Autonomous Differential Equations Research Articles

On Truly Nonlinear Oscillator Equations of Ermakov-Pinney Type

In this paper we present a general class of differential equations of Ermakov-Pinney type which may serve as truly nonlinear oscillators. We show the existence of periodic solutions by exact integration after the phase plane analysis. The related quadratic Lienard type equations are examined to show for the first time that the Jacobi elliptic functions may be solution of second-order autonomous non-polynomial differential equations.

Ring of flows of one-dimensional differential equations

In this paper, we construct a ring of flows, where we can decompose one-dimensional autonomous differential equations into smaller parts, then solve each part and finally put everything together to obtain the exact solution of these equations.

Pandemic model with data-driven phase detection, a study using COVID-19 data

The recent COVID-19 pandemic has promoted vigorous scientific activity in an effort to understand, advice and control the pandemic. Data is now freely available at a staggering rate worldwide. Unfortunately, this unprecedented level of information contains a variety of data sources and formats, and the models do not always conform to the description of the data. Health officials have recognised the need for more accurate models that can adjust to sudden changes, such as produced by changes in behaviour or social restrictions. In this work we formulate a model that fits a “SIR”-type model concurrently with a statistical change detection test on the data. The result is a piece wise autonomous ordinary differential equation, whose parameters change at various points in time (automatically learned from the data). The main contributions of our model are: (a) providing interpretation of the parameters, (b) determining which parameters of the model are more important to produce changes in the spread of the disease, and (c) using data-driven discovery of sudden changes in the evolution of the pandemic. Together, these characteristics provide a new model that better describes the situation and thus, provides better quality of information for decision making.

Local and global robustness at steady state

We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate xi, which means that the tangent space is contained in a hyperplane of the form xi = c, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form xi = c. We clarify and formalize the relation between the two approaches. In particular, we show that ACR implies zero sensitivity and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR.

Pseudo almost automorphic solutions to non autonomous third order differential equation with multiple deviating arguments

The pseudo almost automorphic solutions of non-autonomous third-order differential equation with multiple delays is investigated. By some sufficient conditions, we obtain the uniformly bounded of the solutions. Moreover, by Banach fixed point theorem we obtain the existence and uniqueness of the solutions. In addition, by suitable Laypunov functions, we give the globally exponential stability of the pseudo almost automorphic solutions. Finally, a numerical example is given to illustrate our results.

Exact meromorphic solutions of Schwarzian differential equations

This paper studies exact meromorphic solutions of the autonomous Schwarzian differential equations. All transcendental meromorphic solutions of five canonical types (among six) of the autonomous Schwarzian differential equations are constructed explicitly. In particular, the solutions of four types are shown to be elliptic functions. Also, all transcendental meromorphic solutions that are locally injective or possess a Picard exceptional value are characterized for the remaining canonical type.

Fuzzy Homotopy Analysis Method For Solving Fuzzy Autonomous Differential Equation

In this paper, we have presented the theory and applications of the fuzzy homotopy analysis method to find the fuzzy semi-analytical solutions of the second order fuzzy autonomous differential equation. This method allows for the solution of the Fuzzy initial value problems to be calculated in the form of an infinite fuzzy series in which the fuzzy components can be easily calculated. Some numerical results are given to illustrate the used method. The obtained numerical results are compared with the fuzzy exact-analytical solutions.

Derivation of the Redfield Quantum Master Equation and Corrections to It by the Bogoliubov Method

Following the ideas N. N. Bogoliubov used to derive the classical and quantum nonlinear kinetic equations, we derive the Redfield quantum linear master equation, which is widely used in the theory of open quantum systems. This method allows one to calculate the joint state of a system and a reservoir at every instant of time, as well as to obtain quantum master equations as autonomous differential equations in an arbitrary order of perturbation theory. We prove that under certain conditions the expressions for the corrections of all orders are well defined. We also discuss the question of whether the quantum dynamics described by the derived quantum master equations is Markovian.

Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases.

