Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations

Nonlinear Dyn (2012) 69:577–587DOI 10.1007/s11071-011-0288-8

O R I G I NA L PA P E R

Dynamics at infinity and other global dynamical aspectsof Shimizu–Morioka equations

Marcelo Messias · Márcio R. Alves Gouveia ·Claudio Pessoa

Received: 31 August 2011 / Accepted: 7 November 2011 / Published online: 22 December 2011© Springer Science+Business Media B.V. 2011

Abstract We present some global dynamical aspectsof Shimizu–Morioka equations given by

x = y, y = x − λy − xz, z = −αz + x2,

where (x, y, z) ∈ R3 are the state variables and λ,α

are real parameters. This system is a simplified modelproposed for studying the dynamics of the well-knownLorenz system for large Rayleigh numbers. Using thePoincaré compactification of a polynomial vector fieldin R

3, we give a complete description of the dynamicsof Shimizu–Morioka equations at infinity. Then usinganalytical and numerical tools, we investigate for thecase α = 0 the existence of infinitely many singularlydegenerate heteroclinic cycles, each one consisting ofan invariant set formed by a line of equilibria togetherwith a heteroclinic orbit connecting two of these equi-libria. The dynamical consequences of the existence ofthese cycles are also investigated. The present study is

M. Messias (�) · M.R.A. GouveiaDepartamento de Matemática, Estatística e Computação,Faculdade de Ciências e Tecnologia, FCT, Univ EstadualPaulista—UNESP, Presidente Prudente, SP, Brazile-mail: [email protected]

M.R.A. Gouveiae-mail: [email protected]

C. PessoaDepartamento de Matemática, Instituto de Biociências,Letras e Ciências Exatas—IBILCE, Univ EstadualPaulista—UNESP, São José do Rio Preto, SP, Brazile-mail: [email protected]

part of an effort aiming to describe global properties ofquadratic three-dimensional vector fields with chaoticdynamical behavior, as made for instance in (Dias etal. in Nonlinear Anal. Real World Appl. 11(5):3491–3500, 2010; Kokubu and Roussarie in J. Dyn. Dif-fer. Equ. 16(2):513–557, 2004; Llibre and Messias inPhysica D 238(3):241–252, 2009; Llibre et al. in J.Phys. A, Math. Theor. 41:275210, 2008; Llibre et al.in Int. J. Bifurc. Chaos Appl. Sci. Eng. 20(10):3137–3155, 2010; Lorenz in J. Atmos. Sci. 20:130–141,1963; Lü et al. in Int. J. Bifurc. Chaos Appl. Sci. Eng.14(5):1507–1537, 2004; Mello et al. in Chaos Soli-tons Fractals 37:1244–1255, 2008; Messias in J. Phys.A, Math. Theor. 42:115101, 2009; Messias et al. inTEMA Tend. Mat. Apl. Comput. 9(2):275–285, 2008).

Keywords Shimizu–Morioka equations · Poincarécompactification · Dynamics at infinity · Singularlydegenerate heteroclinic cycles · Chaotic dynamics

1 Introduction

In this paper by using analytical and numerical tools,we perform a global dynamical analysis of the Shi-mizu–Morioka equations given by

x = y, y = x − λy − xz, z = −αz + x2, (1)

where (x, y, z) ∈ R3 are the state variables and λ,α

are real parameters. System (1) has been proposed in

mailto:[email protected]



578 M. Messias et al.

Fig. 1 (Color online) Lorenz-like strange attractor ofthe Shimizu–Morioka system (1) with λ = 0.75 andα = 0.45. Initial conditions: (0.01,−0.01,0), (−0.01,0.01,0);t ∈ [−2,200]; stepsize = 0.05

[15] as a simplified model for studying the dynamicsof the well-known Lorenz system [9]

x = σ(y − x), y = rx − y − xz,

z = −bz + xy,(2)

for large values of Rayleigh number, represented bythe parameter r in system (2). Later on, the systemgained interest by itself due to its rich dynamical be-havior and potential physical applications pointed out,for example, in [5, 14–16, 19, 20]. Indeed, several pa-pers have appeared in the literature concerning the dy-namical properties of system (1), mainly the chaoticbehavior of its solutions and strange attractor forma-tion. It was shown in [14] among other properties thatsystem (1) presents Lorenz-like strange attractors, e.g.,taking α = 0.45 and λ = 0.75 (see Fig. 1).

