Autonomous Differential Systems Research Articles

On the Darboux integrability of a three—dimensional forced—damped differential system

In 2011 Pehlivan proposed a three—dimensional forced—damped autonomous differential system which can display simultaneously unbounded and chaotic solutions. This untypical phenomenon has been studied recently by several authors. In this paper we study the opposite to its chaotic motion, i.e. its integrability, mainly the existence of polynomial, rational and Darboux first integrals through the analysis of its invariant algebraic surfaces and its exponential factors.

Averaging methods of arbitrary order, periodic solutions and integrability

In this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems. This result extends and improves results on averaging theory of periodic analytic differential systems, and it unifies many different kinds of averaging methods.Applying our theory to autonomous analytic differential systems, we obtain some conditions on the existence of limit cycles and integrability.For polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues, we prove that there always exists a positive number N such that if its first N averaging functions vanish, then all averaging functions vanish, and consequently there exists a neighborhood of the origin filled with periodic orbits. Consequently if all averaging functions vanish, the origin is a center for n=2.Furthermore, in a punctured neighborhood of the origin, the system is C∞ completely integrable for n>2 provided that each periodic orbit has a trivial holonomy.Finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities.

Stability regions for fractional differential systems with a time delay

The paper investigates stability and asymptotic properties of autonomous fractional differential systems with a time delay. As the main result, necessary and sufficient stability conditions are formulated via eigenvalues of the system matrix and their location in a specific area of the complex plane. These conditions represent a direct extension of Matignon’s stability criterion for fractional differential systems with respect to the inclusion of a delay. For planar systems, our stability conditions can be expressed quite explicitly in terms of entry parameters. Applicability of these results is illustrated via stability investigations of the fractional delay Duffing’s equation.

Properties of Rational General Solutions for First Order Multivariate Autonomous Rational Differential Systems

This paper studies the properties of rational general solutions for first order multivariate autonomous rational differential systems. We obtain a necessary and sufficient conditions for existence of rational general solution of first order multivariate autonomous rational differential system if the degree bound of rational solutions of this system is given. Due to the problem for computing a rational solution of the multivariate rational differential system can be reduced to finding a linear rational solution of an autonomous differential equation, we also prove that the linear rational solvability of the resulting autonomous differential equation does not depend on the choice of proper parametrizations of invariant algebraic space curves. In addition, two different rational solutions corresponding to the same invariant algebraic space curve are proved to be related by a shifting of the variable.

The Completely Integrable Differential Systems are Essentially Linear Differential Systems

Let $$\dot{x}=f(x)$$ be a $$C^k$$ autonomous differential system with $$k\in {\mathbb {N}}\cup \{\infty ,\omega \}$$ defined in an open subset $$\varOmega $$ of $${\mathbb {R}}^n$$ . Assume that the system $$\dot{x}=f(x)$$ is $$C^r$$ completely integrable, i.e., there exist $$n-1$$ functionally independent first integrals of class $$C^r$$ with $$2\le r\le k$$ . As we shall see, we can assume without loss of generality that the divergence of the system $$\dot{x}=f(x)$$ is not zero in a full Lebesgue measure subset of $$\varOmega $$ . Then, any Jacobian multiplier is functionally independent of the $$n-1$$ first integrals. Moreover, the system $$\dot{x}=f(x)$$ is $$C^{r-1}$$ orbitally equivalent to the linear differential system $$\dot{y}=y$$ in a full Lebesgue measure subset of $$\varOmega $$ . Additionally, for integrable polynomial differential systems, we characterize their type of Jacobian multipliers.

Divergence and Poincaré–Liapunov constants for analytic differential systems

We consider a planar autonomous real analytic differential system with a monodromic singular point p. We deal with the center problem for the singular point p. Our aim is to highlight some relations between the divergence of the system and the Poincaré–Liapunov constants of p when these are defined.

The inverse integrating factor for some classes of n-th order autonomous differential systems

The inverse integrating factor for some classes of higher order autonomous differential systems is studied in this paper. Some sufficient conditions on the existence of the inverse integrating factors of certain classes of n-th order autonomous differential systems are presented. Also, the explicit forms of the inverse integrating factors can be shown. Simultaneously, some related examples are given to illustrate the feasibility and the effectiveness of the proposed results.

Регулятор полного успокоения для линейной автономной дифференциальной системы с запаздыванием

Регулятор полного успокоения для линейной автономной дифференциальной системы с запаздыванием

Integrability and generalized center problem of resonant singular point

In this paper, integrability and generalized center condition of resonant singular point for a broad class of complex autonomous polynomial differential system are studied. A new method—integrating factor method of determining integrability of resonant singular point is obtained for any rational resonance ratio. At the same time, the relations of the first integral method and the integrating factor method with the normal form method are obtained.

Solution procedure of residue harmonic balance method and its applications

This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems. In this solution procedure, no small parameter is assumed. The harmonic residue of balance equation is separated in two parts at each step. The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement. The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again. Three kinds of different differential equations involving general, fractional and delay ordinary differential systems are given as numerical examples respectively. Highly accurate limited cycle frequency and amplitude are captured. The results match well with the exact solutions or numerical solutions for a wide range of control parameters. Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems. Moreover, the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.

