Qualitative Theory of Dynamical Systems 

About: Qualitative Theory of Dynamical Systems is an academic journal published by Springer Science+Business Media. The journal publishes majorly in the area(s): Computer science & Nonlinear system. It has an ISSN identifier of 1575-5460. Over the lifetime, 937 publications have been published receiving 5192 citations.

A survey of isochronous centers

Abstract: In this survey we give an overview of the results obtained in the study of isochronous centers of vector fields in the plane. This paper consists of two parts. In the first one (sections 2--8), we review some general techniques that proved to be useful in the study of isochronicity. In the second one (sections 9--16), we try to give a picture of the state of the art at the moment this review was written. In section 2, we give some basic definitions about centers, isochronous centers, first integrals, integrating factors, particular algebraic solutions, and other related concepts. In this sections we also give some general theorems about centers and isochronous centers, and we give a brief account of the evolution of the researches in this field. In the successive sections we focus on various methods that have been used in attacking the isochonicity problem. We start with linearizations in section 3, stating Poincar\'e's classical theorem and some of its consequences. Section 4 is devoted to describe the procedure that leads to define and compute isochronous constants. In section 5, commutators are introduced, and basic facts about couples of commuting systems are described. Classical theorems about systems obtained from complex ordinary differential equations are collected in section 6. Hamiltonian systems are considered in section 7, where their connection to the study of the Jacobian Conjecture is showed, too. Section 8 is concerned with systems having constant angular speed with respect to some coordinate system. The second part starts with section 9, that is devoted to recent results about second order differential equations not immediately reducible to hamiltonian systems. This section also contains the characterization of isochronous centers of reversible Li\'enard systems. In section 10 we list all fundamental results about isochronous centers of quadratic systems. Next section contains results about cubic systems with homogeneous nonlinearities. Sections 12 is devoted to cubic reversible systems. In section 13 we collect results about quartic and quintic systems with homogeneous nonlinearities. A class of particular cubic systems, with degenerate infinity is considered in section 14. Finally, section 15 is devoted to Kukles system. All the sections of the second part, and some of the first part, contain tables, where the main features of the considered systems are collected. When possible, for every class of systems we have written the system in rectangular and polar coordinates, and we have reported a first integral, a commutator, a linearization and a reciprocal integrating factor. The bibliography contains references both to papers devoted to the study of isochronicity and to papers concerned with integrability of plane systems and the study of the period function of centers. We have tried to make the bibliography so complete as possible for what is concerned with isochronicity. We have made no effort to make it complete for papers about integrability and the study of the period function. We apologize for possible mistakes and encourage the readers to communicate us any corrections.

Algebraic aspects of integrability for polynomial systems

Abstract: We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

Non-Linear Dynamics with Non-Standard Lagrangians

Abstract: Two new actions being of a non-natural class \({S = \int {e^{L(q, \dot {q}, t)}dt}}\) and \({S = \int {L^{1 + \gamma }(q, \dot {q}, t)dt}, \gamma \in {\mathbb{R}}}\) with non-standard Lagrangians are introduced. It is demonstrated that nonlinear systems holding new dynamical properties may be obtained. Several constrained Lagrangians systems have been identified to possess attractive properties. Additional features are explored and discussed in some details.

New Results on the Study of Zq-Equivariant Planar Polynomial Vector Fields

Abstract: In this paper, we introduce some new results on the study of Z q -equivariant planar polynomial vector fields. The main conclusions are as follows. (1) For the Z 2-equivariant planar cubic systems having two elementary foci, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. All possible first integrals are given. Under small Z 2-equivariant cubic perturbations, the conclusion that there exist at most 12 small-amplitude limit cycles with the scheme $${\langle 6\amalg 6\rangle}$$ is proved. (2) On the basis of mentioned work in (1), by considering the bifurcation of a global limit cycle from infinity, we show that under small Z 2-equivariant cubic perturbations, such bi-center cubic system has at least 13 limit cycles with the scheme $${\langle 1 \langle 6\amalg 6\rangle\rangle}$$ , i.e., we obtain that the Hilbert number H(3) ≥ 13. (3) For the Z 5-equivariant planar polynomial vector field of degree 5, we shown that such system has at least five symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z 5-equivariant perturbations, the conclusion that perturbed system has at least 25 limit cycles with the scheme $${\langle 5\amalg 5\amalg 5\amalg 5\amalg 5\rangle}$$ is rigorously proved. (4) For the Z 6-equivariant planar polynomial vector field of degree 5, we proved that such system has at least six symmetric centers if and only if it is a Hamltonian vector field. The characterization of a center problem is completely solved. The shortened expressions of the first four Lyapunov constants are given. Under small Z 6-equivariant perturbations, the conclusion that perturbed system has at least 24 limit cycles with the scheme $${\langle 4\amalg 4\amalg 4\amalg 4\amalg 4\amalg4\rangle}$$ is rigorously proved. Two schemes of distributions of limit cycles are given.

Exact Iterative Solution for an Abstract Fractional Dynamic System Model for Bioprocess

Abstract: In this paper, we study a singular nonlocal fractional dynamic system arising in the abstract model for bioprocess. Conditions for the exact iterative solution to the problem are established, followed by development of an iterative technique for generating approximate solution to the problem. The iterative technique has been proved to give sequences converging uniformly to the exact solution, and formulate for estimation of the approximation error and the convergence rate have been derived. An example is also given in the paper to demonstrate the application of our theoretical results.