In this note, we study some global dynamical as-pects of system (1) solutions aiming to give a contri-bution to the understanding of its complex behavior.More precisely, we give a complete description of itsphase portrait at infinity via the Poincaré compactifi-cation. We also numerically show the existence of in-finitely many singularly degenerate heteroclinic cyclesof system (1) with α = 0 and explore some dynamicalconsequences of such an existence. The obtained re-sults make part of an effort aiming to describe globalproperties of quadratic three-dimensional systems offirst-order ordinary differential equations with chaoticdynamical behavior, recently studied by several au-thors; see, for instance, [4–8, 10–13].

Fig. 2 Phase portrait of system (1) on the Poincaré sphere atinfinity

1.1 Statement of the main results

As any polynomial vector field, the Shimizu–Moriokasystem (1) can be extended to an analytic system de-fined on a closed ball of radius one, whose interior isdiffeomorphic to R

3 and its invariant boundary, the 2-dimensional sphere S

2 = {(x, y, z) | x2 + y2 + z2 = 1}plays the role of the infinity. This ball shall be de-noted here by B and called the Poincaré ball, sincethe technique for doing this extension is the Poincarécompactification for a polynomial vector field in R

n,which is described in detail in [3, 18]. In Sect. 2 be-low, a summary of this technique for polynomial vec-tor fields in R

3 is given aiming to make the rest ofthis paper more easily readable. The boundary of B

is called the Poincaré sphere and represents the pointsof R

3 at infinity. By using this compactification tech-nique, we obtained the following result.

Theorem 1 For all values of the parameters α,λ ∈ R

the phase portrait of system (1) on the Poincaré sphereat infinity is as shown in Fig. 2: there is a circle of equi-libria at the endpoints of the yz-plane and two ellipticsectors contained in the portions of S

2 with x > 0 andx < 0.

In [5], Kokubu and Roussarie proved for a familyof three-dimensional systems of ordinary differentialequations that contains as its subfamily the Shimizu–Morioka system (1), the existence of one specific typeof heteroclinic cycle, which they called singularly de-generate heteroclinic cycle. It consists of an invariant

Global dynamical aspects of Shimizu-Morioka equations 579

Fig. 3 (Color online)(a) Singularly degenerateheteroclinic cycle of system(1) with α = 0, λ = 2;(b) its projection in thexz-plane. A saddle at thepoint (0,0,−100) isconnected to a focus at thepoint (0,0,700)

approximately

Fig. 4 (Color online)(a) Singularly degenerateheteroclinic cycles ofsystem (1) with α = 0,λ = 2; (b) their projectionsin the xz-plane. Each one ofthe saddles at Q� isconnected to one focus atP� forming infinitely manysingularly degenerateheteroclinic cycles

set formed by a line of equilibria together with a het-eroclinic orbit connecting two of these equilibria (seeFig. 3).

More precisely, by applying blowing-up techniquesthe authors in [5] prove analytically the existence ofonly one of these cycles, for which the line of equilib-ria is given by the z-axis, one of the points which formthe cycle is a saddle Q� at the origin (0,0,0) and theother one is a stable focus at the point P� = (0,0,G�),

for some large G� (in fact, G� located near infinity).Here, we show in part analytically and in part numer-ically that there exist in fact infinitely many of suchsingularly degenerate heteroclinic cycles for system(1) with α = 0 (see Fig. 4). Indeed, we have the fol-lowing results (the notation Q� and P� used above andbelow to denote the saddles and foci on the z-axis wasinspired by the notation used in [5]).

Theorem 2 The following statements hold for theShimizu–Morioka system (1).

(a) For α = 0 and all λ ∈ R with λ �= 0, the z-axis is filled with semi-hyperbolic equilibria. Con-sider λ > 0. Then for z < 1, the equilibria Q� =(0,0, z) are saddles normally hyperbolic to thez-axis, that is the Jacobian matrix of (1) at Q�

has two real eigenvalues with opposite signs andthe corresponding one-dimensional stable and un-stable manifolds are transversal to the z-axis; forz > (λ2 + 4)/4 the equilibria P� = (0,0, z) arestable foci normally hyperbolic to the z-axis, thatis the Jacobian matrix of (1) at P� has complexconjugate eigenvalues with negative real part andcorresponding two-dimensional stable manifoldstransversal to the z-axis. Now for 1 < z < (λ2 +


Fig. 5 Bifurcation diagramshowing the change in thestability index of the originof system (1) when α

crosses the value α = 0,illustrating the statementsof Theorem 2. Parametersign: α < 0 (left); α = 0(center); α > 0 (right)

4)/4 the equilibria on the z-axis are stable nodes.For λ < 0, the same analysis holds, but the fociand the node mentioned above become unstable.

