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Localization of periodic orbits of polynomial vector fields of even degree by linear functions
Chaos, Solitons and Fractals 25 (2005) 621–627 www.elsevier.com/locate/chaos
Localization of periodic orbits of polynomial vector fields of even degree by linear functions Konstantin E. Starkov
*
CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC, Mexico Accepted 23 November 2004
Abstract This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered. 2005 Published by Elsevier Ltd.
1. Introduction The problem of locating periodic orbits of polynomial multi-dimensional continuous-time systems has drawn much attention in the last decade, see papers [1,2,6–8,11,12,14–18] and others. In [18] it was proposed to use quadratic polynomials for getting ellipsoidal localization of periodic orbits of systems with cubic right sides and quadratic right sides satisfying one special ‘‘first integral’’ condition. Namely, if we have a system x_ ¼ l þ Ax þ bðxÞ, with b be a vector-polynomial of degree 2, and we apply a function h(x) = Cx + x(x), with x be a quadratic form, x 5 0, then we impose the condition Lbx = 0. Obviously, this condition is very restrictive. In our paper we apply linear functions for the solution of the localization problem of periodic orbits of polynomial right-side systems of even degree. Though in this case we cannot obtain the ellipsoidal localization, in many cases we get a localization of periodic orbits in unbounded sets of a simple structure like as semispaces, their finite intersections, etc. Our methods are based on the solution of the conditional extremum problem introduced in [7] and applying homogeneous polynomial forms of even degree. As examples, we consider the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator. 2. Some notations and useful results Let us introduce a system x_ ¼ F ðxÞ; *
Address: CITEDI-IPN, 2498 Roll Drive #757, San Diego, 92154 CA, USA. Tel.: +52 66231344; fax: +52 66231388. E-mail addresses: [email protected], [email protected]
0960-0779/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2004.11.052
ð1Þ
622
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
where F(x) = l + Ax + f(x) is a polynomial vector field of even degree 2d. Here d P 1; l is a constant vector; A is a constant (n · n)-matrix; x 2 Rn is the state vector. The vector field f is expanded into a sum of homogeneous vector fields f ¼
2d X
f½s :
s¼2
If f is a homogeneous quadratic vector field then (1) is called a general quadratic system. Further, we consider a general differentiable right-side system x_ ¼ vðxÞ:
ð2Þ
For any set B in Rn we denote by C{B} its complement. Let h(x) be a differentiable function such that h is not the first integral of (1) or (2). The function h will be used in the solution of the localization problem of periodic orbits and will be called localizing. By hjB we denote the restriction of h on a set B Rn. By N1(v, h) we denote the set {x 2 RnjLvh(x) = 0}. Let W be a subset in Rn. Let us define hinf(W) :¼ inf{h(x)jx 2 W}; hsup(W) :¼ sup{h(x)jx 2 W}. In [7,8] it was proposed to apply numbers hinf(N1(v, h)); hsup(N1(v, h)) for studying a location of periodic orbits of the system (2). Namely, we have Proposition 1 [7,8] Each periodic orbit C of the system (2) is contained in the set K h ¼ fhinf ðN1 ðv; hÞÞ 6 hðxÞ 6 hsup ðN1 ðv; hÞÞg:
ð3Þ
The bounds hinf (N1(v, h)) and hsup(N(v, h)) can be computed by using the Lagrange multiplier method. Lemma 2 [18]. Let hinf (N1(v, h)) = hsup (N1(v, h)) = c. Then every periodic orbit is contained in the set N1(v, h). Below we shall find estimates for hinf (N1(v, h)) and hsup(N1(v, h)) for localizing of periodic orbits. Finally, we remark that if all periodic orbits are located in the set A1 and in the set A2 then they are located in A1 \ A2 as well.
