Open access peer-reviewed chapter

Phase Portraits of Cubic Dynamic Systems in a Poincare Circle

By Irina Andreeva and Alexey Andreev

Submitted: October 24th 2017Reviewed: February 16th 2018Published: May 23rd 2018

DOI: 10.5772/intechopen.75527

Downloaded: 421

Abstract

In the proposed chapter, we are going to outline the results of a study on an arithmetical plane of a broad family of dynamic systems having polynomial right parts. Let these polynomials be of cubic and square reciprocal forms. The task of our investigation is to find out all the different (in the topological sense) phase portraits in a Poincare circle and indicate the coefficient criteria of their appearance. To achieve this goal, we use the Poincare method of central and orthogonal consecutive displays (or mappings). As a result of this thorough investigation, we have constructed more than 250 topologically different phase portraits in total. Every portrait we present using a special table called a descriptive phase portrait. Each line of such a special table corresponds to one invariant cell of the phase portrait and describes its boundaries, as well as a source of its phase flow and a sink of it.

Keywords

  • dynamic systems
  • phase portraits
  • phase flows
  • Poincare sphere
  • Poincare circle
  • singular points
  • separatrices
  • trajectories

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 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.

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¯x,y2[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.

dxdt=Xxy,dydt=YxyE1

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¯x,y2plane) on a sphere ∑:X2+Y2+Z2=1with identified diametrically opposite points, and second, the orthogonal mapping of a lower enclosed semi-sphere of a sphere ∑ to a circle Ω¯: x2+y21with 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].

2. Basic definitions and notation

φtp,p=xya fixed point: = a solution (a motion) of Eq. (1) – system with initial data 0p.

Lp:φ=φtp,tImax, − a trajectory of motion φtp.

Lp+:= + (−) – a semi-trajectory of a trajectory Lp.

O-curve of a system := the system’s semi-trajectory Lsp(p ≠ O, s+) adjoining to a point O under a condition such that st+.

O+- curve of a system: = the system’s O-curve Lp+.

O+-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 TO-curves Lsp, where s+is a fixed index, pᴧ, ᴧ a simple open arc, Lsp=p.

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.

PuX1up0+p1u+p2u2+p3u3,
QuY1ua+bu+cu2.

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.

3. Topological type (T-type) of a singular point O(0, 0)

In order to find all O-curves and to split their totality into the bundles N, S, let us use the method of exceptional directions of a system in the point O [1]. According to this method, the equation of exceptional directions for the point O of the Eq. (1) system has the form.

xYxyxax2+bxy+cy2=0.

For this, the following cases are possible:

  1. When db24ac>0,this equation defines simple straight lines x=0and.

    y=qix,i=1,2,q1<q2

  • When d=0, this equation defines the straight line x=0and the double straight line.

    y=qx,q=b2c

  • When d<0,the equation defines only the straight line x=0.

  • is true for the aforementioned cases [5].

    Words AO and BO, which define a topological type (T-type) of a singular point O(0, 0) of the Eq. (1) system:

    1. in the case of d>0, depending on signs of values P(qi)=1,2,have forms, indicated in a Table 1;

    2. in the case of d=0depending on signs of values qand P(q), they have forms, indicated in a Table 2,

    3. in the case of d<0they have forms: AO = S0S0, BO = HH (Table 1).

    Let us clarify the meaning of the new symbols introduced in Theorem 1.

    rP (q1)P (q2)AOBO
    1, 4++S0S+1N+2S0N1S2=S0S+1NS2PH2
    2__S0N+1S+2S0S1N2=NS+2S0S+1PH2
    3, 6_+S0N+1N+2S0S1S2PEPH3
    5+_S0S+1S+2S0N1N2H3PEP

    Table 1.

    Т-type of a singular point when d > 0 (r =1,6¯).

    S0 (S0) means a bundle S, adjoining to point O(0,0) from the domain x>0along a semi-axis x=0,y<0, when t +(along a semi-axis x=0,y>0,when t).

    1. The lower sign index “ + ” or “–” on every bundle N or S, different from S0 and S0, indicates whether the bundle consists of O+-curves or of O-curves. Upper index 1 or 2 on every such a bundle indicates whether its O-curves are adjoining to point O along a straight line y=q1xor along a straight line y=q2x.