(b) For α = 0 and z = (λ2 + 4)/4, the equilibria P =(0,0, z) is a normally hyperbolic improper node,which is stable if λ > 0 and unstable if λ < 0, thatis, the Jacobian matrix of (1) at P has a multiplic-ity two eigenvalue equal to −λ/2, whose corre-sponding two-dimensional stable (unstable) man-ifold are transversal to the z-axis.

(c) For α �= 0, system (1) has two infinite heteroclinicconnections, one of them consisting by the origin(0,0,0), the positive portion of the z-axis, andby one equilibrium on the sphere at infinity (theendpoint of the positive z-axis); the other consistsof the origin, the negative part of the z-axis, andof one equilibrium point on the sphere at infinity(the endpoint of the negative z-axis). Moreover, ifα > 0, the origin is asymptotically stable alongthe z-axis while for α < 0 the origin is unstablealong this axis (see Fig. 5).

Based on the results stated in Theorem 2 and on theanalytical results stated in [5], we performed a detailednumerical study of system (1) with α = 0 from whichwe state the following (numerical) result.

Numerical Result 3 For α = 0 and λ > 0 the one-dimensional unstable manifolds Wu(Q�) of each nor-mally hyperbolic saddle Q� of system (1) given inpart (a) of Theorem 2 tend asymptotically toward oneof the normally hyperbolic stable foci or nodes P�

as t → +∞, forming a singularly degenerate hetero-clinic cycle (see Figs. 3 and 4).

Combining the numerical result 3 with Theorem 2,we have that Shimizu–Morioka system with α = 0 hasinfinitely many singularly degenerate heteroclinic cy-cles, connecting the saddles Q� with the foci P� de-scribed in Theorem 2(a). Some of these cycles are il-lustrated in Fig. 4.

From the analytical and numerical results obtained,one can say that, beyond the existence of the singu-larly degenerate heteroclinic cycles themselves, thereis a mechanism which plays an important role in thecreation of strange attractors of system (1) with α > 0small. This mechanism is the change in the stabilityindex of the saddle at the origin when the parame-ter α crosses the zero value and the degenerate cy-cles are created. In fact, it is easy to check that, forα < 0, the origin is a saddle with a two-dimensionalunstable and a one-dimensional stable manifold (thatis, stability index equals to 1); in this case, the ori-gin is unstable along the invariant z-axis, and conse-quently the nonvanishing solutions on this axis escapeto infinity as t → +∞ (see Fig. 5 left). For α = 0,

the z-axis becomes a line of equilibria and the sin-gularly degenerate heteroclinic cycles are created (seeFig. 5 center). For α > 0, the origin is a saddle witha two-dimensional stable and a one-dimensional un-stable manifold (that is, stability index equals to 2); inthis case, the origin becomes stable along the invariantz-axis (see Fig. 5 right).

Furthermore, for α > 0, two new symmetric sta-ble equilibria given by (

√α, 0, 1) and (−√

α, 0, 1)

arise beyond the origin. The dynamical changes tak-ing place when the parameter α crosses the origin anddescribed above can be called a degenerate Pitchforkbifurcation, following [11].

By varying the parameter α near zero, we found forα > 0 small strange attractors near the singularly de-generate heteroclinic cycles described above. An ex-ample of this kind of attractor is shown in Fig. 6.

This mechanism leading to the creation of strangeattractors has already been described in [12] for theLorenz system (2), when the parameter b crossesb = 0. Moreover, the existence of infinitely many sin-gularly degenerate heteroclinic cycles and the oc-currence of strange attractors nearby them has beenpointed out for the Lorenz and Shimizu–Morioka sys-


Fig. 6 (Color online)(a) Strange attractor ofsystem (1) with α = 0.0261and λ = 0.88 (near thedegenerate heterocliniccycles existing for α = 0).(b) Its projection on theyz-plane. Note the spiralingbehavior of the solutionsaround the z-axis. Initialconditions:(0.0001,0.0001,0) and(−0.0001,−0.0001,0).Time interval: t ∈ [−6,950]

tems in [5] and recently described in [4, 7, 11–13] inthe study of other quadratic systems in R

3.In order to prove the results stated above, the paper

is organized as follows. In Sect. 2, we give a synthesisof the Poincaré compactification technique for a poly-nomial vector field in R

3, and use it to prove Theo-rem 1. In Sect. 3, we prove Theorem 2 by a local lin-ear stability analysis of the equilibria contained in thez-axis and present some numerical simulations, fromwhich we state the numerical result 3. In Sect. 4, wedo a numerical study of system (1) with α > 0 small,showing the existence of strange attractors near thesingularly degenerate heteroclinic cycles. Section 5brings some final remarks, ending the paper.