3. Main results In what follows, our goal is to sharpen Proposition 1 for the case of linear localizing functions hðxÞ ¼ Cx:
ð4Þ n
For any vector-row C 2 R and b 2 R let us define a homogeneous polynomial G of degree 2d by the formula GðxÞ ¼ Cf ½2d ðxÞ þ bðCxÞ2d :
ð5Þ
The main result of this paper consists in Theorem 3. Suppose that there is a vector C 2 Rn and b 2 R such that the homogeneous polynomial G is positive definite, i.e. G P 0 and G(x) = 0 if and only if x = 0. Let us take a matrix A0 2 Rn·n such that C is its eigenvector with some real nonzero eigenvalue k. Then each periodic orbit C of (1) is contained in the set fCl þ n þ khðxÞ 6 bhðxÞ2d g;
ð6Þ
for some real number n. Proof. We start from Lemma 4. Assume that the homogeneous polynomial h(x) of degree 2d is positive definite. Then the polynomial H :¼ h + d has a global minimum for any polynomial d(x) of degree <2d. Proof. It is sufficient to prove this assertion for H considered outside the ball Bn of the unit radius centered at the origin. We take any x 2 C{Bn}. Then x = ay, with a > 1 and y 2 Sn1; here Sn1 is the (n1)-dimensional sphere of the unit radius centered at the origin. By l we denote miny2Sn1 jhðyÞj; obviously, l > 0. By j we denote max
y2Sn1 ;j¼1;...;2d1
jdj ðyÞj;
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
where dðyÞ ¼
623
P2d1
dj ðyÞ is the expansion of d into a sum of homogeneous parts. In these notations we have for a > 1 2d1 2d1 X X jH ðayÞj ¼ a2d hðyÞ þ aj dj ðyÞ P a2d l j aj ¼ a2d ðl jða 1Þ1 Þ þ jaða 1Þ1 : j¼1 j¼1 j¼i
Now tending a to +1 we conclude that the polynomial H has the global minimum.
h
We compute LFh (x) = Cl + CAx + Cf(x) = Cl + kh(x) + C(A A0)x + Cf (x). By applying this lemma to polynomials G and 2d1 X dðxÞ :¼ CðA A0 Þx þ C f½s ðxÞ; s¼2
we establish that there is a real number n not depending on x such that Cf ½2d ðxÞ þ bðCxÞ2d þ CðA A0 Þx þ C
2d1 X
f½s ðxÞ P n:
ð7Þ
s¼2
Therefore we get the inequality in the formula (6) provided x 2 N1. Hence, by Proposition 1, we deduce the desirable result. h If f = f[2d] in (1), CA = kC for some real k 5 0, with nonzero C 2 Rn and G[2d](x) is positive semidefinite, i.e. G[2d](x) P 0 for some b, then it is clear that we can take n = 0 in (6). The simple analysis of the formula (5) with G(x) 0 leads to the following observation. Suppose that Cf ðxÞ ¼ gðCxÞ;
ð8Þ
for some polynomial g of even degree and CA = kC for some nonzero real k. Then LFh(x) = Cl + CAx + Cf(x) = Cl + kh(x) + g(h(x)). Again, taking x 2 N1 we can conclude that all periodic orbits are contained in the set defined by the equation Cl + kh(x) + g(h(x)) = 0. We obtain that if the equation Cl + kt + g(t) = 0 has no real roots then our system has no periodic orbits. Otherwise, let f1, . . ., fl be all real distinct roots of this equation. If l = 1 then all periodic orbits are contained in the plane Cx = f1. Now if l > 1 and fmax = max{fs; s = 1, . . ., l}; fmin = min{fs; s = 1, . . ., l} then N1 ¼ [ls¼1 h1 ðfs Þ and all periodic orbits are contained in the frustum {fmin 6 h(x) 6 fmax}. Obviously, it is followed from Proposition 1 and Lemma 2. We remark that if the system (1) is given in the observer form, see e.g. in [5], with the observation function h(x) = Cx, then the condition (8) holds. Let us apply our results for the general quadratic system. In this case f½2 ðxÞ ¼ ðxT S 1 x; . . . ; xT S n xÞT ; where matrices S i ¼ ðsijk Þj;k¼1;...;n; i ¼ 1; . . . ; n, are symmetric. Corollary 5. Suppose that there are a positive semidefinite (n · n)-matrix P and a nonzero vector-row C 2 Rn such that P and C satisfy the system of equations n X ci sijk þ bcj ck ¼ pjk ; j; k ¼ 1; . . . ; n ð9Þ i¼1
and C is an eigenvector of some (n · n) matrix A0 with a real nonzero eigenvalue k. Then each periodic orbit C is contained in the set (6) with d =1 for some real n. Proof. By (9) we get: LF h(x) = Cl + CAx + Cf[2](x) = Cl + kh(x) + C(A A0)x bh(x)2 + xTPx = Cl + kh(x) bh(x)2 + (x x*)TP(x x*) + n for some vector x* and real scalar n. Therefore if x 2 N1 then Cl + n + kh 6 bh2. It remains to apply Proposition 1. h Let P = 0 in (9). If the quadratic equation bh2 kh Cl ¼ 0;
ð10Þ
has no real roots respecting h considered as an indeterminate then the general quadratic system has no periodic orbits. If (10) has the unique real root f (of multiplicity 2) then each periodic orbit C is contained in the set {h(x) = f}. If (10) has two real roots f1 < f2 then each periodic orbit C is contained in the frustum {f1 6 h(x) 6 f2}.
