1. Introduction

A dynamic system appears to be a mathematical model of some process or phenomenon, in which fluctuations and other so-called statistical events are not taken into consideration. It can be characterized by its initial state and a law according to which the system goes into a different state. A phase space of a dynamic system is the totality of all admissible states of this system.

It is necessary to distinguish dynamic systems with the discrete time and with the continuous time. For dynamic systems with the discrete time (they are called cascades), a system's behavior

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 dynamics.

center (0, 0, 1) of a sphere ∑), augmented with a line at infinity (i.e., R<sup>2</sup>

in more detail.

data 0ð Þ ; p :

Lþ �ð Þ

(x < 0).

curves L<sup>s</sup>

touches some ray in it.

sphere, respectively [1].

2. Basic definitions and notation

under a condition such that st ! þ∞:

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

Oþ �ð Þ- curve of a system: = the system's O-curve 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>∑</sup>: <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Y</sup><sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>¼</sup> 1 with identified diametrically opposite points, and second, the orthogonal mapping of a lower enclosed semi-sphere of a sphere <sup>∑</sup> to a circle <sup>Ω</sup>: <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>≤</sup> 1 with identified diametrically opposite points of its boundary Г. We will now describe this process

The circle Ω and the sphere ∑ in this process are called the Poincare circle and the Poincare

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

<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

TO-curve of a system: = the system's O-curve, which, being supplemented by a point O,

A nodal bundle of NO-curves of a system := an open continuous family of the system's TO-

A saddle bundle of SO-curves of a system, a separatrix of the point O:= a fixed TO-curve,

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.,

P uð Þ <sup>≔</sup> <sup>X</sup>ð Þ� <sup>1</sup>; <sup>u</sup> <sup>p</sup><sup>0</sup> <sup>þ</sup> <sup>p</sup>1<sup>u</sup> <sup>þ</sup> <sup>p</sup>2u<sup>2</sup> <sup>þ</sup> <sup>p</sup>3u<sup>3</sup>

Q uð Þ <sup>≔</sup> <sup>Y</sup>ð Þ� <sup>1</sup>; <sup>u</sup> <sup>a</sup> <sup>þ</sup> bu <sup>þ</sup> cu<sup>2</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>

which is not included in some bundle of NO-curves of a system.

counterclockwise, starting with some of them.

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

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

x, <sup>y</sup> plane) on a sphere

67

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

http://dx.doi.org/10.5772/intechopen.75527

<sup>p</sup>(p 6¼ O, s∈ fþ; �g) adjoining to a point O

<sup>p</sup> ∩ ᴧ ¼ f gp :

,

:

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 differential equations.

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

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 of motion under small changes of parameters.

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 arithmetical plane R<sup>2</sup> x, <sup>y</sup> [1]. Inspired by the great Poincare's works, mathematicians of the next 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 particular kinds of systems.

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

$$\frac{d\mathbf{x}}{dt} = \mathbf{X}(\mathbf{x}, y),\\\frac{dy}{dt} = \mathbf{Y}\left(\mathbf{x}, y\right) \tag{1}$$

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 center (0, 0, 1) of a sphere ∑), augmented with a line at infinity (i.e., R<sup>2</sup> x, <sup>y</sup> plane) on a sphere <sup>∑</sup>: <sup>X</sup><sup>2</sup> <sup>þ</sup> <sup>Y</sup><sup>2</sup> <sup>þ</sup> <sup>Z</sup><sup>2</sup> <sup>¼</sup> 1 with identified diametrically opposite points, and second, the orthogonal mapping of a lower enclosed semi-sphere of a sphere <sup>∑</sup> to a circle <sup>Ω</sup>: <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>≤</sup> 1 with identified diametrically opposite points of its boundary Г. We will now describe this process in more detail.

The circle Ω and the sphere ∑ in this process are called the Poincare circle and the Poincare sphere, respectively [1].
