Autonomous Equations Research Articles

Analytical solutions for autonomous differential equations with weighted derivatives

In this work, we introduce a new definition of weighted derivatives along with corresponding integral operators, which aim to facilitate the solution of both linear and non-linear differential equations. A significant finding is that the fractional derivative of Caputo–Fabrizio type is a special case within this framework, allowing us to build upon existing research in this area. Additionally, we provide closed-form analytical solutions for autonomous and logistic equations using our newly defined derivatives and integrals. We thoroughly explore the properties associated with these weighted derivatives and integrals. To demonstrate the reliability and practical applicability of our results, we include several examples and applications that highlight the effectiveness of our approach.

Dynamical system approach of interacting dark energy models in $f(R,T^{\phi})$ gravity

Abstract We have examined an isotropic and homogeneous cosmological model in $f(R,T^{\phi})$ gravity, where $R$ represents the Ricci scalar and $T^{\phi}$ exhibits the energy momentum tensor's trace. We examine the stability criteria by performing the dynamical system analysis for our model $f(R,T^{\phi})=R+2(aT^{\phi}+b)$, where $a$ $\&$ $b$ are the constants. We derive a set of autonomous equations and find their solutions by assuming a flat potential $V_{0}$. We assess the equilibrium points from these equations and find the eigenvalues. We analyze the physical interpretation of the phase space for this system. We obtain three stable equilibrium points. We also examine the interaction between scalar field and dark energy, represented by $Q=\psi H\rho_{de}$ and determine the equilibrium points for this interaction. We identify four stable equilibrium points for this interaction. We calculate the values of the physical parameters for both scenarios at each equilibrium point, indicating the Universe's accelerated expansion.

About tautochronic movements

It is shown that a material point, under the influence of an attractive linear force and a repulsive force inversely proportional to the cube of the distance from the center of attraction, performs a periodic motion, the period of which does not depend on the initial data (tautochronic motion). The problem is reduced to a nonlinear autonomous second-order equation, the general solution of which is expressed in terms of elementary functions. It has also been proven that for other power laws of repulsive force, except for degrees 0, 1 and –3, the movement of a material point is not tautochronous.

Update (3.0) to LyapXool: Quadratic programming

This is an update to PII: S2352711020303290, LyapXool is a C＋＋ program to compute complete Lyapunov functions and their orbital derivatives for any dynamical system expressed by an autonomous ordinary differential equation. New methods and improvements made in the third version are discussed. In particular, the computation of complete Lyapunov functions within quadratic programming methods are now computed by the software. These new approaches can be used to better analyse the behaviour of dynamical systems expressed as systems of autonomous equations. The structure of the program is object-oriented rather than procedural and the output formatting has been restructured to make it easier to understand. The program maintains its user friendliness. This paper describes how the code is organised, how it can be used to compute complete Lyapunov functions and their subsequent properties for dynamical systems, and it provides an interesting example of its application.

Dynamic bifurcation of nonautonomous evolution equations under Landesman–Lazer condition with cohomology methods

In this article we study the dynamic bifurcation of nonautonomous evolution equations by using cohomology methods. First, we construct a homotopy equivalence relation between the nonautonomous system and a product flow. Then, we slightly extend some continuation theorems on bifurcations for autonomous equations, and prove some new cohomology consequences on the reduced singular groups. Based on this homotopy equivalence relation and these conclusions, we establish some typical results on the dynamic bifurcation from infinity of the abstract nonautonomous evolution equation. Finally, we consider the parabolic equation ut−Δu=λu+f(x,u)+g(x,t) associated with the Dirichlet boundary condition, where f(x,u) satisfies the appropriate Landesman–Lazer type condition. Some new results on the dynamical behaviors of this equation near resonance of the equation are derived.

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Existence and stability of mild solutions to non autonomous impulsive stochastic neutral integrodifferential equations

Phase space analysis of interacting and non-interacting models in f(Q, C) gravity

Abstract Using a dynamical system method, we have studied a Friedmann-Robertson-Walker (FRW) cosmological model within the context of $f(Q,C)$ gravity, where $Q$ is the non-metricity scalar and $C$ represents the boundary term, considering both interacting and non-interacting models. A set of autonomous equations is derived, and solutions are calculated accordingly. We assessed the critical points obtained from these equations, identified their characteristic values, and explored the physical interpretation of the phase space for this system. Two types of $f(Q,C)$ are assumed to be $(i)$ $f(Q,C)=Q+\alpha Q+\beta C logC$ and $(ii)$ $f(Q,C)=Q+\alpha Q+\frac{\beta}{C}$, where $\alpha$ and $\beta$ are the parameters. In Model I, we obtain two stable critical points, while in Model II, we identified three stable critical points for both interacting and non-interacting models. We look into how the phase space trajectories behave at every critical points. We calculate the values of the physical parameters for both systems at each critical point, indicating the Universe's accelerated expansion.