    2. In Table 2, row 5, 6, a bundle N does not have a lower sign index because it contains both O+-curves and O-curves simultaneously.

    From Theorem 1, it follows, that Eq. (1) systems do not have limit cycles on the R2x;y plane.

    qP (q)AOBO
    ++S0S+N+S0H2P
    __S0N+S+S0PH2
    +_S0S0SNH2P
    _+S0S0NSPH2
    0+S0S+NSH2P
    0_NS+S0SPH2

    Table 2.

    T-type of the singular point O(0, 0) when d = 0.

    Indeed, such a cycle could surround a singular point O (0,0) of an Eq. (1) system, and then the Poincare index of this singular point must be equal to 1 [1]. However, Bendixon’s formula for the index of an isolated singular point of a smooth dynamic system is as follows:

    IO=1+eh2

    where ehis the number of elliptical (hyperbolic) O-sectors of the system. This formula and our Theorem 1 give: for the singular point O (0, 0) of every Eq. (1) system, Poincare index I(O) = 0.

    For the singular point O (0, 0) of an Eq. (1) system, 11 different topological types (T-types) are possible, and from the analysis of these 11 T-types we can conclude:

    for every Eq. (1) system, the singular point O(0, 0) has not more than four separatrices (actually 2, 3, or 4 ones).

    4. Infinitely remote singular points (IR points)

    Now it is time to discuss the behavior of trajectories of the Eq. (1) systems in a neighborhood of infinity. For the investigation of this question we use the method of Poincare consecutive transformations, or mappings [1].

    The first Poincare transformation

    x=1z,y=uz(u=yx,z=1x).

    unambiguously maps a phase plane R2x,y of the Eq. (1) system onto a Poincare sphere ∑: x2+y2+z2=1(where z=Z1) with the diametrically opposite points identified, which is considered without its equator E, and an infinitely remote straight line of a plane Rx,y2¯. The first Poincare transformation maps onto the equator E of the sphere ∑; the diametrically opposite points are also considered to be identified.

    The Eq. (1) system in this mapping transforms into a system, which in the Poincare coordinates u,zafter a time change dt=z2looks likethe following:

    du=PuuQuz,dz=Puz,

    where Pu:X1uandQu:Y1uare reciprocal polynomials.

    This new system is determined on the whole sphere ∑, including its equator, and on the whole uz− plane α, which is tangent to a sphere ∑ at point C = (1, 0, 0). We shall study this system, namely on a plane Ru,z2¯, and project the received results onto a closed circle Ω¯,sequentially mapping, first, a plane R2u,z onto the sphere ∑ from its center, and second, its lower semi-sphere H¯onto the Poincare circle Ω¯,i. e., onto a closed unit circle of a plane R2x,y through the orthogonal mapping.

    For our new system, the axis z=0is invariant (consists of this system’s trajectories). On this axis, lie its singular points Oiui0,i=0,m,¯whereui,i=1,m¯are all real roots of the polynomial Pu,and u0=0; at the same time, there may exist i01m: ui0= 0. Let us call such points IR points of the first kind for the Eq. (1) system.

    The second Poincare transformation

    x=vz,y=1zv=xyz=1y

    also unambiguously maps a phase plane R2x,y onto a Poincare sphere ∑ with the diametrically opposite points identified, considered without its equator. Every Eq. (1) system transforms into a system, which in the coordinates τ,v,zlooks like the following:

    dv=Xv1+Yv1vz,dz=Yv1z2.

    This last system is determined on the whole sphere ∑, and on the whole vzplaneα̂,which is tangent toasphereatpointD=010[1]. A set z=0is invariant for this last system. On this set, lie its singular points v00,where v0is any real root of the polynomial Xv1p3+p2v+p1v2+p0v3.It would be natural to call such points IR points of the second kind for Eq. (1) systems, but each of these points, for which v00,obviously coincides with one of the IR-points of the first kind, namely with the point 1v00,

    while v0=0is not a root of the polynomial X(x, 1), because X(0, 1) = p30 for the Eq. (1) system. Consequently, the following corollary is correct.

    The infinitely remote singular points of any Eq. (1) system are only IR-points of the first kind.