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Remark 1. It is clear that we get the localization of periodic orbits as well if we replace positive definiteness (semidefiniteness) properties by the negative definiteness (semidefiniteness) correspondingly in assertions formulated above. Remark 2. Conditions for the positive definiteness (semidefiniteness) of a quadratic form are well-known, see e.g. in [4, Chapter 7.2]. Necessary and sufficient conditions for the positive definiteness of a homogeneous polynomial form of an even degree (P4) are described in the paper [21] for dimensions 2, 3 and 4. One algorithm for checking positive semidefiniteness of a homogeneous polynomial form of an even degree (P4) is contained in [11]. From now up to the end of this section we replace the condition that C in (4) is an eigenvector of A by the weaker condition. Namely, we have Proposition 6. Assume that there are a (n · n)-matrix A0 and a vector-row C 2 Rn such that (1) CA0 = kC, with real k 5 0; (2) there is a polynomial w(x) such that Cf(x) = w(C(A A0)x). Then hinf ðN1 ðv; hÞÞ ¼ inf ðk1 ðCl þ y þ wðyÞÞÞ; y
ð11Þ
hsup ðN1 ðv; hÞÞ ¼ supðk1 ðCl þ y þ wðyÞÞÞ: y
Proof. Indeed, let us take h(x) = Cx. We compute LF hðxÞ ¼ Cl þ CAx þ Cf ðxÞ ¼ Cl þ khðxÞ þ CðA A0 Þx þ wðCðA A0 ÞxÞ: By using y :¼ C(A A0)x we get that the set N1 can be defined by the equation Cl + kh + y + w(y) = 0. Now it is clear that the last formula leads to localization bounds (11). h Finally, we provide computations for two special cases. Case 1. By A(i) we denote the matrix A in which the ith column is replaced by the zero column. Further, let D :¼ A A(i). Let us assume that CA(i) = kC, with real k 5 0, and Cf(x) = w(xi) for some polynomial w. Therefore we establish that n X cj aji xi þ wðxi Þ; LF hðxÞ ¼ Cl þ CAðiÞ x þ CDðxÞ þ Cf ðxÞ ¼ Cl þ khðxÞ þ j¼1
and we get localization bounds hinf ¼ inf k1 Cl þ y
n X
!! cj aji y þ wðyÞ
;
j¼1
hsup ¼ sup k
1
Cl þ
y
n X
!! cj aji y þ wðyÞ
:
j¼1
Case 2. Suppose that there is a (n · n)-matrix A0 and a vector-row C 2 Rn such that (1) CA0 = kC, with real k 5 0; (2) X cj xT S j x ¼ cxT ðA A0 ÞT C T CðA A0 Þx; j
for some real c 5 0. Then hinf = k1Cl + c1k1/4; hsup = 1 provided ck1 < 0 and hsup = k1Cl + c1k1/4; hinf = 1 provided ck1 > 0. Indeed, let us take h(x) = Cx. We compute LF hðxÞ ¼ Cl þ CAx þ Cf ðxÞ ¼ Cl þ khðxÞ þ CðA A0 Þx þ cxT ðA A0 ÞT C T CðA A0 Þx: By using y :¼ C(A A0)x we get that N1 can be defined by Cl þ kh þ y þ cy 2 ¼ 0: The last formula leads to localization bounds (11) where we take w(y) = cy2.
