Autonomous Differential Equations Research Articles

Dynamical system approach of non-minimal coupling in holographic cosmology

We study the dynamical system approach of non-minimally coupled scalar field to induced gravity on the brane in the framework of holographic cosmology. In this context, we derive the modified Friedmann equation and the equation of motion. The dynamics of this model are studied by rewriting the cosmological field equations in the form of a system of autonomous differential equations. In particular, the analysis is considered by investigating an exponential potential and a monomial form of the non-minimal coupling function. We show that, for sufficient conditions, a past de Sitter attractor solution is obtained in the case of a minimal coupling, meanwhile a future de Sitter attractor solutions is obtained for a conformal coupling and for a non-minimal coupling.

Construction of Finsler‐Lyapunov functions with meshless collocation

AbstractWe study the stability of invariant sets such as equilibria or periodic orbits of a Dynamical System given by a general autonomous nonlinear ordinary differential equation (ODE). A classical tool to analyse the stability are Lyapunov functions, i.e. scalar‐valued functions, which decrease along solutions of the ODE. An alternative to Lyapunov functions is contraction analysis. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler‐Lyapunov function. This has the advantage that the invariant set plays no special role and does not need to be known a priori.In this paper, we propose a method to numerically construct a Finsler‐Lyapunov function by solving a first‐order partial differential equation using meshless collocation. Depending on the expected attractor, the contraction only takes place in certain directions, which can easily be implemented within the method. In the basin of attraction of an exponentially stable equilibrium or periodic orbit, we show that the PDE problem has a solution, which provides error estimates for the numerical method. Furthermore, we show how the method can also be applied outside the basin of attraction and can detect the stability as well as the stable/unstable directions of equilibria. The method is illustrated with several examples.

“Reverse Engineering” in Econophysics

The work presented here is a paradigm of EconoPhysics, i.e. of research in the area of finance and economics by applying physical models, in this case chaos theory. A specific analysis of a macroeconomic model proposed by Vosvrda is presented. The Vosvrda model is an idealized macroeconomic model, combining the savings of households, Gross Domestic Product and the foreign capital inflow. It is simulated by three autonomous differential equations. According to this model, there are six parameters, having their values regulating the system behavior (parameters of Vosvdra). Using artificial noisy data for simulating real data and using an inverse modelling procedure, the authors have fitted and tuned the parameters of Vosvdra differential equations to achieve more accurate solutions. The relevant resultant evaluation showed that the system is a chaotic one, even though for the same values proposed by Vosvrda. Finally, this chaotic behavior has provided the capability to expand the time horizon of the solution, thus achieving reliable forecasting for the system.

Construction of a contraction metric by meshless collocation

<p style='text-indent:20px;'>A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium. <p style='text-indent:20px;'>The contraction metric is described by a matrix-valued function <inline-formula><tex-math id="M1">\begin{document}$ M(x) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M2">\begin{document}$ M(x) $\end{document}</tex-math></inline-formula> is positive definite and <inline-formula><tex-math id="M3">\begin{document}$ F(M)(x) $\end{document}</tex-math></inline-formula> is negative definite, where <inline-formula><tex-math id="M4">\begin{document}$ F $\end{document}</tex-math></inline-formula> denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation <inline-formula><tex-math id="M5">\begin{document}$ F(M)(x) = -C(x) $\end{document}</tex-math></inline-formula>. We then use a construction method based on meshless collocation, developed in the companion paper [<xref ref-type="bibr" rid="b12">12</xref>], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.

Switching Regimes in Economics: The Contraction Mapping and the &amp;amp;omega;-Limit Set

This paper considers a dynamical system defined by a set of ordinary autonomous differential equations with discontinuous right-hand side. Such systems typically appear in economic modelling where there are two or more regimes with a switching between them. Switching between regimes may be a consequence of market forces or deliberately forced in form of policy implementation. Stiefenhofer and Giesl [1] introduce such a model. The purpose of this paper is to show that a metric function defined between two adjacent trajectories contracts in forward time leading to exponentially asymptotically stability of (non)smooth periodic orbits. Hence, we define a local contraction function and distribute it over the smooth and nonsmooth parts of the periodic orbits. The paper shows exponentially asymptotical stability of a periodic orbit using a contraction property of the distance function between two adjacent nonsmooth trajectories over the entire periodic orbit. Moreover it is shown that the ω-limit set of the (non)smooth periodic orbit for two adjacent initial conditions is the same.