Age evolution in the mean field forest fire model via multitype branching processes

We study the distribution of ages in the mean field forest fire model introduced by Ráth and Tóth. This model is an evolving random graph whose dynamics combine Erdős–Rényi edge-addition with a Poisson rain of lightning strikes. All edges in a connected component are deleted when any of its vertices is struck by lightning. We consider the asymptotic regime of lightning rates for which the model displays self-organized criticality. The age of a vertex increases at unit rate, but it is reset to zero at each burning time. We show that the empirical age distribution converges as a process to a deterministic solution of an autonomous measure-valued differential equation. The main technique is to observe that, conditioned on the vertex ages, the graph is an inhomogeneous random graph in the sense of Bollobás, Janson and Riordan. We then study the evolution of the ages via the multitype Galton–Watson trees that arise as the limit in law of the component of an identified vertex at any fixed time. These trees are critical from the gelation time onwards.

Bifurcation analysis for a flow of viscoelastic fluid due to peristaltic activity

In this article, bifurcation analysis is performed to study the qualitative nature of stagnation points and various flow regions for a peristaltic transport of viscoelastic fluid through an axisymmetric tube. The rheological behavior of viscoelastic fluid is characterized by the simplified Phan–Than–Tanner fluid model. An analytic solution in a wave frame is obtained subject to the low Reynolds number and long wavelength approximations. The stagnation points and their bifurcations (critical conditions) are explored by developing a system of autonomous differential equations. The dynamical system theory is employed to examine the nature and bifurcations of obtained stagnation points. The ranges of various flow phenomena and their bifurcations are scrutinized graphically through global bifurcation diagrams. This analysis reveals that the bifurcation in the flow is manifested at large flow rate for high extensional parameter and Weissenberg number. Backward flow phenomenon enhances and trapping diminishes with an increase in the Weissenberg number. At the end, the results of present analysis are verified by making a comparison with the existing literature.

First integrals and solutions for a class of third-order nonlinear autonomous differential equations

A class of third-order nonlinear autonomous differential equations is analyzed and solved. The first integral method suitable for the third-order autonomous differential equation is introduced and the first integrals of the equation are obtained by this method. Besides this, the functional transformations and Lie group theory are used to solve the equation.

Graph-based multi-core higher-order time integration of linear autonomous partial differential equations

Modern high-performance computing (HPC) systems rely on increasingly complex nodes with a steadily growing number of cores and matching deep memory hierarchies. In order to fully exploit them, algorithms must be explicitly designed to exploit these features. In this work we address this challenge for a widely used class of application kernels: polynomial-based time integration of linear autonomous partial differential equations.We build on prior work [1] of a cache-aware, yet sequential solution and provide an innovative way to parallelize it, while addressing cache-awareness across a large number of cores. For this, we introduce a dependency graph driven view of the algorithm and then use both static graph partitioning and dynamic scheduling to efficiently map the execution to the underlying platform. We implement our approach on top of the widely available Intel Threading Building Blocks (TBB) library, although the concepts are programming model agnostic and can apply to any task-driven parallel programming approach.We demonstrate the performance of our approach for a 2nd, 4th and 6th order time integration of the linear advection equation on three different architectures with widely varying memory systems and achieve an up to 60% reduction of wall clock time compared to a conventional, state-of-the-art non-cache-aware approach.

Dual feedback IRC ring for chaotic waveform generation

The authors have proposed an inverter resistor capacitor (IRC)-based simple chaotic ring oscillator with dual feedback for the generation of the chaotic waveform. The proposed chaotic system uses three coupled autonomous first-order differential equations and it can be electrically demonstrated using the complementary metal oxide semiconductor (CMOS) inverters with few passive RC elements for the generation of the high-frequency strange attractor. The basic dynamical characteristics and multistability property with complex bifurcation pattern in a wide range of parameters are well observed through theoretical as well as numerical simulation analysis using scaled inverse tangent function. In addition, a simple IRC chaotic model design is investigated using 0.18μm MOS transistor parameters with grounded capacitors and an equivalent CMOS-based scaled inverse tangent function. Moreover, the Cadence post layout simulation gives useful information about the circuit sustainability for IC design with high frequency (MHz), low power dissipation (940 μW) and small chip area (826.3 μm2).

