2. Basic definitions and notation

is described with a sequence of its states. For dynamic systems with continuous time (which are called flows), a state of the system is defined for each moment of time on a real or an imaginary axis. Cascades and flows are the main subject of study in symbolic and topological

Dynamic systems, both with discrete and continuous time, can be usually described by an autonomous system of differential equations, defined in a certain domain and satisfying in it the conditions of the Cauchy theorem of existence and uniqueness of solutions of the differen-

Singular points of differential equations correspond to equilibrium positions of dynamic systems, and periodical solutions of differential equations correspond to closed phase curves of

The main task of the theory of dynamic systems is a study of curves, defined by differential equations. This process includes splitting of a phase space into trajectories and studying their limit behavior—finding and classifying the equilibrium positions, and revealing the attracting and repulsive manifolds (i.e., attractors and repellers; sinks and sources). The most important notions of the theory of dynamic systems are the notion of stability of equilibrium states, which means the ability of a system under considerably small changes of initial data to remain near an equilibrium state (or on a given manifold) for an arbitrary long period of time, as well as the notion of roughness of a system (i.e., the saving of a system's properties under small changes of a model itself). A rough dynamic system is a system that preserves its qualitative character

The research methods proposed in this chapter are new and effective; they can also be used for the study of applied dynamic systems of the second order with polynomial right parts.

According to Jules H. Poincare, a normal autonomous second-order differential system with polynomial right parts, in principle, allows its full qualitative investigation on an extended

generations, including contemporary researchers, have studied some of such systems, for example, quadratic dynamic systems [2], ones containing nonzero linear terms, homogeneous cubic systems, and dynamic systems with nonlinear homogeneous terms of the odd degrees (3, 5, 7) [3], which have a center or a focus in a singular point O (0, 0) [4], as well as other

dy

such that X (x, y), Y (x, y) are reciprocal forms of x and y, X is a cubic, Y a square form, and X (0,1) > 0, Y (0, 1) > 0. Our objective is to depict in a Poincare circle all kinds (different in the topological sense) of possible for systems phase portraits for Eq. (1), and also to indicate the criteria of every portrait realization close to coefficient ones. With this aim, we apply Poincare's method of consecutive mappings: first, the central mapping of a plane x, y (from a

We consider in the present chapter a family of dynamic systems on a real plane x, y.

dt <sup>¼</sup> X xð Þ ; <sup>y</sup> ,

dx

x, <sup>y</sup> [1]. Inspired by the great Poincare's works, mathematicians of the next

dt <sup>¼</sup> Y xð Þ ; <sup>y</sup> (1)

dynamics.

66 Differential Equations - Theory and Current Research

tial equations.

dynamic systems.

arithmetical plane R<sup>2</sup>

particular kinds of systems.

of motion under small changes of parameters.

φð Þ t; p , p ¼ ð Þ� x; y a fixed point: = a solution (a motion) of Eq. (1) – system with initial data 0ð Þ ; p :

Lp:φ ¼ φð Þ t; p , t ∈Imax, � a trajectory of motion φð Þ t; p :

Lþ �ð Þ <sup>p</sup> := + (�) – a semi-trajectory of a trajectory Lp:

O-curve of a system := the system's semi-trajectory Ls <sup>p</sup>(p 6¼ O, s∈ fþ; �g) adjoining to a point O under a condition such that st ! þ∞:

Oþ �ð Þ- curve of a system: = the system's O-curve Lþ �ð Þ <sup>p</sup> :

Oþ �ð Þ-curve of a system: = the system's O-curve adjoining to a point O from a domain x > 0 (x < 0).

TO-curve of a system: = the system's O-curve, which, being supplemented by a point O, touches some ray in it.

A nodal bundle of NO-curves of a system := an open continuous family of the system's TOcurves L<sup>s</sup> p, where <sup>s</sup> <sup>∈</sup>fþ; �g is a fixed index, <sup>p</sup> <sup>∈</sup>ᴧ, <sup>ᴧ</sup> a simple open arc, <sup>L</sup><sup>s</sup> <sup>p</sup> ∩ ᴧ ¼ f gp :

A saddle bundle of SO-curves of a system, a separatrix of the point O:= a fixed TO-curve, which is not included in some bundle of NO-curves of a system.

E, H, P-O-sectors of a system: an elliptical, a hyperbolic, a parabolic sector.

A topological type (T-type) of a singular point O of a system:= a word AO consisting of letters N, S (a word BO consisting of letters E, H, P), which describes a circular order of bundles N, S of its O-curves (of its O-sectors E, H, P) when traversing the point O in the " + "-direction, i. e., counterclockwise, starting with some of them.

$$P(\mu) \coloneqq X(1, \mu) \equiv p\_0 + p\_1\mu + p\_2\mu^2 + p\_3\mu^3,$$

$$Q(\mu) \coloneqq Y(1, \mu) \equiv a + b\mu + c\mu^2.$$

Note 1. For every Eq. (1) system:

1) T-type of a singular point O in its form BO is easy to construct using its Т-type in the form AO, and going backward (we need to determine both forms, see Corollary 1);

2) Real roots of a polynomial P(u) (polynomial Q(u)) are in fact angular coefficients of isoclines of infinity (isoclines of a zero));

3) When we write out the real roots of the system's polynomials P(u), Q(u), separately or all together, we always number the roots of each one of them in an ascending order.