Periodic free vibrations of composite laminates with curvilinear fibres and CNTs

This article addresses the combined effect of using curvilinear fibres and carbon nanotubes (CNTs) reinforcements in the non-linear modes of vibration of laminated composite plates. To arrive at the material properties of the three-phase composite material, a two-step hierarchic procedure is followed. A modified version of the Halpin–Tsai model is employed to predict the Young’s modulus of the CNT enriched resin and expressions, deduced from equilibria of a unit cell where a fibre is embedded in resin, are applied to obtain the diverse elasticity moduli of the three-phase composite. Moderately large displacements are considered, with von Kármán strain–displacement relations. Although the presented model is an equivalent single layer one, it applies to thick plates, because a Third-order Shear Deformation Theory (TSDT) is followed. The set of autonomous non-linear equations of motion is reduced using static condensation and a modal basis with selected modes, chosen after a convergence analysis. The reduced set of equations of motion is solved by the shooting method. Numerical tests considering plates with diverse curvilinear fibre paths, CNT contents and thicknesses are carried out. The results obtained are thoroughly analysed.

Quantum Bernoulli noises approach to quantum master equations and applications to Ehrenfest-type theorems

This paper investigates the regularity properties of quantum master equations with unbounded coefficients within the space of square-integrable complex-valued Bernoulli functionals. Initially, we demonstrate the existence of a unique regular solution to the linear stochastic Schrödinger equation driven by cylindrical Brownian motions. Subsequently, we establish the existence of regular solutions for the autonomous linear quantum master equation and provide a probabilistic representation of this solution in terms of the stochastic Schrödinger equation. Finally, by taking the expectation of some observables with respect to the solution of the quantum master equation, we derive a system of differential equations that describe the evolution of the mean values of certain quantum observables.

Theory on Linear L-Fractional Differential Equations and a New Mittag–Leffler-Type Function

The L-fractional derivative is defined as a certain normalization of the well-known Caputo derivative, so alternative properties hold: smoothness and finite slope at the origin for the solution, velocity units for the vector field, and a differential form associated to the system. We develop a theory of this fractional derivative as follows. We prove a fundamental theorem of calculus. We deal with linear systems of autonomous homogeneous parts, which correspond to Caputo linear equations of non-autonomous homogeneous parts. The associated L-fractional integral operator, which is closely related to the beta function and the beta probability distribution, and the estimates for its norm in the Banach space of continuous functions play a key role in the development. The explicit solution is built by means of Picard’s iterations from a Mittag–Leffler-type function that mimics the standard exponential function. In the second part of the paper, we address autonomous linear equations of sequential type. We start with sequential order two and then move to arbitrary order by dealing with a power series. The classical theory of linear ordinary differential equations with constant coefficients is generalized, and we establish an analog of the method of undetermined coefficients. The last part of the paper is concerned with sequential linear equations of analytic coefficients and order two.

Hyers–Ulam stability of integral equations with infinite delay

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay.

Evolutionary equations are G-compact

We prove a compactness result related to G-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for G-convergence for time-dependent equations solely in terms of static equations.

The Dynamical Equations of the Restricted Three-Body Problem with Poynting-Robertson Drag Force and Variable Masses

The restricted three-body problem (R3BP) is a formulation which defines the motion of a passively gravitating test particle having infinitesimal mass and moving in the gravitational environment of two bodies, called primaries. The R3BP is still an exciting and active research field that has been getting attention of scientists and astronomers because of its applications in dynamics of the solar and stellar systems, lunar theory, and artificial satellites. The equations of motion are usually the starting point in the investigations of the dynamical predictions of the infinitesimal mass. Therefore, in this paper, we examine the derivations of the dynamical equations of the R3BP with Poynting-Robertson (P-R) Drag force and variable masses. In this model formulation, both primaries are assumed to vary their masses under the combined Mestschersky law (CML) and they move in the frame of the GyldenMestschersky equation (GME). Further, the bigger primary is assumed to be emitting radiation force, which is a component of the radiation pressure and the P-R drag. The non- autonomous dynamical equations of the model are derived and converted into the autonomized equations with constant coefficients using the Mestschersky transformation (MT), the CML, the particular solutions of the GMP, and a transformation for the time dependent velocity of light. We observed that the P-R drag of the bigger primary depends on the mass parameter, radiation pressure, velocity of light and the mass variation constant . The derived systems of equations with variable and constant coefficients can be used to model the long-term motion of satellites and planets in binary systems.