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
625
4. Examples Example 7. Let us consider the Lanford system examined by Hopf in [3] x_ 1 ¼ ðv 1Þx1 x2 þ x1 x3 ; x_ 2 ¼ x1 þ ðv 1Þx2 þ x2 x3 ; x_ 3 ¼ vx3 x21 þ x22 x23 ;
v > 0:
We choose h(x) = x3. Then f1 = 0; f2 = v and all periodic orbits are located in the semispace {v 6 x3 6 0}. Remark 3. This type of localization which is called the semispace localization can be easily established if there is a number j, 1 6 j 6 n, such that Fj(x) = lj + axj + xTPx, with a 5 0 and, P be a positive or negative semidefinite quadratic form. There are many three-dimensional quadratic right-side systems satisfying this condition which are considered in the existing literature on this subject. For example, one can mention the system described by Rucklidge in [13], the modified Ro¨ssler system with a quadratic nonlinearity [20], the system described by Liu et al. in [9], some of quadratic Sprott systems, see Table 1, Examples E; F; L; M; P in [13] and others. Example 8. Let us take the jerky system with a quadratic monomial x_ 1 ¼ x2 ; x_ 2 ¼ x3 ; 3 X x_ 3 ¼ a3s xs þ bx2j : s¼1
Here j = 1 or 2 or 3; a3s; s = 1, 2, 3; b are parameters; b 5 0. As an example of this system, we recall the system proposed in [19] by Tesi, Abed, Genesio and Wang, with a31 = 1; a32 = a; a33 = l; b = 1; j = 1. Below for the sake of brevity let us impose the condition a3s 5 0; s = 1, 2, 3. We take h(x) = Cx and compute LF hðxÞ ¼ c3 a31 x1 þ ðc1 þ c3 a32 Þ þ ðc2 þ c3 a33 Þx3 þ c3 bx2j . Let c3 a31 ¼ kc1 ;
c1 þ c3 a32 ¼ kc2 ;
c2 þ c3 a33 ¼ kc3 :
By using (12) we have: LF hðxÞ ¼ khðxÞ þ 1
hjN1 ¼ k
c3 bx2j .
ð12Þ
Now let j = 1 or 2. Then we obtain that
c3 bx2j :
By taking c1 = 1 in (12) we get c33 a231 c23 a33 a31 c3 a32 ¼ 1;
c2 ¼ c33 a231 c3 a33 :
ð13Þ
Let H be a set of real roots (c2*, c3*) of (13). Then all periodic orbits are located in the semispace AðC Þ ¼ fxjC x P 0; k1 c3 b < 0; C ¼ ð1; c2 ; c3 Þg; with (c2*, c3*) 2 H, or in the semispace AðC Þ ¼ fxjC x 6 0; k1 c3 b > 0; C ¼ ð1; c2 ; c3 Þg; with (c2*, c3*) 2 H, by the remark in the end of Section 2, they are contained in the set \ðc2 ;c3 Þ 2 HAðC Þ: If j = 3 then by successive applying LF to the formula LFh(x) = h(k + c3bh) we get that {h = 0} and {k + c3bh = 0} are invariant planes and by Proposition 2.2 in [16] all periodic orbits are located in these planes. Example 9. We take the quartically perturbed harmonic oscillator, [10] x_ 1 ¼ x2 þ ðax1 þ bx1 x2 þ dx31 þ ex41 Þ; x_ 2 ¼ x1 þ gx41 ; here ; a; b; d; e; g are some nonzero parameters. Let e 5 1. We take h(x) = x1 and form the function GðxÞ ¼ bðc1 x1 þ c2 x2 Þ4 þ c1 ex41 þ c2 gx41 . By taking b = 1 and c1e + c2g = 1 we get G as a positive definite.