    With the orthogonal projection of a closed lower semi-sphere H¯of a Poincare sphere ∑ onto a plane x, y, its open part H one-to-one maps onto an open Poincare circle Ω,while its boundary E (an equator of the Poincare sphere ∑) maps onto the boundary of the Poincare circle Γ=∂Ω, which implies the following. 1) Trajectories of any Eq.(– (including its singular point O (0, 0)) are displayed in a circle Ω, filling it.

    2) Such a system’s infinitely remote trajectories (including IR points) are displayed on the boundary Γ of a circle Ω, filling it.

    Following Poincare, we call the first trajectories of the Eq. (1) system in Ω, and the second, we call trajectories of the Eq. (1) system on Γ.

    As it follows from the aforementioned conclusions, to each IR point Oiui0, of the Eq. (1) system,i1m,correspond two diametrically opposite points situated on the Γ circle.

    Oi±ui0:Oi+OiΓ+Γx>0x<0.

    i1mfor the pointOi+Oi,we shall introduce the following notation.

    1. Let a Oi+Oicurve be a semi-trajectory of the Eq. (1) system in Ω, starting in an ordinary point pΩand adjacent to a point Oi+.

    2. A notation for bundles N, S, adjacent to a point Oi+Oifrom the circle Ω, similar to the notation introduced for the point O (0, 0).

    3. A notation of a word Ai+(Ai)consisting of letters N,S, which fixes an order of bundles of Oi+Oi-curves at a semi-circumvention of the point Oi+Oiin the circle Ω in the direction of increasing u.

    We shall describe a T-type of a point Oi+Oiwith a word Ai+(Ai),and a T-type of a point Oiwith wordsAi±.

    T-types of IR points O0±00of Eq. (1) systems are described in the following theorem.

    Let a number u=0be the multiplicity k03of the root of a polynomial Puof the Eq. (1) system. Then, words A0±,which determine the topological types (T-types) of IR points O0±00of this system, depending on the value of k and a sign of a number apk (where a and pk are coefficients of the system), have the forms as shown in Table 3 [5].

    IR points O0±of any Eq.1systemdo not have separatrices.

    kapkA0+A0
    00NN
    0, 2+ (−)N+NNN+
    1, 3+ (−)NN+NN+

    Table 3.

    T-types of IR points O0±00.

    T-types of IR points Oiui0Oo00,i=1,m¯,of Eq. (1) systems are described in the following theorem.

    Let a real number ui(0) be a multiplicity ki123of the root of a polynomial Puof an Eq. (1) system. Then for this system, a value gi = P (ki)(ui)Q(ui) 0 and words Ai±,which determine topological types (T-types) of IR points Oi±ui0of this system, depending on the value of ki and signs of numbers ui and gi, have forms as shown in Table 4 [5].

    As can be seen from Theorems 2 and 3, for the IR points of Eq. (1) systems, only a finite number (13) of different T-types are possible. The investigation of these T-types shows that IR-points of each Eq. (1) system have only m separatrices: one separatrice for every singular point Oiui0,i=1,m¯.

    1. In Tables 3 and 4, the lower sign index “ + ” or “–” on every bundle N or S, indicates whether the bundle adjusts to the point Oi+or to the pointOifrom the side u>uior from the side u<uiof the isocline u=ui.

    2. In Table 3, row 1, a bundle N does not have a lower sign index because as the detailed study of this case shows, it contains Oi+-curves (Oi-curves) in every domain u> 0 [5].

    uikigiAi+Ai
    +(−)1, 3+N+NSS+
    +(−)1, 3_SS+N+N
    +(−)2+SN+NS+
    +(−)2_NS+SN+

    Table 4.

    T-types of IR points Oi±ui0,i1m.

    5. Systems containing 3 and 2 multipliers in their right parts

    In this section, we present a solution to the main assigned problem for those Eq. (1) systems whose decompositions of forms X (x, y), Y (x, y) into real forms of lower degrees contain 3 and 2 multipliers, respectively:

    Xxy=p3yu1xyu2xyu3x,Yxy=cyq1xyq2xE2

    where p3>0,c>0,u1<u2<u3, q1<q2,uiqjfor each i and j.

    The solution process contains the follows steps.

    5.1. Basic concepts and notation

    The following notations are introduced for the arbitrary system under consideration in the Section 5.

