Autonomous Differential Equations Research Articles

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.

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.

The linear stability and basic reproduction numbers for autonomous FDEs

In this paper, we first prove the stability equivalence between a linear autonomous and cooperative functional differential equation (FDE) and its associated autonomous and cooperative system without time delay. Then we present the theory of basic reproduction number $ \mathcal{R}_0 $ for general autonomous FDEs. As an illustrative example, we also establish the threshold dynamics for a time-delayed population model of black-legged ticks in terms of $ \mathcal{R}_0 $.

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.

New insights from GW170817 in the dynamical system analysis of Einstein Gauss–Bonnet gravity

In this work, we explore the dynamical system phase space of Einstein–Gauss–Bonnet theory in the cosmological minisuperspace in the light of GW170817. This approach binds the main features of the theory through a system of autonomous differential equations, in the context of a flat Friedmann–Lemaître–Robertson–Walker spacetime. We analyze the critical points that feature in this system to assess their stability criteria. The phase space of this form of scalar-tensor gravity is very rich due to the fourth-order contributions of the Gauss–Bonnet invariant together with the second order contribution of the scalar field together with their coupling dynamics. We find additional critical points as compared with previous works in the literature which may be important for understanding the larger evolution of standard background cosmology within this class of gravitational models.

Modifying an existing SIMIODE project to create an in-depth project requiring a written report

This article considers starting with an existing SIMIODE modeling scenario [Winkel, B. (2015). 1-031-CoolIt-ModelingScenario. SIMIODE (Version 2.0). QUBES Educational Resources. https://doi.org/10.25334/3WG8-EC31] that develops Newton's law of cooling by considering data on the cooling of a beaker of water in a room, and expanding upon it to create a longer project modeling the temperature of a building with an internal heat pump that is also subject to a sinusoidal varying outside temperature. The proposed project was undertaken early on in the course, since little background was required, and a longer project format was utilised so that students could return to the project and see direct applications of new concepts, such as autonomous vs. nonautonomous ordinary differential equations, slope fields, and Euler's method as they were introduced. The project prompt is provided as well as a detailed discussion of the motivation behind the questions. Finally, the pros and cons of requiring the students to submit a written lab report for the project are reflected upon and a sample rubric is provided.

Nonlinear Singularly Perturbed Cauchy Problem With Inner Layer

In the article, the asymptotics of the solution of the Cauchy problem for a nonlinear autonomous ordinary differential equation of the first order, which is a simple mathematical model with an inner layer, is constructed. For all theoretical calculations, evidence-based calculations in the Maple system are given.

Evolution of coupled weakly driven waves in a dissipative plasma

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. The obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.

Exact Solutions of Nonlinear Second-Order Autonomous Ordinary Differential Equations: Application to Mechanical Systems

Many physical processes can be described via nonlinear second-order ordinary differential equations and so, exact solutions to these equations are of interest as, aside from their accuracy, they may reveal beforehand key properties of the system’s response. This work presents a method for computing exact solutions of second-order nonlinear autonomous undamped ordinary differential equations. The solutions are divided into nine cases, each depending on the initial conditions and the system’s first integral. The exact solutions are constructed via a suitable parametrization of the unknown function into a class of functions capable of representing its behavior. The solution is shown to exist and be well-defined in all cases for a general nonlinear form of the differential equation. Practical properties of the solution, such as its period, time to reach an extreme value or long-term behavior, are obtained without the need of computing the solution in advance. Illustrative examples considering different types of nonlinearity present in classical physical systems are used to further validate the obtained exact solutions.

A geometrical index for the group Sn and its applications in periodic solutions of differential and differential-difference equations

In this paper, we construct a new index theory, Sn index theory, which is an improvement of the S1 index theory given by Benci in [1]. Also S1 index theory is a powerful tool in the study of the multiplicity of periodic orbits of autonomous differential equations, the definition of index confined the functional space being composed by functions whose mean values must zero. Under the condition that the functional Φ is even, our new index theory can be applied to study the multiplicity of periodic orbits in Hilbert space of functions without the restriction of mean value zero. After the shift of the origin it can be applied to the multiplicity of periodic orbits with a nonzero mean value. Besides, we give in this paper a delay differential system as a complete example to show the calculation of the multiplicity of periodic orbits.

Reconstructing the parameter space of nonanalytical cosmological fixed points

Dynamical system theory is a widely used technique in the analysis of cosmological models. Within this framework, the equations describing the dynamics of a model are recast in terms of dimensionless variables, which evolve according to a set of autonomous first-order differential equations. The fixed points of this autonomous set encode the asymptotic evolution of the model. Usually, these points can be written as analytical expressions for the variables in terms of the parameters of the model, which allows a complete characterization of the corresponding parameter space. However, a thoroughly analytical treatment is impossible in some cases. In this work, we give an example of a dark energy model, a scalar field coupled to a vector field in an anisotropic background, where not all the fixed points can be analytically found. Then, we put forward a general scheme that provides a numerical description of the parameter space. This allows us to find interesting accelerated attractors of the system with no analytical representation. This work may serve as a template for the numerical analysis of highly complicated dynamical systems.

Second‐order modified positive and elementary stable nonstandard numerical methods for n$$ n $$‐dimensional autonomous differential equations

A novel technique is presented to construct and analyze second‐order modified positive and elementary stable nonstandard (SOPESN) numerical methods for solving two general classes of ‐dimensional autonomous ordinary differential equations. The new SOPESN methods are second‐order generalizations of the explicit Euler's method, thereby improving the order of accuracy of the underlying numerical method. Numerical simulations are presented to validate the theoretical results and highlight the advantages of the proposed new modified nonstandard methods over existing standard and nonstandard numerical methods.