2 Poincaré compactification and proof ofTheorem 1

2.1 The Poincaré compactification in R3

A polynomial vector field X in Rn can be extended

to an unique analytic vector field on the sphere Sn.

The technique for making such an extension is calledthe Poincaré compactification and allows us to studya polynomial vector field in a neighborhood of in-finity, which corresponds to the equator S

n−1 of thesphere S

n. Poincaré introduced this compactificationfor polynomial vector fields in R

2 and its extension toR

n for n > 2 can be found in [3, 18]. In this section,we present a summary of the Poincaré compactifica-tion for polynomial vector fields in R

3, which will beused in the next subsections to prove Theorem 1.

Consider in R3 the polynomial differential system

x = P 1(x, y, z), y = P 2(x, y, z),

z = P 3(x, y, z),

or equivalently its associated polynomial vector fieldX = (P 1,P 2,P 3). The degree n of X is definedas n = max{deg(P i) : i = 1,2,3}. Let S

3 = {y =(y1, y2, y3, y4) ∈ R

4 : ‖y‖ = 1} be the unit sphere inR

4, and S+ = {y ∈ S3 : y4 > 0} and S− = {y ∈ S

3 :y4 < 0} be the northern and southern hemispheres ofS

3, respectively. The tangent space to S3 at the point

y is denoted by TyS3. Then the tangent plane

T(0,0,0,1)S3 = {

(x1, x2, x3,1) ∈ R4 : (x1, x2, x3) ∈ R

3}

can be identified with R3.

We consider the central projections f+ : R3 =

T(0,0,0,1)S3 −→ S+ and f− : R

3 = T(0,0,0,1)S3 −→ S−

defined by f±(x) = ±(x1, x2, x3,1)/�x, where �x =(1 + ∑3

i=1 x2i )1/2. Through these central projections

R3 is identified with the northern and southern hemi-

spheres. The equator of S3 is S

2 = {y ∈ S3 : y4 = 0}.

Clearly, S2 can be identified with the infinity of R

3.The maps f+ and f− define two copies of X on

S3, one Df+ ◦ X in the northern hemisphere and the

other Df− ◦ X in the southern one. Denote by X thevector field on S

3 \ S2 = S+ ∪ S−, which restricted

to S+ coincides with Df+ ◦ X and restricted to S−coincides with Df− ◦ X. Now we can extend analyt-ically the vector field X(y) to the whole sphere S

3

by p(X)(y) = yn−14 X(y). This extended vector field

p(X) is called the Poincaré compactification of X

on S3.

As S3 is a differentiable manifold in order to

compute the expression for p(X), we can consider


the eight local charts (Ui,Fi), (Vi,Gi), where Ui ={y ∈ S

3 : yi > 0} and Vi = {y ∈ S3 : yi < 0} for

i = 1,2,3,4; the diffeomorphisms Fi : Ui → R3 and

Gi : Vi → R3 for i = 1,2,3,4 are the inverses of

the central projections from the origin to the tangenthyperplanes at the points (±1,0,0,0), (0,±1,0,0),(0,0,±1,0) and (0,0,0,±1), respectively. Now wedo the computations on U1. Suppose that the ori-gin (0,0,0,0), the point (y1, y2, y3, y4) ∈ S

3 and thepoint (1, z1, z2, z3) in the tangent hyperplane to S

3

at (1,0,0,0) are collinear. Then we have 1/y1 =z1/y2 = z2/y3 = z3/y4, and consequently F1(y) =(y2/y1, y3/y1, y4/y1) = (z1, z2, z3) defines the coor-dinates on U1. As

DF1(y) =⎛

⎝−y2/y

21 1/y1 0 0

−y3/y21 0 1/y1 0

−y4/y21 0 0 1/y1

⎞

⎠

and yn−14 = (z3/�z)n−1, the analytical vector field

p(X) in the local chart U1 becomes

zn3

(�z)n−1

(−z1P1 + P 2,−z2P

1 + P 3,−z3P1),

where P i = P i(1/z3, z1/z3, z2/z3).In a similar way, we can deduce the expressions of

p(X) in U2 and U3. These are

zn3

(�z)n−1

(−z1P2 + P 1,−z2P

2 + P 3,−z3P2),

where P i = P i(z1/z3,1/z3, z2/z3) in U2, and

zn3

(�z)n−1

(−z1P3 + P 1,−z2P

3 + P 2,−z3P3),

where P i = P i(z1/z3, z2/z3,1/z3) in U3.The expression for p(X) in U4 is zn+1

3 (P 1,P 2,P 3).The expression for p(X) in the local chart Vi is thesame as in Ui multiplied by (−1)n−1, where n is thedegree of X.