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K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
Let us take A0 in the form ! ð0Þ a11 1 1 0 and C ¼ ð1; ð1 eÞg1 Þ:
ð14Þ
Then C is the eigenvector of A0 with the corresponded eigenvalue k ¼ ð1 eÞ1 g; ð0Þ a11
ð15Þ 1
1
as (1 e) g + (1 e)g . Therefore the corresponded matrix A A0 is ! 2 a ð1 eÞ1 g ð1 eÞg1 0 : 0 0
provided we assign
In order to compute the bound n we should form the polynomial dðxÞ ¼ ð2 a ð1 eÞ1 g ð1 eÞg1 Þx1 þ bx1 x2 þ 2 dx31 : Further, we apply the necessary extremum conditions to the function G + d. With this goal we form the system of equations g þ bx2 þ 32 dx21 þ 4ðx1 þ ð1 eÞg1 x2 Þ3 þ 4x31 ¼ 0; bx1 þ 4ð1 eÞg1 ðx1 þ ð1 eÞg1 x2 Þ3 ¼ 0;
ð16Þ
with g = 2a (1 e)1g (1 e)g1. The system (16) has the real solution(s). In order to find them we substitute the expression for (x1 + (1 e)g1x2)3 taken from the second equation of (16) into the first equation and after this we express x2 as a polynomial of x1. We substitute the expression obtained for x2 into the second equation of (16). As a result, we get the algebraic equation of the 9th degree respecting x1. Substituting real roots for x1 of this equation into the first equation of (16) we obtain corresponding roots for x2. Thus we have found the set X of real solutions of (16). Now we deduce that the bound n from (7) is given by n ¼ minðGðxÞ þ dðxÞÞ: x2X
ð17Þ
Hence all periodic orbits are contained in the set defined by {n + kCx 6 (Cx)4}, with C, k and n introduced in formulae (14), (15) and (17).
5. Conclusions In this paper we study the localization problem of periodic orbits of multi-dimensional continuous-time systems with a polynomial right side of even degree. We present new results concerning domains containing all periodic orbits enlarging the class of systems examined in [18]. Though we cannot provide the localization of all periodic orbits into a compact domain we localize all periodic orbits in some unbounded sets of a simple structure: semispaces, their finite intersections, etc. The principal idea of our approach consists in solving the conditional extremum problem described in [7] and applying homogeneous polynomial forms of even degree. Our results are used in the analysis of a location of periodic orbits of the Lanford system, the jerky system with one quadratic monomial and the quartically perturbed harmonic oscillator.
References [1] [2] [3] [4] [5] [6]
Babloyantz A, Krishchenko A, Nosov A. Analysis and stabilization of nonlinear systems. Comput Math Appl 1997;34:356–69. Hardy Y, Steeb WH. The Rikitake two-disk dynamo system and domains with periodic orbits. Int J Theoret Phys 1999;38:2413–7. Hopf E. A mathematical example displaying the features of turbulence. Comm Pure Appl Math 1948;1:303–22. Horn RA, Johnson CR. Matrix Analysis Cambridge: Cambridge University Press; 1986. Isidori A. Nonlinear Control Systems, third ed., Berlin: Springer; 1995. Kocarev L, Parlitz U, Hu B. Lie derivatives and dynamical systems. Chaos, Solitons & Fractals 1998;9:1359–66.
