Visualization of four normal size limit cycles in two-dimensional polynomial quadratic system (16th Hilbert problem)
نویسنده
چکیده
This paper is devoted to analytical and numerical investigation of limit cycles in twodimensional polynomial quadratic systems. The appearance of modern computers permits one to use a numerical simulation of complicated nonlinear dynamical systems and to obtain new information on a structure of their trajectories. However the possibilities of naive approach, based on the construction of trajectories by numerical integration of the considered differential equations, turns out to be highly limited. In the paper the effective analytical-numerical methods for investigation of limit cycles in two-dimensional polynomial quadratic system are discussed. Estimations of domains of parameters, corresponding to existence of different configurations of large limit cycles, are obtained and visualization of four large limit cycles in quadratic system is presented.
منابع مشابه
Four Limit cycles from perturbing quadratic integrable Systems by quadratic polynomials
The well-known Hilbert’s 16th problem has remained unsolved since Hilbert proposed the 23 mathematical problems at the Second International Congress of Mathematics in 1900 [Hilbert, 1902]. Recently, a modern version of the second part of the 16th problem was formulated by Smale [1998], chosen as one of the 18 challenging mathematical problems for the 21st century. To be more specific, consider ...
متن کاملGlobal phase Portraits of quadratic Polynomial differential Systems with a Semi-Elemental Triple Node
Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert’s 16th problem [Hilbert, 1900, Hilbert, 1902], are still open for this family. In this article we make a global study of the familyQTN ...
متن کاملLimit cycles for a quadratic perturbation of a quadratic polynomial system
The weak Hilbert 16th problem was solved completely in the quadratic case; that is, the least upper bound of the number of zeros of the Abelian integrals associated with quadratic perturbations of quadratic Hamiltonian systems is known. See [3, 4, 5, 8, 10] and the references therein. The next natural step is to consider the same problem for quadratic integrable but non-Hamiltonian systems. Mos...
متن کاملTopological Classification of Quadratic Polynomial Differential Systems with a Finite Semi-Elemental Triple Saddle
The study of planar quadratic differential systems is very important not only because they appear in many areas of applied mathematics but due to their richness in structure, stability and questions concerning limit cycles, for example. Even though many papers have been written on this class of systems, a complete understanding of this family is still missing. Classical problems, and in particu...
متن کاملAround Hilbert –Arnol′d Problem
H(n) = uniform bound for the number of limit cycles of (1) . One way to formulate the Hilbert 16th problem is the following: Hilbert 16th Problem (HP). Estimate H(n) for any n ∈ Z+. To prove that H(1) = 0 is an exercise, but to find H(2) is already a difficult unsolved problem (see [DRR,DMR] for work in this direction). Below we discuss two of the most significant branches of research HP has ge...
متن کامل