Residue harmonic balance solution procedure to nonlinear delay differential systems

This paper develops the residue harmonic balance solution procedure to predict the bifurcated periodic solutions of some autonomous delay differential systems at and after Hopf bifurcation. In this solution procedure, the zeroth-order solution employs just one Fourier term. The unbalanced residues due to Fourier truncation are considered by solving linear equation iteratively to improve the accuracy. The number of Fourier terms is increased automatically. The well-known sunflower equation and van der Pol equation with unit delay are given as numerical examples. Their solutions are verified for a wide range of system parameters. Comparison with those available shows that the residue harmonic balance method is effective to solve the autonomous delay differential equations. Moreover, the present method works not only in determining the amplitude but also the frequency at bifurcation.

On the Limit-Properties of Structural Change

Many economic theories deal with some sort of “structure” and with the changes in this structure. We provide mathematical meta-models for studying the limit-properties of continuous structural change. The limit properties of processes are important information in economic modelling. For example, they indicate whether a process can generate long-run growth effects. We show that continuous structural change is a relatively “simple” process. In most cases it is transitory or cyclical. Our meta-models cover different modelling techniques (autonomous, quasi-autonomous and non-autonomous differential equation systems) and many economic notions of structure (e.g. income-distribution, cross-sector labour allocation and investment structure).

Zero-Hopf bifurcation for a class of Lorenz-type systems

We apply the averaging theory to a class of three-dimensional autonomous quadratic polynomial differential systems known as Lorenz-type systems, and we prove the existence of a small amplitude periodic orbit bifurcating from a degenerate zero-Hopf equilibrium of these systems.

Multiple Hopf bifurcation in [formula omitted] and inverse Jacobi multipliers

In this paper we study the maximum number of limit cycles that can bifurcate from a singular point of saddle-focus type of an analytic, autonomous differential system in R3 under any analytic perturbation that keeps the location and nature of the singularity. We only consider those foci on center manifolds having associated two nonzero purely imaginary and one nonzero real eigenvalues. Our approach is different from the classical one in the sense that we do not use any center manifold reduction to compute Poincaré–Lyapunov constants. Instead, we study the multiple Hopf bifurcation first doing a Lyapunov–Schmidt reduction to the associated Poincaré map, obtaining in this way an analytic reduced displacement map. Next we prove that the order of this displacement map coincides with the vanishing multiplicity (denoted m) of any locally smooth and non-flat inverse Jacobi multiplier. Finally the cyclicity of the focus is given in terms of m.

Computation of universal unfolding of the double-zero bifurcation in -symmetric systems by a homological method

We present a homological method and explicit computational formulae for normal form and universal unfolding of the double-zero bifurcation in any autonomous ordinary differential equation system with -symmetry. We take the Chua's circuit as an example to show advantages of this method.

Computation of universal unfolding of the double zero bifurcation in the Z2-symmetric system

We present explicit formulae of universal unfolding of the double zero bifurcation for any autonomous ordinary differential equation systems (ODEs) with Z2-symmetry. These formulae only need to use coefficients of the Taylor expansion of the vector field at its equilibria and are equally suitable for both numerical and symbolic computations, and they construct the relationship between original parameters and topological structures for the double zero bifurcation analysis in any autonomous ODEs. They can be directly used to check whether there exist the original parameters such that a system can exhibit the versal double zero bifurcation. Moreover, we can use them to compute the corresponding bifurcation curves with high precision. We take Chua's circuit as an example to demonstrate the advantages of these formulae.

Stability regions for linear fractional differential systems and their discretizations

This paper concerns with basic stability properties of linear autonomous fractional differential and difference systems involving derivative operators of the Riemann–Liouville type. We derive stability regions for special discretizations of the studied fractional differential systems including a precise description of their asymptotics. Our analysis particularly shows that discretizations based on backward differences can retain the key qualitative properties of underlying fractional differential systems. In addition, we introduce the backward discrete Laplace transform and employ some of its properties as the main proof tool.

Spectral reducibility of delay differential controlled systems in the regular case

For pointwise controlled autonomous linear differential systems with commensurate delays, a dynamic controller is designed that reduces the original system to a difference-differential system with a finite spectrum. The results are illustrated by an example.

An algebraic approach to proving the global stability of a class of epidemic models

The global stability of an autonomous differential equation system is an important issue for ecological, epidemiological and virus dynamical models. By means of the direct Lyapunov method and the LaSalle’s Invariance Principle, an algebraic approach to proving the global stability is presented in this paper. This approach gives a logic and possibly programming method on how to choose coefficients ai based on the classic Lyapunov function of the form ∑i=1nai(xi−xi∗−xi∗lnxi/xi∗) such that the derivative of the Lyapunov function is negative definite or semidefinite. As an application, the global stability of an SVS-SEIR epidemic model with vaccination and the latent stage is examined. The generality of the approach is also shown by discussing certain cases.

Preservation of Stability and Synchronization of a Class of Fractional‐Order Systems

We present sufficient conditions for the preservation of stability of fractional‐order systems, and then we use this result to preserve the synchronization, in a master‐slave scheme, of fractional‐order systems. The systems treated herein are autonomous fractional differential linear and nonlinear systems with commensurate orders lying between 0 and 2, where the nonlinear ones can be described as a linear part plus a nonlinear part. These results are based on stability properties for equilibria of fractional‐order autonomous systems and some similar properties for the preservation of stability in integer order systems. Some simulation examples are presented only to show the effectiveness of the analytic result.