    P(u), Q(u) – the system’s polynomials P, Q:

    P(u)=X1up3uu1)uu2(uu3,Qu=Y1ucuq1uq2

    RSP (RSQ) – an ascending sequence of all real roots of then system’s polynomial P(u) (Q(u)),RSPQ – an ascending sequence of all real roots of both the system’s polynomials P(u), Q(u).

    5.2. The double change (DC) transformation

    Let us call a double change of variables in this dynamic system: (t, y) → (−t, −y). The double change transformation transforms the system under consideration into another such system, for which numberings and signs of roots of polynomials P(u), Q(u), as well as the direction of motion upon trajectories with the increasing of t are reversed. Let us agree to call a pair of different Eq. (2) systems mutually inversed in relation to the DC transformation, if this transformation appears to convert one into another, and call them independent of a DC transformation in the opposite case.

    Clearly, 10 different types of RSPQ are possible for an arbitrary Eq. (2) system, as C52=5!3!2!= 10.

    As we can conclude using the DC transformation of Eq. (2) systems, six of the RSPQs appear to be independent in pairs. Similarly, each of the remaining four systems has the mutually inversed one among the first six Eq. (2)-systems.

    Let us assign a specific number r110to each one of the different RSPQs of the Eq. (2) system in such a manner that RSPQr = 1,6¯are independent in pairs, while RSPQsequences with numbers r = 7,10¯are mutually inversed to RSPQ`s which have numbers r = 1,4.¯

    It is time to introduce the important notion of a family number r of Eq. (2) systems.

    An r family of Eq. (2) systems =the totality of systems (belonging to Eq. (2) family) having the RSPQ number r.

    Now following a single plan, we consistently investigate the families of Eq. (2)systems that have numbers r = 1,6.¯For families having numbers r = 7,10,¯we obtain data through the DC-transformation of families, r = 1,4¯.

    A plan of the investigation of each selected Eq. (2) family contains the follows items.

    1. We determine a list of singular points of systems of the fixed family in a Poincare circle Ω¯.They appear to be a point O (0, 0)Ω and points Oi±(ui, 0) Г, i =0,3¯, u0= 0. For every point in the list, we use the notions of a saddle (S) and node (N) bundles adjacent to this point’s semi-trajectories, of a separatrix of the singular point, and of a topodynamical type of the singular point (TD type).

    2. Further, we split the family under consideration to subfamilies with numbers s = 1,7.¯For every subfamily, we reveal topodynamical types of singular points and separatrices of them.

    3. We investigate the separatrices’ behavior for all singular points of systems belonging to the chosen subfamily s{1, …, 7}. Very important are the following questions: a question of a uniqueness of a continuation of every given separatrix from a small neighborhood of a singular point to all the lengths of this separatrix, as well as a question about a mutual arrangement of all separatrices in a Poincare circle Ω. We answer these questions for all families of systems under consideration.

    4. As a result of all previous studies, we depict phase portraits of dynamic systems of a given family and outline the criteria of every portrait appearance [5, 6].

    From this section, we can conclude the following:

    Systems of the family number r = 1 have 25 different types of phase portraits.

    Systems of families number 2 and 3: there are 9 types of phase portraits per family.

    Systems of families 4 and 5: there exist 7 types of phase portraits per family.

    Systems belonging to the family number r = 6 show 36 different types of phase portraits.

    Hence, we have obtained 93 different types in total for the systems described in this section—a lot of possible types at first glance. However, it is important to keep this in mind: every given family includes an uncountable number of differential systems.

    6. Two classes of systems containing various combinations of two different multipliers in both right parts: an A-class

    In Sections 6 and 7, the problem has been solved for an Eq. (3) family. The Eq. (3) family of Eq. (1) systems is as follows—the family consists of a totality of all Eq. (1) systems; for each of them, decompositions of forms X (x, y), Y (x, y) into real multipliers of the lowest degrees contain two multipliers each:

    Xxy=pyu1xk1yu2xk2,Yxy=qyq1xyq2xE3

    where p, q, u1,u2,q1,q2R,p>0,q>0,u1<u2, q1<q2,uiqjfor each i,j 12,k1,k2N, k1+k2=3.

    It is natural to distinguish two classes of Eq. (3) systems. The A class contains systems with k1=1,k2=2;and the B class contains systems with k1=2,k2=1.