When we work with the expression of the compact-ified vector field p(X) in the local charts, we usuallyomit the factor 1/(�z)n−1. We can do that through arescaling of the time variable.

In what follows, we shall work with the orthogo-nal projection of p(X) from the closed northern hemi-sphere to y4 = 0, we continue denoting this projectedvector field by p(X). Note that the projection of theclosed northern hemisphere is a closed ball B of ra-dius one, whose interior is diffeomorphic to R

3 andwhose boundary S

2 corresponds to the infinity of R3.

Of course p(X) is defined in the whole closed ballB in such a way that the flow on the boundary is in-variant. This new vector field on B will be called thePoincaré compactification of X, and B will be calledthe Poincaré ball.

Remark 1 All the points on the invariant sphere S2 at

infinity in the coordinates of any local chart Ui andVi have z3 = 0. Also, the points in the interior of thePoincaré ball, which is diffeomorphic to R

3, are givenin the local charts U1, U2, and U3 by z3 > 0 and inthe local charts V1, V2, and V3 by z3 < 0. These half-spaces will be considered in the study of the flow ofsystem (1) in a neighborhood of the infinite spherelater on.

2.2 Behavior of system (1) near and at infinity

In this section, we shall make an analysis of the flowof system (1) near and at infinity by analyzing thePoincaré compactification of the system in the localcharts Ui and Vi , i = 1,2,3.

Compactification in the local charts U1 and V1 Fromthe results of Sect. 2.1 the expression of the Poincarécompactification p(X) of system (1) in the local chartU1 is given by

z1 = −z2 + z3 − λz1z3 − z21z3,

z2 = 1 − z1z2z3 − αz2z3, (3)

z3 = −z1z23.

For z3 = 0 (which correspond to the points on thesphere S

2 at infinity) system (3) has no equilibriumpoint and the phase portrait is as shown in Fig. 7.

The flow in the local chart V1 is the same as the flowin the local chart U1 reversing the time, because thecompactified vector field p(X) in V1 coincides withthe vector field p(X) in U1 multiplied by −1 (for de-tails see Sect. 2.1).

Compactification in the local charts U2 and V2

Again using the results stated in Sect. 2.1, we obtainthe expression of the Poincaré compactification p(X)

of system (1) in the local chart U2 as

z1 = z3 + λz1z3 + z21z2 − z2

1z3,

z2 = z21 − αz2z3 + λz2z3 + z1z

22 − z1z2z3, (4)

z3 = z3(λz3 + z1z2 − z1z3).


Fig. 7 Phase portrait of system (1) on the Poincaré sphere atinfinity in the local chart U1

If z3 = 0, system (4) reduces to

z1 = z21z2, z2 = z2

1 + z1z22, (5)

which has a line of equilibria given by the z2-axis. Thelinear part of the system at these equilibria has nulleigenvalues. Considering the invariance of z1z2-planeunder the flow of (4), we can completely described thedynamics on the sphere at infinity, which is shown inFig. 8.

The flow in the local chart V2 is the same as the flowin the local chart U2, because the compactified vectorfield p(X) in V2 coincides with the vector field p(X)

in U2 multiplied by −1. Hence, the phase portrait onthe chart V2 is the same shown in Fig. 8, reversingappropriately the time direction.

Compactification in the local charts U3 and V3 Theexpression of the Poincaré compactification p(X) ofsystem (1) in the local chart U3 is

z1 = αz1z3 + z2z3 − z31,

z2 = −z1 + z1z3 + αz2z3 − λz2z3 − z21z2, (6)

z3 = −z3(z21 − αz3).

For z3 = 0, system (6) reduces to

z1 = z31, z2 = −z1 − z2

1z2, (7)

which also has a line of equilibria on z2-axis. Con-sidering the invariance of z1z2-plane we numerically



study the phase portrait of system (7), which corre-sponds to the phase portrait of system (1) at infinity onthe chart U3. It is shown in Fig. 9.

Again the flow at infinity in the local chart V3 isthe same as the flow in the local chart U3 reversingappropriately the time.


We now study system (6) in a neighborhood of theinfinite sphere on the chart U3, by considering z3 > 0small, since we are interested in the behavior of thesolutions which tend to infinity on the z-axis. Indeedit will be used in the proof of Theorem 2 in the nextsection.