Разрешимость неоднородного автономного дифференциального уравнения с последействием на отрицательной полуоси

Разрешимость неоднородного автономного дифференциального уравнения с последействием на отрицательной полуоси

ABSOLUTE STABILITY CONDITIONS FOR DIFFERENTIAL EQUATIONS WITH DISTRIBUTED DELAY

The problem of asymptotic stability for autonomous functional differential equations is studied on the basis of the investigation of the roots of the characteristic function. We apply D-decomposition method for obtaining the sharp boundaries of stability domains. We obtain necessary and sufficient conditions of asymptotic stability for two families of linear autonomous differential equations with distributed delay and power kernels. These criteria of stability are formulated in terms of the parameters of the original problem. Based on the criteria, we find absolute stability conditions for each of the families.

Sliding mode disturbance observer control based on adaptive synchronization in a class of fractional‐order chaotic systems

SummaryIn this paper, a fractional‐order Dadras‐Momeni chaotic system in a class of three‐dimensional autonomous differential equations has been considered. Later, a design technique of adaptive sliding mode disturbance‐observer for synchronization of a fractional‐order Dadras‐Momeni chaotic system with time‐varying disturbances is presented. Applying the Lyapunov stability theory, the suggested control technique fulfils that the states of the fractional‐order master and slave chaotic systems are synchronized hastily. While the upper bounds of disturbances are unknown, an adaptive regulation scheme is advised to estimate them. The recommended disturbance‐observer realizes the convergence of the disturbance approximation error to the origin. Finally, simulation results are presented in one example to demonstrate the efficiency of the offered scheme on the fractional‐order Dadras‐Momeni chaotic system in the existence of external disturbances.

Stabilization of Oscillations in a Periodic System by Choosing Appropriate Couplings

We study a model containing coupled subsystems (MCCS) defined by a system of ordinary differential equations, where subsystems are systems of autonomous ordinary differential equations. The model splits into unrelated systems when the numerical parameter that characterizes couplings is e = 0, and the couplings are given by time-periodic functions. We solve the natural stabilization problem which consists in finding relationships that simultaneously guarantee the existence and asymptotic stability of MCCS oscillations. We generalize results previously obtained for the case of two coupled subsystems each of which is defined on its own plane.

The Behavior of Solutions to a Special Abel Equation of the Second Kind near a Nodal Singular Point

The propagation of a diffusion–reaction plane traveling wave (for example, a flame front), the charge distribution inside a heavy atom in the Thomas–Fermi model, and some other models in natural sciences lead to bounded solutions of a certain autonomous nonlinear second-order ordinary differential equation reducible to an Abel equation of the second kind. In this study, a sufficient condition is obtained under which all solutions to a special second-kind Abel equation that pass through a nodal singular point of the equation can be represented by a convergent power series (in terms of fractional powers of the variable) in a neighborhood of this point. Under this condition, new parametric representations of bounded solutions to the corresponding autonomous nonlinear equation are derived. These representations are efficient for numerical implementation.

Oscillatory Solutions of Some Autonomous Partial Differential Equations with a Parameter

We study a class of evolutionary partial differential equations depending on a parameter τ (stemming from the problems of groundwater flows). The existence of an open interval 𝒯0 of the parameter τ and of a function τ ⟼ Θ(τ), Θ: 𝒯0 ⟼(0, + ∞), is proved with the property that any nonzero global solution u:ℝ+ × Ω → ℝ of the equation cannot remain nonnegative (nonpositive) throughout the set J × Ω; where J ⊂ ℝ+ is any interval whose length is greater than Θ (τ). In other words, these solutions are globally oscillatory and Θ (τ) is the uniform oscillatory time. The interval 𝒯0 and the function Θ are explicitly determined.

