Autonomous Differential Systems Research Articles

Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems

A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system.Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka–Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena.In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability.From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.

Influence of the Effective Reproduction Number on the SIR Model with a Dynamic Transmission Rate

In this paper, we examine the epidemiological model B-SIR, focusing on the dynamic law that governs the transmission rate B. We define this dynamic law by the differential equation B′/B=F⊕−F⊖, where F⊖ represents a reaction factor reflecting the stress proportional to the active group’s percentage variation. Conversely, F⊕ is a factor proportional to the deviation of B from its intrinsic value. We introduce the notion of contagion impulse f and explore its role within the model. Specifically, for the case where F⊕=0, we derive an autonomous differential system linking the effective reproductive number with f and subsequently analyze its dynamics. This analysis provides new insights into the model’s behavior and its implications for understanding disease transmission.

An improved algebraic approach to proving global stability of autonomous polynomial differential systems with applications to epidemic models

Global stability of autonomous polynomial differential systems is an important issue in applications. By means of the Lyapunov direct method, an improved algebraic approach to proving the global stability of such systems in the positive cone is presented in this paper. This approach is based on an important finding that the derivative of a Volterra-type Lyapunov function along a polynomial differential system can be expressed as a linear combination of some Volterra-type functions. Applications to some epidemic models demonstrate the effectiveness and universality of the approach. Moreover, according to the property of the Volterra-type function and the relation between the endemic equilibrium and the disease-free equilibrium, the applicable Lyapunov function for the disease-free equilibrium can be derived by reformulating the Lyapunov function for the endemic equilibrium. The approach proposed here greatly simplifies the discussion on global stability of the feasible equilibria of some models described by polynomial differential systems.

Examples of Autonomous Differential Systems with Contrasting Combinations of Lyapunov, Perron, and Upper-Limit Stability Measures

Examples of Autonomous Differential Systems with Contrasting Combinations of Lyapunov, Perron, and Upper-Limit Stability Measures

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Examples of autonomous differential systems with contrast combination of measures of Lyapunov, Perron, and upper-limit stability

Examples of autonomous differential systems with contrast combination of measures of Lyapunov, Perron, and upper-limit stability

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.

A weakened Markus–Yamabe condition for planar polynomial differential systems of degree

AbstractFor a general autonomous planar polynomial differential system, it is difficult to find conditions that are easy to verify and which guarantee global asymptotic stability, weakening the Markus–Yamabe condition. In this paper, we provide three conditions that guarantee the global asymptotic stability for polynomial differential systems of the form $x^{\prime}=f_1(x,y)$, $y^{\prime}=f_2(x,y)$, where f1 has degree one, f2 has degree $n\ge 1$ and has degree one in the variable y. As a consequence, we provide sufficient conditions, weaker than the Markus–Yamabe conditions that guarantee the global asymptotic stability for any generalized Liénard polynomial differential system of the form $x^{\prime}=y$, $y^{\prime}=g_1(x) +y g_2(x)$ with g1 and g2 polynomials of degrees n and m, respectively.

Collective dynamics of two coupled Hopfield inertial neurons with different activation functions: theoretical study and microcontroller implementation

This work deals with the regular and chaotic dynamics of a system made up of two Hopfield-type neurons with two different activation functions: the hyperbolic tangent function and the Crespi function. The mathematical model is in the form of an autonomous differential system of order four with odd symmetry. The analysis highlights nine equilibrium points and four of these points experience a Hopf bifurcation at the same critical value of a control parameter which can be either the diss1ipation parameter or one of the coupling coefficients. This makes plausible the presence of four parallel bifurcation branches as well as the coexistence of multiple attractors in the behavior of the system. One of the highlights revealed in this work is the coexistence of three double-scroll type attractors of particular topology as well as the presence of a four-spiral attractor. Furthermore, the coexistence of both self-excited and hidden dynamics is also reported. All this plethora of dynamics is elucidated by making use of the usual tools for analyzing nonlinear systems such as bifurcation diagrams, the maximum of Lyapunov exponent, basins of attractions as well as phase portraits. A physical implementation of the microcontroller-based system is envisaged in order to confirm the plethora of behaviors observed theoretically.

A survey on local integrability and its regularity

In this survey paper, we summarize our results and also some related ones on local integrability of analytic autonomous differential systems near an equilibrium. The results are on necessary conditions related to existence of local analytic or meromorphic first integrals, on existence of analytic normalization of local analytically integrable system, and also on some sufficient conditions for existence of local analytic first integrals. Among which the results are also on regularity of the local first integrals, including analytic and Gevrey smoothness. We also present some open questions for further investigation.

On the [formula omitted] non-integrability of the autonomous differential systems

In the study of the dynamics of the autonomous differential systems to know the existence or non-existence of first integrals is a relevant fact. These last decades the meromorphic non-integrability of the autonomous differential systems have been studied intensively using the Ziglin’s and the Morales–Ramis’ theories. Here we study the C1 non-integrability of the autonomous differential systems, these studies goes back to Poincaré.It is known that the semiclassical Jayne–Cummings differential system of dimension five has only two independent meromorphic first integrals, namely H and F, and of course any meromorphic function in the variables H and F. Here we illustrate how to study the C1 non-integrability of the autonomous differential systems showing that the semiclassical Jayne–Cummings differential system of dimension five has only two independent C1 first integrals H and F, and of course any C1 function in the variables H and F.