The z3-axis is invariant under the flow of system(6), since for z1 = z2 = 0 the system reduces to

z1 = 0, z2 = 0, z3 = αz23,

hence the origin is asymptotically stable (resp. unsta-ble) along the z3-axis if α < 0 (resp. α > 0). Further-more, if α = 0, then the system has a line of equilibriain the z3-axis and the following proposition holds.

Proposition 1 The equilibrium points (0,0, z3),z3 > 0, of system (6), with α = 0 and λ > 0, are stablespirals normally hyperbolic to the z3-axis, that is, thelinear part of the system at each equilibria (0,0, z3)

has complex conjugate eigenvalues with negative realpart and corresponding 2-dimensional stable mani-folds transversal to the z3-axis, provided the followingcondition holds:

z3 <4

4 + λ2. (8)

Proof For α = 0, the Jacobian matrix of system (6) atthe equilibrium point (0,0, z3) is given by⎛

⎝0 z3 0

z3 − 1 −λz3 0

0 0 0

⎞

⎠ ,

which has the eigenvalues

λ1,2 = −λz3

2±

√λ2z2

3 + 4(z23 − z3)

2, λ3 = 0

with corresponding eigenvectors

v1,2 =(

1,−λz3 ±

√λ2z2

3 + 4(z23 − z3)

2z3,0

)

,

v3 = (0,0,1),

from which the proposition follows, since we are con-sidering λ > 0 and z3 > 0. �

As the flow in the local chart V3 is the same as theflow in the local chart U3 reversing the time, with thesame type of analysis and taking into account that nearthe infinity in the local chart V3 we have to considerz3 < 0 (see Remark 1 in Sect. 2.1) we can prove thefollowing proposition.

Fig. 10 Orientation of the local charts Ui, i = 1, . . . ,3 in thepositive endpoints of the coordinate axis x, y, and z, used todraw the phase portrait of system (1) on the Poincaré sphere atinfinity (Fig. 2). The charts Vi, i = 1, . . . ,3 are diametricallyopposed to Ui , in the negative endpoints of the coordinate axis

Proposition 2 The equilibrium points (0,0, z3),z3 < 0, of system (6) with α = 0 and reversed time(that is in the local chart V3) are saddles normallyhyperbolic to the z3-axis, that is the linear part ofthe system at each equilibrium (0,0, z3) has two realeigenvalues with opposite signs and corresponding 1-dimensional stable and unstable manifolds transversalto the invariant z3-axis.

2.3 Dynamics of system (1) on the sphere at infinity

Considering the analysis made in the previous subsec-tion and gluing the flow in the six studied charts, tak-ing into account its orientation, shown in Fig. 10, wehave a global picture of the dynamical behavior of sys-tem (1) on the sphere at infinity: the system has a circleof equilibria contained in the set U2 ∪V2 ∪U3 ∪V3 andtwo elliptic sectors contained in the portions of S

2 withx > 0 and x < 0 (see Fig. 2). This proves Theorem 1.Note that the dynamics at infinity does not depend onthe values of the parameters α and λ.

We observe that the complete description of thephase portrait of system (1) on the Poincaré sphere atinfinity was possible because of the invariance of thisset under the flow of the compactified system. Other-wise, the dynamics near the line of (nonhyperbolic)equilibria could be far more complicated.


3 Heteroclinic orbits and cycles and proof ofTheorem 2

3.1 Singularly degenerate heteroclinic cycles

For α = 0, the Shimizu–Morioka system (1) reducesto

x = y,

y = x − λy − xz,

z = x2,

(9)

which has the line of equilibria {(0,0, z), z ∈ R}. Re-member that we are considering λ > 0. The Jacobianmatrix of system (9) at the equilibrium point (0,0, z)

is given by⎛

⎝0 1 0

1 − z −λ 0

0 0 0

⎞

⎠ ,

which has the eigenvalues

λ1,2 = −λ

2±

√λ2 − 4(z − 1)

2, λ3 = 0

with corresponding eigenvectors

v1,2 =(

1,−λ

2±

√λ2 − 4(z − 1)

2,0

),

v3 = (0,0,1).