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20] [21]
627
Krishchenko AP. Localization of limit cycles. Differ Equ 1995;31:1826–33. Krishchenko AP. Estimation of domains with cycles. Comput Math Appl 1997;34:325–32. Liu C, Liu T, Liu L, Liu K. A new chaotic attractor. Chaos, Solitons & Fractals 2004;5:1031–8. Perko L. In: Differential Equations and Dynamical Systems. second ed., Texts in Applied Mathematics, New York, Berlin: Springer; 1996. Pogromsky A, Nijmeijer H, Rooda K. A negative criterion for a class of hybrid systems. In: Proceedings of 16th international symposium mathematical theory of networks and systems MTNSÕ2004, Leuven, Belgium, 2004. Powers V, Wo¨rmann T. An algorithm for sums of squares of real polynomials. J Pure Appl Algebra 1998;127:99–104. Rucklidge AM. Chaos in models of double convection. J Fluid Mech 1992;237:209–29. Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50:647–50. Starkov K. Some results on periodic orbits of polynomial systems. Stab Control Theory Appl 2002;4:30–42. Starkov K. Localization of periodic orbits of autonomous systems based on high order extremum conditions. Math Probl Eng 2004;3:277–90. Starkov K, Krishchenko AP. Ellipsoidal estimates for domains containing all periodic orbits of general quadratic systems. In: CDROM with proceedings of the 16th international symposium on mathematical theory of networks and systems MTNSÕ2004, Leuven, Belgium, 2004. Starkov K, Krishchenko AP. Localization of periodic orbits of polynomial systems by ellipsoidal estimates. Chaos, Solitons & Fractals 2005;3:981–8. Tesi A, Abed EH, Genesio R, Wang HO. Harmonic balance analysis of period-doubling bifurcations with applications for control of nonlinear dynamics. Automatica 1996;32:1255–71. Thomas R. The Ro¨ssler equations revisited in terms of feedback control. J Biol Syst 1999;7:225–37. Uteshev AYu, Shulyak SG. HermiteÕ method of separation of solutions of algebraic equations and its applications. Linear Algebra Appl 1992;177:49–88.
Localization of periodic orbits of polynomial vector fields of even degree by linear functions Konstantin E. Starkov
*
CITEDI-IPN, Av. del Parque 1310, Mesa de Otay, Tijuana, BC, Mexico Accepted 23 November 2004
Abstract This paper is concerned with the localization problem of periodic orbits of polynomial vector fields of even degree by using linear functions. Conditions of the localization of all periodic orbits in sets of a simple structure are obtained. Our results are based on the solution of the conditional extremum problem and the application of homogeneous polynomial forms of even degrees. As examples, the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator are considered. 2005 Published by Elsevier Ltd.
1. Introduction The problem of locating periodic orbits of polynomial multi-dimensional continuous-time systems has drawn much attention in the last decade, see papers [1,2,6–8,11,12,14–18] and others. In [18] it was proposed to use quadratic polynomials for getting ellipsoidal localization of periodic orbits of systems with cubic right sides and quadratic right sides satisfying one special ‘‘first integral’’ condition. Namely, if we have a system x_ ¼ l þ Ax þ bðxÞ, with b be a vector-polynomial of degree 2, and we apply a function h(x) = Cx + x(x), with x be a quadratic form, x 5 0, then we impose the condition Lbx = 0. Obviously, this condition is very restrictive. In our paper we apply linear functions for the solution of the localization problem of periodic orbits of polynomial right-side systems of even degree. Though in this case we cannot obtain the ellipsoidal localization, in many cases we get a localization of periodic orbits in unbounded sets of a simple structure like as semispaces, their finite intersections, etc. Our methods are based on the solution of the conditional extremum problem introduced in [7] and applying homogeneous polynomial forms of even degree. As examples, we consider the Lanford system, the jerky system with one quadratic monomial and a quartically perturbed harmonic oscillator. 2. Some notations and useful results Let us introduce a system x_ ¼ F ðxÞ; *
Address: CITEDI-IPN, 2498 Roll Drive #757, San Diego, 92154 CA, USA. Tel.: +52 66231344; fax: +52 66231388. E-mail addresses: [email protected], [email protected]
0960-0779/$ - see front matter 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2004.11.052
ð1Þ
622
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
where F(x) = l + Ax + f(x) is a polynomial vector field of even degree 2d. Here d P 1; l is a constant vector; A is a constant (n · n)-matrix; x 2 Rn is the state vector. The vector field f is expanded into a sum of homogeneous vector fields f ¼
2d X
f½s :
s¼2
If f is a homogeneous quadratic vector field then (1) is called a general quadratic system. Further, we consider a general differentiable right-side system x_ ¼ vðxÞ:
ð2Þ
For any set B in Rn we denote by C{B} its complement. Let h(x) be a differentiable function such that h is not the first integral of (1) or (2). The function h will be used in the solution of the localization problem of periodic orbits and will be called localizing. By hjB we denote the restriction of h on a set B Rn. By N1(v, h) we denote the set {x 2 RnjLvh(x) = 0}. Let W be a subset in Rn. Let us define hinf(W) :¼ inf{h(x)jx 2 W}; hsup(W) :¼ sup{h(x)jx 2 W}. In [7,8] it was proposed to apply numbers hinf(N1(v, h)); hsup(N1(v, h)) for studying a location of periodic orbits of the system (2). Namely, we have Proposition 1 [7,8] Each periodic orbit C of the system (2) is contained in the set K h ¼ fhinf ðN1 ðv; hÞÞ 6 hðxÞ 6 hsup ðN1 ðv; hÞÞg:
ð3Þ
The bounds hinf (N1(v, h)) and hsup(N(v, h)) can be computed by using the Lagrange multiplier method. Lemma 2 [18]. Let hinf (N1(v, h)) = hsup (N1(v, h)) = c. Then every periodic orbit is contained in the set N1(v, h). Below we shall find estimates for hinf (N1(v, h)) and hsup(N1(v, h)) for localizing of periodic orbits. Finally, we remark that if all periodic orbits are located in the set A1 and in the set A2 then they are located in A1 \ A2 as well.
