality of the problem before starting the modeling to allow for wider range of possible behaviors in the final model. One way to increase the dimensionality is to delay-embed the time series data but this approach can be extended to allow the use of an autoencoder neural network that would associate a higher-dimensional vector to each point in the time series and will allow the modeling to be performed in higher dimension.

Delay-driven phase transitions in an epidemic model on time-varying networks.

A complex networked system typically has a time-varying nature in interactions among its components, which is intrinsically complicated and therefore technically challenging for analysis and control. This paper investigates an epidemic process on a time-varying network with a time delay. First, an averaging theorem is established to approximate the delayed time-varying system using autonomous differential equations for the analysis of system evolution. On this basis, the critical time delay is determined, across which the endemic equilibrium becomes unstable and a phase transition to oscillation in time via Hopf bifurcation will appear. Then, numerical examples are examined, including a periodically time-varying network, a blinking network, and a quasi-periodically time-varying network, which are simulated to verify the theoretical results. Further, it is demonstrated that the existence of time delay can extend the network frequency range to generate Turing patterns, showing a facilitating effect on phase transitions.

Shadowing, Hyers–Ulam stability and hyperbolicity for nonautonomous linear delay differential equations

It is known that hyperbolic nonautonomous linear delay differential equations in a finite dimensional space are Hyers–Ulam stable and hence shadowable. The converse result is available only in the special case of autonomous and periodic linear delay differential equations with a simple spectrum. In this paper, we prove the converse and hence the equivalence of all three notions in the title for a general class of nonautonomous linear delay differential equations with uniformly bounded coefficients. The importance of the boundedness assumption is shown by an example.

Investigation of the stability of trajectories for three-dimensional nonlinear dynamic system using an accompanying coordinate reference

The development of methods of stability analysis and the study of various types of dynamic systems stability belongs to an urgent scientific direction. Among the problems solved in the framework of this direction, an important place is occupied by the problems of finding conditions for stability and stabilization in the sense of N.E. Zhukovsky of trajectories of dynamic systems modeled by systems of three nonlinear differential equations of the first order. The aim of the work is to study the stability in the Zhukovsky sense of the trajectories of a dynamic system described by three autonomous nonlinear first-order differential equations using a differential geometric method called the accompanying coordinate reference method. The formulation of the stability problem in the Zhukovsky sense of the trajectories of dynamic systems described by systems of three nonlinear differential equations of the first order is considered. The definitions of stability of trajectories of a three-dimensional nonlinear system are specified. Using the accompanying coordinate reference, the conditions of exponential stability and instability according to Zhukovsky are obtained. The results can be used in studying the stability of processes in systems of natural science and technology, as well as in solving theoretical and applied problems of mathematical modeling and qualitative research of trajectories of nonlinear dynamic systems.

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Machine learning methods for autonomous ordinary differential equations

Machine learning methods for autonomous ordinary differential equations

Theoretical analysis of co-existing periodic orbits in sparse binary neural networks

A sparse binary neural network is a discrete-time recurrent neural network characterized by local binary connections and signum-type neuron models. The dynamics is described by an autonomous difference equation of binary state variables. Depending on the parameters, the network generates various periodic orbits of binary vectors. Multiple periodic orbits can co-exist and the network exhibits one of them depending on initial condition. Real/potential applications of the periodic orbits include control signals of switching circuits and basic signals for time-series approximation. The network is well suited for theoretical analysis and simple hardware implementation. This paper considers shift-type periodic orbits in the networks. As the main theoretical results, we clarify the number, period, and stability of the periodic orbits for key parameters. As a first step to the applications in the future, we present an FPGA-based hardware prototype and confirm typical periodic orbits experimentally.

Periodic Center Manifolds for Nonhyperbolic Limit Cycles in ODEs

In this paper, we deal with a classical object, namely, a nonhyperbolic limit cycle in a system of smooth autonomous ordinary differential equations. While the existence of a center manifold near such a cycle was assumed in several studies on cycle bifurcations based on periodic normal forms, no proofs were available in the literature until recently. The main goal of this paper is to give an elementary proof of the existence of a periodic smooth locally invariant center manifold near a nonhyperbolic cycle in finite-dimensional ordinary differential equations by using the Lyapunov–Perron method. In addition, we provide several explicit examples of analytic vector fields admitting (non)-unique, (non)-[Formula: see text]-smooth and (non)-analytic periodic center manifolds.

Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey

This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.

Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks

In this work, we present a fast, globally convergent, iterative algorithm for computing the asymptotically stable states of nonlinear large-scale systems of quadratic autonomous ordinary differential equations (ODE) modeling, e.g., the dynamic of complex chemical reaction networks. Toward this aim, we reformulate the problem as a box-constrained optimization problem where the roots of a set of nonlinear equations need to be determined. Then, we propose to use a projected Newton’s approach combined with a gradient descent algorithm so that every limit point of the sequence generated by the overall algorithm is a stationary point. More importantly, we suggest replacing the standard orthogonal projector with a novel operator that ensures the final solution to satisfy the box constraints while lowering the probability that the intermediate points reached at each iteration belong to the boundary of the box where the Jacobian of the objective function may be singular. The effectiveness of the proposed approach is shown in a practical scenario concerning a chemical reaction network modeling the signaling network of colorectal cancer cells. Specifically, in this scenario the proposed algorithm is proved to be faster and more accurate than a classical dynamical approach where the asymptotically stable states are computed as the limit points of the flux of the Cauchy problem associated with the ODE system.