Then, for z > (λ2 + 4)/4, the eigenvalues λ1,2 arecomplex with negative real part. Considering alsothe corresponding eigenvectors, which span a planetransversal to the z-axis, we have that the solutions lo-cally spiral toward the equilibrium point Q� = (0,0, z)

on a surface tangent to the plane spanned by the eigen-vectors v1,2, hence in a direction transversal to the z-axis. If z = (λ2 + 4)/4, then the eigenvalues λ1,2 arereal and negative. Hence, trajectories move toward thez-axis without spiraling, since in this case the equilib-rium (0,0, z) is locally an improper node. For z < 1,the eigenvalues λ1,2 are real with opposite signs. Thentaking into account the eigenvectors v1,2, the systemhas a normally hyperbolic saddle at the point P � =(0,0, z). Now, for 1 < z < (λ2 + 4)/4 the eigenvaluesλ1,2 are real with the same signs and so, taking into ac-count the eigenvectors v1,2, the system has a normallyhyperbolic node at the point P � = (0,0, z). With thislinear analysis, the proof of statements (a) and (b) ofTheorem 2 follows.

We perform a detailed numerical study of the solu-tions of system (1) with α = 0, which clearly indicatethat in this case the system presents infinitely manysingularly degenerate heteroclinic cycles. Each one ofthese cycles is formed by one of the 1-dimensional un-stable manifolds of a saddle P �, which connects P �

with a normally hyperbolic focus Q�, as t → +∞. Asthe system presents an infinity of normally hyperbolicsaddles P � and foci Q�, there exist infinitely many sin-gularly degenerate heteroclinic cycles. In Figs. 3 and4 are shown some of them: for each initial conditionsufficiently close to the saddle P � at the z-axis, a sin-gularly degenerate heteroclinic cycle is obtained.

There is nothing special about the values of the pa-rameter λ considered to generate the singularly degen-erate heteroclinic cycles shown in Figs. 3 and 4. Othervalues produce the same type of cycles, provided wehave α = 0.

From the considerations above, the numerical re-sult 3 is computationally verified.

3.2 Infinite heteroclinic orbits

For α �= 0 small and λ > 0, system (1) has a saddlepoint at the origin, which is the unique equilibriumpoint on the z-axis. It is a straightforward calculationto show that this equilibrium point is a saddle with sta-bility index equals to 1 if α < 0. It is easy to check thatthe z-axis is invariant under the flow of system (1) andfrom the calculations above it follows that the originis unstable along this axis. Moreover, the equilibria atthe endpoints of the z-axis on the sphere at infinity,which coincide with the origin in the local charts U3

and V3, are asymptotically stable (see Sect. 2.2). Thus,system (1) has two infinite heteroclinic orbits, one ofthem consisting of the origin, the positive portion ofthe z-axis, and of one equilibrium on the sphere at in-finity (the endpoint of the positive z-axis); the otherone consists of the origin, the negative part of the z-axis and by the endpoint of the negative z-axis (seeFig. 5).

For α > 0, the stability index of the saddle is 2, theorigin is asymptotically stable along the z-axis whilethe endpoints of this axis on the sphere at infinity areunstable (see Sect. 2.2). Again the system has two in-finite heteroclinic orbits on the z-axis, but now the ori-gin is stable along this axis (see Fig. 5).

From these considerations we have proven state-ment (c) of Theorem 2.


Fig. 11 (Color online)Projections on the planesxy (a) and xz (b) of thesystem (1) strange attractorshown in Fig. 6

Fig. 12 (Color online) Coordinates x(t) (in (a)) and y(t) (in(b)) of the solutions of system (1) with λ = 0.88 and α = 0.0261with initial conditions (0.1,0.1,0.1) (red) and (0.1,0.1,0.11)

(blue). The time was taken from t = 100 to t = 400 and show

the exponential divergence of the solutions initiated in veryclose points in R

3, which is a typical behavior of chaotic dy-namics

4 Strange attractors near the singularlydegenerate heteroclinic cycles

We performed several numerical experiments show-ing that there exist strange attractors near the singu-larly degenerate heteroclinic cycles of system (1) withα = 0. Indeed, taking α nearby α = 0, we found solu-tions like the one shown in Fig. 6 of the Introduction.In this section, we present some more numerical sim-ulations of such a solutions, including the exponential

divergence of its coordinates x(t), y(t) and z(t), typi-cal in chaotic dynamics (see Figs. 11 and 12).

5 Concluding remarks

In this paper, we present some global dynamical as-pects of the Shimizu–Morioka equations, includingthe complete description of its phase portrait on thePoincaré sphere at infinity and the existence of in-finitely many singularly degenerate heteroclinic cy-cles. Through the global analysis presented here, we


intend to give a contribution to the understand ofthe complex dynamical behavior of Shimizu–Moriokasystem, whose local properties of singularities andtopological aspects of strange attractor has been stud-ied in several papers. Although applied to a specificsystem, the global analysis presented here may beuseful in the study of other polynomial vector fieldsin R

3. Indeed, it has yet been applied in the globalstudy of the standard Lorenz system [8, 12] and otherthree-dimensional quadratic chaotic systems [6, 7].It could be interesting to extend the techniques pre-sented here to study global properties of the gener-alized Lorenz canonical form presented in [1, 2, 17],since it is a quadratic polynomial system defined in R

3

which holds other chaotic systems (Lorenz, Shimizu–Morioka, Lü, Chen) as its subsystems.