Bifurcation Analysis for Peristaltic Transport of a Power-Law Fluid

Abstract In this work, the streamline topologies and their bifurcations for peristaltic transport of shear-thinning and shear-thickening fluids characterised by power-law model are analysed. The flow is assumed in a two-dimensional symmetric channel. The analytical solution is obtained in a wave frame of reference under low Reynolds number and long wavelength approximations. To study the streamline topologies, a system of non-linear autonomous differential equations is formed and the method of dynamical system is employed to investigate the bifurcations and their changes. Three different types of flow situations occur: backward flow, trapping and augmented flow. The conversions of backward flow to trapping and then trapping to augmented flow correspond to bifurcations. The stability and nature of bifurcations and their topological changes are explained graphically. For this purpose, a global bifurcation diagram is constructed. The backward flow and trapping regions are significantly affected by fluid behaviour index. In fact, the trapping region expands and the backward region shrinks by increasing the fluid behaviour index. Theoretical results are verified by comparing them with the experimental data, which is available in the literature.

Some applications of the generalized Eulerian numbers

In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is related to an autonomous first-order differential equation, solutions of which are used to obtain integral representations of some numbers, including the Bernoulli numbers.

Bianchi I and V cosmologies with Hu-Sawicki f(R) gravity in Palatini formalism

The main objective of this work is the pursuit of a detailed analysis of the cosmological solutions corresponding to each fixed point belonging to an exclusive form of f(R) initially introduced by Hu and Sawicki in the background of Palatini f(R) gravity. Fixed points and their associated solutions for the anisotropic Bianchi type I and V models are derived by utilizing an approach used to solve autonomous ordinary differential equations. Building on this background, an exhaustive study on the phase space is accomplished and the existence of a sequence of evolution that starts from unstable anisotropic universe to a stable phase of late times de Sitter expansion through the standard radiation and matter dominated era has been established.

Development of combined Runge–Kutta Broyden's load flow approach for well‐ and ill‐conditioned power systems

Load flow (LF) is an extensively used tool in planning and operation of power systems. Formulation of LF problem can be assimilated as a set of autonomous ordinary differential equations, therefore, many numeric methods can be used to solve this problem. However, LF methods often need to compute one or more Jacobian matrix inversions in each iteration. Owing to this fact, these methods might not be computationally efficient. In this study, the authors propose combined Runge-Kutta Broyden's LF (RK4B) method in order to reduce the required Jacobian matrix inversion to only one in the first iteration. In this proposed method, Broyden's approach is employed in fourth-order Runge-Kutta method. In addition, two modifications of the proposed method are presented to reduce the number of iterations and improve the computational performance. The proposed method and the two modifications are validated using several well- and ill-conditioned cases. Results show that the combined approach has better computational performance than the classical multistage numeric methods, besides it preserves the robustness features of fourth-order Runge-Kutta method.

On existence and stabilization of the strong solution of the autonomous stochastic partial differential Ito-Skorokhod equation with random parameters

This paper considers the asymptotic behavior of the strong solution of the linear partial stochastic differential Ito–Skorokhod equation in the corresponding space with random parameters. An existence of the strong solution is proved and sufficient conditions for the asymptotic stability and the mean square instability of a strong solution of a similar equation are obtained. The stochastic model of complex systems, which is proposed in this paper, is an attempt to take into consideration the full extent of randomness in the studying of real processes, which are described by differential equations in partial derivatives, on the right side of which a diffuse perturbations of the Brownian process type and random perturbations of other types are taken into consideration.

A critique of Hubbert’s model for peak oil

In 1956, Shell Oil Company geologist M. King Hubbert published a model for the growth and decline over time of the production rates of oil extracted from the land mass of the continental US. Employing an estimate for the amount of ultimately recoverable oil and a logistic curve for the oil production rate, he accurately predicted a peak in US oil production for 1970. His arguments and the success of his prediction have been much celebrated, and the original paper has 1400 publication citations to date. The theory of “peak oil” (and subsequently, of natural resource scarcity in general) has consequently become associated with Hubbert and “Hubbert” curves and models. However, his prediction for the timing of a world peak oil production rate and the subsequent predictions of many others have proven inaccurate. We revisit the Hubbert model for oil extraction and provide an analysis of it and several variants in the language of (time) autonomous differential equations.