Stability and controllability results by n–ary aggregation functions in matrix valued fuzzy n–normed spaces

In the present paper, we apply the n–ary aggregation functions on several special functions (Hypergeometric function, generalized exponential function, and Fox's H–function) to define a class of matrix–valued fuzzy controllers which help us to study the Ulam–Hyers stability for a (non) autonomous fractional differential system in the Hilfer sense, through the fixed point theorem, in a matrix valued fuzzy n–normed space. Next, by the properties of Mittag-Leffler functions, the Laplace transform and the non–standard Gronwall inequality, we propose some efficient conditions on the (asymptotic) stability of the governing model, in matrix fuzzy normed spaces.

Criterion for Lyapunov Reducibility of a Linear Autonomous Differential System to a Linear Autonomous Equation

Criterion for Lyapunov Reducibility of a Linear Autonomous Differential System to a Linear Autonomous Equation

Criteria on the existence of limit cycles in planar polynomial differential systems

We summarize known criteria for the non-existence, existence and on the number of limit cycles of autonomous real planar polynomial differential systems, and also provide new results. We give examples of systems which realize the maximum number of limit cycles provided by each criterion. In particular we consider the class of differential systems of the form ẋ=Pn(x,y)+Pm(x,y),ẏ=Qn(x,y)+Qm(x,y), where n,m are natural numbers with m>n≥1 and (Pi,Qi) for i=n,m, are quasi-homogeneous vector fields.

Darboux polynomials and first integrals of polynomial Hamiltonian systems

We consider an autonomous polynomial Hamiltonian differential system of n degrees of freedom. In this Letter we study relationships between the existence of Darboux polynomials and additional Darboux polynomials for polynomial Hamiltonian systems. Also we extend some ideas of the Darboux theory of integrability for polynomial differential systems to general polynomial Hamiltonian systems taking into account the multiplicity of Darboux polynomials.

Multiple partial integrals of polynomial Hamiltonian systems

We consider an autonomous real polynomial Hamiltonian ordinary differential system. Sufficient conditions for the construction of additional first integrals on polynomial partial integrals and multiple polynomial partial integrals are obtained. Classes of autonomous polynomial Hamiltonian ordinary differential systems with first integrals which analytically expressed by multiple polynomial partial integrals are identified. Also we present examples that illustrate the theoretical results.

Melnikov functions of arbitrary order for piecewise smooth differential systems in [formula omitted] and applications

In this paper we develop an arbitrary order Melnikov function to study limit cycles bifurcating from a periodic submanifold for autonomous piecewise smooth differential systems in Rn with two zones separated by a hyperplane. This result not only extends some of the known results on the Melnikov theory in dimension and order but also compensates for some defects of the averaging theory in studying the limit cycle bifurcation of autonomous systems from a periodic submanifold. To demonstrate the application of our theoretical result and its superiority for some systems to the existing averaging theory, we study the maximum number of limit cycles bifurcating from an n-dimensional periodic submanifold caused by non-smooth centers of the fold-fold type, providing an upper bound for any order piecewise polynomial perturbations of degree m. Concerning the planar case of the unperturbed system, a piecewise Hamiltonian system, we obtain a better upper bound for piecewise polynomial Hamiltonian perturbations up to order two. The realizability of these upper bounds is also discussed.

Studying the Stability of the ψ‐Hilfer Fractional Differential System

This paper devotes to the study on the stability and decay of solution to fractional differential system involving the ψ‐Hilfer fractional derivative of order α ∈ (0,1) and type β ∈ [0,1]. We first derive the solution of linear system by using the generalized Laplace transform, which can be represented by the form of Mittag‐Leffler function. Then, in terms of the asymptotic expansion of the Mittag‐Leffler function, stability properties of linear system are analyzed in more detail. Finally, we construct a linearization theorem and determine the stability near the equilibrium for the autonomous nonlinear differential system with the ψ‐Hilfer derivative.

Fundamental Results on Determining Matrices for a Certain Class of Hereditary Systems

Three major tools are required to investigate the controllability of control systems, namely, determining matrices, index of control systems and controllability Grammian. Determining matrices are the preferred choice for autonomous control systems due to the fact that they are devoid of integral operators in their computations. This article developed the structure of certain parameter-ordered determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for Euclidean controllability of the system. Expressions for the relevant determining matrices were formulated and it was established that the determining matrices for double time-delay linear autonomous functional differential control systems do not exist if one of the time-delays is not an integer multiple of the other paving the way for the investigation of the Euclidean controllability of generic double time-delay control systems.

Global Structure of Determining Matrices for a Class of Differential Control Systems

This paper developed and established unprecedented global results on the structure of determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for the Euclidean controllability of the system. The computational process and implementation of the controllability matrix were demonstrated on the MATLAB platform to determine the controllability disposition of a small-problem instance. Finally, the work examined the computing complexity of the determining matrices.