    In this section, we give a full solution of the assigned task for systems belonging to the A class of the Eq. (3) family, i.e.,

    dxdt=pyu1xyu2x2,dydt=qyq1xyq2xE4

    The process of forming the solution contains steps similar to the ones described in Section 4 of this chapter.

    For an arbitrary Eq.4– system, we introduce the following concepts.

    Let P(u), Q(u) be the system’s polynomials P, Q:

    Pu=X1upuu1uu22,Qu=Y1uquq1uq2,

    and RSP (RSQ) be an ascending sequence of all the real roots of the system’s polynomial, while P(u) (Q(u)),RSPQ is an ascending sequence of all the real roots of both system’s polynomials P(u) and Q(u). There exist 6 different possible variants of RSPQ as C42=4!2!2!= 6. Let us number them from 1 to 6 in some order.

    Now let us put into use an important notion:

    An r-family of Eq.4– systems is the totality of Eq. 4dynamic systems with the RSPQ number r from the list of six allowable variants.

    A consistent research of families of Eq.4dynamic systems.

    The steps of research of every fixed family belonging to Eq. 4dynamic systems are as follows.

    1. For all singular points of a given dynamic system that belongs to the family under consideration, let us introduce notions of S (saddle) and N (node) bundles of semi-trajectories, which are adjacent to a chosen singular point; also let us introduce a notion for its separatrix and a notion for its topodynamical type (TD-type).

    2. Now the considered family must be divided into subfamilies numbered s15.Then it is necessary to determine the TD-types of singular points of systems belonging to the obtained subfamilies, and separatrices of singular points s=1,5¯.

    3. For all five subfamilies, we investigate the separatrices’ of singular points behavior and find an answer to a question concerning a uniqueness of a global continuation of every chosen separatrix from a tiny neighborhood of a singular point to all the lengths of this separatrix in the Poincare circle Ω, as well as an answer to a question of all separatrices’ mutual arrangement in Ω.

    The mutual arrangement of all separatrices in the Poincare circle is invariable when, for a given s, a global continuation of every separatrix of each singular point of the subfamily number s is unique. Consequently, all systems of a chosen subfamily number s have, in a Poincare circle, one common type of phase portrait.

    But in a different situation, when, for a fixed number s, systems of such subfamily have, for example, m separatrices with global continuations that are not unique, this subfamily is divided into m additional subfamilies (so as to say subsubfamilies) of the next order.

    As we could understand conducting their further study, for each of subsubfamilies, the global continuation of every separatrix is unique, and the mutual arrangement of separatrices in the Poincare circle Ω is invariable.

    As a result, the topological type of phase portrait of all systems belonging to this subsubfamily in the Ω¯circle is common for the chosen subsubfamily.

    1. 4. We depict phase portraits in Ω¯for the systems of Eq.4families, r = 1,6¯,in the two possible forms (the table and the graphic ones), and indicate for each portrait close to coefficient criteria of its realization.

    A conclusion for the Section 6 of our chapter is:

    1. Eq. 4–systems belonging to the number 1 family have in the Poincare circle Ω¯, 13 different topological types of phase portraits.

    2. Eq.4– systems of the family number 2 have 7 types.

    3. Family number 3 have 10 types.

    4. Family numbers 4, 5, and 6 have 5 different types of phase portraits per number.

    This means that in total, all large families of Eq.4dynamic systems of the A class may have 45 different topological types of phase portraits in a Poincare circle.

    7. Systems with 2 different multipliers in both right parts, belonging to a B class

    In this section, the full solution of our task for Eq. (3) systems of the B class is given:

    dxdt=pyu1x2yu2x,dydt=qyq1xyq2x.E5

    For an arbitrary Eq.5– system, P(u), Q(u) are the system’s polynomials P, Q.

    Pu=X1upuu12uu2,Qu=Y1uquq1uq2,

    RSPQ shows 6 different variants, because C42=6.

    We can thus conclude that all Eq. 5family of systems is split into 52 different subfamilies, and all systems of each chosen subfamily show in a circle Ω¯,one common type of a phase portrait belonging to this particular subfamily. We have constructed all 52 topologically different phase portraits.

    8. Systems containing 3 and 1 different multipliers in right parts

    In this section, we solve the problem for an Eq. (6) family, i.e., for a family of Eq. (1) systems

    dxdt=p3yu1xyu2xyu3x,dydt=cyq1x2E6
    p3>0,c>0,u1<u2<u3,qRui,i=1,3¯.