Exploring the Dynamics of a Third-Order Phase-Locked Loop Model

We study the dynamics and characterize the bifurcation structure of a phase-locked loop (PLL) device modeled by a third-order autonomous differential equation with sinusoidal phase detector. The development of this work was performed using rigorous analysis and numerical experiments. Through theoretical analysis the bifurcation structures related to two fundamental equilibrium points of the system are described. By using extensive numerical experiments we investigate the intricate organization between periodic and chaotic domains in parameter space (named here parameter plane as the PLL model has only two control parameters) and obtain two following remarkable findings: (i) there are self-organized generic stable periodic structures along specific directions in parameter plane, whose periods are defined by a mathematical rule and, (ii) the existence of transient chaos phenomenon responsible for long chaotic temporal evolution preceding the asymptotic (periodic) dynamics for some particular control parameter pairs is characterized. Our theoretical and numerical results present an astonishing concordance. We believe that the present study, specially the parameter plane analysis, may have a great importance to experimental studies and general applications involving PLL devices when, for example, one would like to avoid the chaotic regimes.

Development of an approach to ensure stability of the traction direct current system

The result of applying the quantitative approach to the calculation of static stability of the traction power system helped us establish that when a train runs along an actual section there emerge zones with lack of stability in terms of voltage. Exact solution to the task of evaluating the stability is extremely difficult because of the need to compute the nonlinear dependences determining the modes of operation of the traction power system and electric rolling stock. In this work, we constructed a system of four autonomous nonlinear differential equations based on experimental data that simulate the behavior of current and voltage in the contact network. We also calculated stability regions for voltage regulators in the traction network, which stabilize voltage at pantographs of electric rolling stock. The obtained stability regions of voltage regulators made it possible to estimate resource of stability and to find the most robust regulators out of those constructed. The study revealed that the non-linear regulator has better robust properties than the linear one. In this case, stability of the linear regulator is very narrow ‒ Δk=0.000004, which is an order of magnitude lower than for the non-linear regulator. When applying the non-linear regulator, voltage in the contact network stabilizes 3 times faster regardless of the place of its location. Application of the devised approach would make it possible to calculate the stability regions for various schematics of the traction network in the implementation of high-speed motion and to narrow the range of voltage fluctuations. The developed dynamic model of power consumption processes, as well as the voltage regulator, could be used when constructing an intelligent, adaptive traction power system for high-speed motion.

State variable mapping method for studying initial-dependent dynamics in memristive hyper-jerk system with line equilibrium

Memristive hyper-jerk system has been reported to exhibit the initial-dependent multistability phenomenon, whose dynamical mechanism and control strategy deserve further in-depth investigations. However, a more effective method is required for the existence of line equilibrium. Inspired by the dimensionality decreasing flux-charge analysis method, we handle these problems via state variable mapping (SVM) method of integral transformation. With this method, the hyper-jerk system characterized by four autonomous differential equations can be reconstructed and a dimensionality decreasing model is obtained. In the newly formulated system, the initial conditions of the original system are mapped as standalone system parameters. Thus, the initial-dependent multi-stability phenomenon is transformed into the easily controlled parameter-relied dynamics, which facilitates physical direction of multi-stable system's operation mode in hardware circuit. Specially, the line equilibrium point exhibited in the voltage-current model is mapped as two determined equilibria. Thus, the dynamics of coexisting infinitely many attractors and their forming mechanisms are readily investigated in the newly constructed system. Moreover, a hardware circuit is designed and fabricated, whose operation mode is rigorously controlled by applying additional DC signals which are proportional to the specified initial conditions. Through circuit simulations and experimental measurements, it is demonstrated that the SVM method offers a direction for interpretation and control of multistability in memristive systems without any determined equilibrium.