Acknowledgements The first author is supported by CNPq-Brazil—Project 305204/2009-2 and by CAPES/MECD–TQEDII. The second and third authors are supported by ProgramaPrimeiros Projetos—PROPe/UNESP.

References

1. Celikovsky, S., Chen, G.: On a generalized Lorenz canoni-cal form of chaotic systems. Int. J. Bifurc. Chaos Appl. Sci.Eng. 12, 1789–1812 (2002)

2. Celikovsky, S., Chen, G.: On the generalized Lorenz canon-ical form. Chaos Solitons Fractals 26, 1271–1276 (2005)

3. Cima, A., Llibre, J.: Bounded polynomial vector fields.Trans. Am. Math. Soc. 318, 557–579 (1990)

4. Dias, F.S., Mello, L.F., Zhang, J.-G.: Nonlinear analysis ina Lorenz-like system. Nonlinear Anal., Real World Appl.11(5), 3491–3500 (2010)

5. Kokubu, H., Roussarie, R.: Existence of a singularly de-generate heteroclinic cycle in the Lorenz system and its dy-namical consequences: Part I. J. Dyn. Differ. Equ. 16(2),513–557 (2004)

6. Llibre, J., Messias, M.: Global dynamics of the Rikitakesystem. Physica D 238(3), 241–252 (2009)

7. Llibre, J., Messias, M., da Silva, P.R.: On the global dynam-ics of the Rabinovich system. J. Phys. A, Math. Theor. 41,275210 (2008)

8. Llibre, J., Messias, M., da Silva, P.R.: Global dynamics ofthe Lorenz system with invariant algebraic surfaces. Int. J.Bifurc. Chaos Appl. Sci. Eng. 20(10), 3137–3155 (2010)

9. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos.Sci. 20, 130–141 (1963)

10. Lü, J., Chen, G., Cheng, D.: A new chaotic system andbeyond: the generalized Lorenz-like system. Int. J. Bifurc.Chaos Appl. Sci. Eng. 14(5), 1507–1537 (2004)

11. Mello, L.F., Messias, M., Braga, D.C.: Bifurcation analysisof a new Lorenz-like chaotic system. Chaos Solitons Frac-tals 37, 1244–1255 (2008)

12. Messias, M.: Dynamics at infinity and the existence of sin-gularly degenerate heteroclinic cycles in the Lorenz system.J. Phys. A, Math. Theor. 42, 115101 (2009)

13. Messias, M., Nespoli, C., Dalbelo, T.M.: Mechanics forthe creation of strange attractors in Rössler’s second sys-tem. TEMA Tend. Mat. Apl. Comput. 9(2), 275–285 (2008)(Portuguese)

14. Shilnikov, A.L.: On bifurcations of the Lorenz attractorin the Shimizu-Morioka model. Physica D 62, 338–346(1993)

15. Shimizu, T., Morioka, N.: On the bifurcation of a symmet-ric limit cycle to an asymmetric one in a simple model.Phys. Lett. A 76(3,4), 201–204 (1980)

16. Tigan, G., Turaev, D.: Analytical search for homoclinicbifurcations in the Shimizu–Morioka model. Physica D240(12), 985–989 (2011)

17. Van Gorder, R.A.: Emergence of chaotic regimes in thegeneralized Lorenz canonical form: a competitive modesanalysis. Nonlinear Dyn. 66, 153–160 (2011)

18. Velasco, E.A.G.: Generic properties of polynomial vec-tor fields at infinity. Trans. Am. Math. Soc. 143, 201–221(1969)

19. Vladimirov, A.G., Volkov, D.Y.: Low-intensity chaotic op-erations of a laser with a saturable absorber. Opt. Commun.100, 351–360 (1993)

20. Yu, S., Tang, W.K.S., Lü, J., Chen, G.: Generationof n × m-wing Lorenz-like attractors from a modifiedShimizu–Morioka model. IEEE Trans. Circuits Syst.55(11), 1168–1172 (2008)

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Dynamics at infinity and other global dynamical aspects of Shimizu–Morioka equations