The effect of a small bounded noise on the hyperbolicity for autonomous semilinear differential equations

In this work, we study permanence of hyperbolicity for autonomous differential equations under nonautonomous random/stochastic perturbations. For the linear case, we study robustness and existence of exponential dichotomies for nonautonomous random dynamical systems. Next, we establish a result on the persistence of hyperbolic equilibria for nonlinear differential equations. We show that for each nonautonomous random perturbation of an autonomous semilinear problem with a hyperbolic equilibrium there exists a bounded random hyperbolic solution for the associated nonlinear nonautonomous random dynamical systems. Moreover, we show that these random hyperbolic solutions converge to the autonomous equilibrium. As an application, we consider a semilinear differential equation with a small nonautonomous multiplicative white noise, and as an example, we apply the abstract results to a strongly damped wave equation.

A Two-Scale Analysis for a Spherical Pendulum with a Vertically Vibrating Pivot

Summary In this article, we consider the behaviour of a simple undamped spherical pendulum subject to high-frequency small amplitude vertical oscillations of its pivot. We use the method of multiple scales to derive an autonomous ordinary differential equation describing the slow time behaviour of the polar angle which generalises the Kapitza equation for the plane problem. We analyse the phase plane structure of this equation and show that for a range of parameter values there are conical orbits which lie entirely above the horizontal. Going further, we identify a family of quasi-conical orbits some of which may lie entirely above the pivot and establish that initial conditions can be chosen so that precession is eliminated for these orbits. For the general initial value problem, we show that the leading order solutions for the polar and azimuthal angles diverge significantly from their exact counterparts. However, by consolidating the slow scale error term into the leading order structure we may construct extremely accurate solutions for the slow scale evolution of the system. These solutions, confirmed by exact numerical simulations, show that by suitable choice of initial data orbital precession can be eliminated.

Polynomial differential equations over the quaternions

We study some fundamental properties of autonomous polynomial differential equations over the quaternions H. After showing that every polynomial differential equation in R4 can be rewritten as a differential equation over H, we specialize the investigation to quaternionic differential equations with coefficients in C⊆H, which retain some properties of polynomial differential equations in C. We provide a detailed characterization of this class and discuss their stationary points and periodic orbits, including those at infinity. As an application of our theory we investigate the dynamics of generic quadratic systems of this type and of a special class of quaternionic Riccati differential equations with complex coefficients. Here we obtain a new type of Liouville–Arnold integrable system.

An application of a novel geometric criterion to global-stability problems of a nonlinear SEIVS epidemic model.

This work applies a novel geometric criterion for nonlinear autonomous differential equations developed by Lu and Lu (NARWA 36:20–43, 2017) to a nonlinear SEIVS epidemic model with temporary immunity and achieves its threshold dynamics. Specifically, global-stability problems for the SEIVS model of Cai and Li (AMM 33:2919–2926, 2009) are effectively solved. The corresponding optimal control system with vaccination, awareness campaigns and treatment is further established and four different control strategies are compared by numerical simulations to contain hepatitis B. It is concluded that joint implementation of these measures can minimize the numbers of exposed and infectious individuals in the shortest time, so it is the most efficient strategy to curb the hepatitis B epidemic. Moreover, vaccination for newborns plays the core role and maintains the high level of population immunity.

Semi-Dynamical Systems Generated by Autonomous Caputo Fractional Differential Equations

An autonomous Caputo fractional differential equation of order α ∈ (0,1) in a finite dimensional space whose vector field satisfies a global Lipschitz condition is shown to generate a semi-dynamical system in the function space \(\mathfrak {C}\) of continuous functions with the topology uniform convergence on compact subsets. This contrasts with a recent result of Cong and Tuan (J. Integral Equ. Appl.: 29, 585–608, 2017), which showed that such equations do not, in general, generate a dynamical system on the state space.

Computation and verification of contraction metrics for exponentially stable equilibria

The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases. The Riemannian metric can be expressed by a matrix-valued function on the phase space.The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF). However, so far no rigorous verification that the computed metric is indeed a contraction metric has been provided.In this paper, we combine the RBF method to compute a contraction metric with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification. We apply the method to two examples.