Dynamical System Approach and Thermodynamical Perspective of Hořava‐Lifshitz Gravity

AbstractThe authors have examined a Friedmann Robertson Walker cosmological model in Hořava‐Lifshitz gravity by using a dynamical system approach. A set of autonomous equations is derived and their solutions are calculated. The critical points from these equations and find the characteristics values with the analysis of the physical interpretation of the phase space for this system are assessed. Three stable critical points are found and the values of the physical parameters and the scale factor's expressions at each critical points are displayed in Tables 1, 2, and 3. A hybrid scale factor to develop the model, which results in a phase transition from deceleration to acceleration is used. The suitable values of the parameters are governed by applying the Monte Chain Monte Carlo method technique to the Hubble 46 and joint Hubble 46 and Baryon Acoustic Oscillations 15 datasets. In contrast to the negative behavior of pressure, the positive behavior of energy density and illustrate the Universe's acceleration epoch and the model is represented by the EoS parameter . The authors investigated that the energy conditions and their model violates the strong energy condition. Utilizing diagnostic test, it is found that the model represents phantom behavior. The thermodynamical perspective for the model is also examined. The model accurately explained the Universe's propagation history and fits well with contemporary cosmic data.

Ring-like partially nonlocal extreme wave of a (3+1)-dimensional NLS system with partially nonlocal nonlinearity and external potential

This paper aims to study extreme waves localized in (3+1)-dimensional space based on a (3+1)-dimensional nonautonomous partially nonlocal NLS system under the linear and parabolic potentials, which is simplified into a (2+1)-dimensional autonomous equation via a converting formula. Based on solutions of the autonomous NLS system, an approximate solution of ring-like extreme wave is found. Characteristics and evolution of ring-like extreme wave are explored in an exponential system. The influence of the magnitude and exponential parameters of diffraction on ring-like extreme wave is studied. Moreover, the impact of the Hermite polynomial parameter q, the radius parameter R and the thickness parameter w on ring-like extreme wave is also discussed.

New Classes of Solutions for Euclidean Scalar Field Theories

This paper presents new classes of exact radial solutions to the nonlinear ordinary differential equation that arises as a saddle-point condition for a Euclidean scalar field theory in D-dimensional spacetime. These solutions are found by exploiting the dimensional consistency of the radial differential equation for a single massless scalar field, which allows it to transform into an autonomous equation. For massive theories, the radial equation is not exactly solvable, but the massless solutions provide useful approximations to the results for the massive case. The solutions presented here depend on the power of the interaction and on the spatial dimension, both of which may be noninteger. Scalar equations arising in the study of conformal invariance fit into this framework, and classes of new solutions are found. These solutions exhibit two distinct behaviors as D→2 from above.

Asymptotically autonomous stability of kernel sections for lattice plate equations with nonlinear damping

We develop the theory of kernel sections of non-autonomous dynamical systems. Under some sufficient conditions, we establish some abstract results on the asymptotically autonomous stability of kernel sections, which suggests that the time-section of a kernel section is asymptotically stable to a global attractor of the autonomous dynamical system. A lattice plate equation with nonlinear damping and non-autonomous forcing is considered as an application. It is shown that the time-section of kernel sections of the non-autonomous lattice plate equations are nonempty, uniformly compact, pullback attracting, and asymptotically stable to a global attractor of the autonomous lattice plate equations.

On the reduction of an autonomous nonlinear boundary value problem, unsolved with respect to the derivative, to the first-order critical case

Constructive solvability conditions and a scheme for constructing solutions of an autonomous nonlinear boundary value problem unsolved with respect to the derivative have been found. A convergent iterative scheme for finding approximations to the solutions of a nonlinear autonomous nonlinear boundary value problem unsolved with respect to the derivative was constructed by reducing the problem to the first-order critical case. As an example of application of the constructed iterative scheme, approximations to the solutions of a periodic boundary value problem for the autonomous Lotka-Volterra equation were determined.

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.

Multiplicity and stability of normalized solutions to non-autonomous Schrödinger equation with mixed non-linearities

AbstractThis paper first studies the multiplicity of normalized solutions to the non-autonomous Schrödinger equation with mixed nonlinearities \begin{equation*} \begin{cases} -\Delta u=\lambda u+h(\epsilon x)|u|^{q-2}u+\eta |u|^{p-2}u,\quad x\in \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2\,\textrm{d}x=a^2, \end{cases} \end{equation*} where $a, \epsilon, \eta \gt 0$, q is L2-subcritical, p is L2-supercritical, $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and h is a positive and continuous function. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when ϵ is small enough. The solutions obtained are local minimizers and probably not ground state solutions for the lack of symmetry of the potential h. Secondly, the stability of several different sets consisting of the local minimizers is analysed. Compared with the results of the corresponding autonomous equation, the appearance of the potential h increases the number of the local minimizers and the number of the stable sets. In particular, our results cover the Sobolev critical case $p=2N/(N-2)$.