    The solution process includes the follows steps. Let us break the Eq. (6) family into subfamilies numbered r = 1,4.¯

    Each of these is a totality of systems with an RSPQ number r, where r is the system’s number in the list of possible RSPQs.

    1. u1,u2,u3,q,

    2. u1,u2,q, u3,

    3. u1,q,u2,u3,

    4. q,u1,u2,u3.

    Applying to the Eq. (6) system, a double change of variables (DC): (t, y) → (−t,-y), we reveal that it transforms families of these systems having the numbers r = 1, 2, 3, 4, into their families with numbers r = 4, 3, 2, 1 respectively, and backward. We emphasize: this fact means that families of Eq. (6) systems having numbers 1 and 2 are not connected with the DC transformation, and that families having numbers 3 and 4 are not related to each other; at the same time, family number 3 is mutually inversed by the DC transformation to the family number 2, and family number 4 is mutually inversed to the family number 1 correspondingly. This conclusion follows from the consideration of their RSPQ sequences [5, 6].

    1. We study alternately the families of systems, r = 1,2, following the common program of Eq. (1) systems study [5], i.e.:

    1. We fix r 12,then webreak the chosen family into subfamilies numbered s [5, 6], s = 1,9,¯and find the topodynamical types (TD-types) of singular points of these systems.

    2. We construct for the systems of a fixed subfamily s=1,9,¯the so-called off-road map (ORM) [5, 6, 7]. The ORM helps us to find anαωlimit set of every αωseparatrix. It also lets us describe the mutual arrangement of all separatrices in the Poincare circle Ω.

    3. We depict all possible topologically different phase portraits for Eq. (6) systems.

    1. 2. We investigate consistently families of Eq. (6) systems, r = 3, 4, using the DC transformation of the results obtained for families, r = 2, 1. Then, we depict all types of existing phase portraits for the families 3 and 4.

    Then, we conclude the following.

    For families of Eq. (6) systems with numbers 1, 2, 3, and 4, there exist

    15 + 11 + 11 + 15 = 52

    different topological types of phase portraits in a Poincare circle Ω¯.

    9. Systems containing 2 and 1 different multipliers in right parts

    In this section, we give the full solution of the problem for Eq. (7) systems, i.e., for the Eq. (1) systems of the kind

    ẋ=p0x3+p1x2y+p2xy2+p3y3p3yu1x2yu2xE7
    ẏ=x2+bxy+cy2cyqx2,

    where p3>0,c>0,u1<u2, qRu1,2.

    The process of study of these systems is quite similar to that previously described for other families of Eq. (1) systems. For an arbitrary Eq. (7) system, P(u), Q(u) are the system’s polynomials P, Q:

    Pu=X1up3uu12uu2,Qu=Y1ucuq2,

    and there exists 3 different variants for their RSPQs.

    A conclusion from our research for this particular type of systems is the following.

    We’ve revealed, that for every possible family of Eq. (7) systems, 7 different topological types of their phase portraits are being implemented. This means that for all three existing families of such systems, r = 1,3¯,the number of different phase portraits is 21 [8, 9].

    10. Conclusions

    The presented work is devoted to the original study.

    The main task of the work was to depict and describe all the different, in the topological meaning, phase portraits in a Poincare circle, possible for the dynamical differential systems belonging to a broad family of Eq. (1) systems, and to its numerical subfamilies. The authors have constructed all such phase portraits in two ways—in a descriptive (table) and in a graphic form. Each table contains 5–6 rows. Every row describes one invariant cell of the phase portrait in detail—it describes its boundary, source, and sink of its phase flow. The table was the descriptive phase portrait.

    The second objective of this work was to develop, outline, and successfully apply some new effective methods of investigation [8, 9, 10].

    This was a theoretical work, but due to aforementioned new methods, the chapter may be useful for applied studies of dynamic systems of the second order with polynomial right parts. The authors hope that this work may be interesting and useful for researchers and for both students and postgraduates.

    © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Irina Andreeva and Alexey Andreev (May 23rd 2018). Phase Portraits of Cubic Dynamic Systems in a Poincare Circle, Differential Equations - Theory and Current Research, Terry E. Moschandreou, IntechOpen, DOI: 10.5772/intechopen.75527. Available from:

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