3. Main results In what follows, our goal is to sharpen Proposition 1 for the case of linear localizing functions hðxÞ ¼ Cx:
ð4Þ n
For any vector-row C 2 R and b 2 R let us define a homogeneous polynomial G of degree 2d by the formula GðxÞ ¼ Cf ½2d ðxÞ þ bðCxÞ2d :
ð5Þ
The main result of this paper consists in Theorem 3. Suppose that there is a vector C 2 Rn and b 2 R such that the homogeneous polynomial G is positive definite, i.e. G P 0 and G(x) = 0 if and only if x = 0. Let us take a matrix A0 2 Rn·n such that C is its eigenvector with some real nonzero eigenvalue k. Then each periodic orbit C of (1) is contained in the set fCl þ n þ khðxÞ 6 bhðxÞ2d g;
ð6Þ
for some real number n. Proof. We start from Lemma 4. Assume that the homogeneous polynomial h(x) of degree 2d is positive definite. Then the polynomial H :¼ h + d has a global minimum for any polynomial d(x) of degree <2d. Proof. It is sufficient to prove this assertion for H considered outside the ball Bn of the unit radius centered at the origin. We take any x 2 C{Bn}. Then x = ay, with a > 1 and y 2 Sn1; here Sn1 is the (n1)-dimensional sphere of the unit radius centered at the origin. By l we denote miny2Sn1 jhðyÞj; obviously, l > 0. By j we denote max
y2Sn1 ;j¼1;...;2d1
jdj ðyÞj;
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
where dðyÞ ¼
623
P2d1
dj ðyÞ is the expansion of d into a sum of homogeneous parts. In these notations we have for a > 1 2d1 2d1 X X jH ðayÞj ¼ a2d hðyÞ þ aj dj ðyÞ P a2d l j aj ¼ a2d ðl jða 1Þ1 Þ þ jaða 1Þ1 : j¼1 j¼1 j¼i
Now tending a to +1 we conclude that the polynomial H has the global minimum.
h
We compute LFh (x) = Cl + CAx + Cf(x) = Cl + kh(x) + C(A A0)x + Cf (x). By applying this lemma to polynomials G and 2d1 X dðxÞ :¼ CðA A0 Þx þ C f½s ðxÞ; s¼2
we establish that there is a real number n not depending on x such that Cf ½2d ðxÞ þ bðCxÞ2d þ CðA A0 Þx þ C
2d1 X
f½s ðxÞ P n:
ð7Þ
s¼2
Therefore we get the inequality in the formula (6) provided x 2 N1. Hence, by Proposition 1, we deduce the desirable result. h If f = f[2d] in (1), CA = kC for some real k 5 0, with nonzero C 2 Rn and G[2d](x) is positive semidefinite, i.e. G[2d](x) P 0 for some b, then it is clear that we can take n = 0 in (6). The simple analysis of the formula (5) with G(x) 0 leads to the following observation. Suppose that Cf ðxÞ ¼ gðCxÞ;
ð8Þ
for some polynomial g of even degree and CA = kC for some nonzero real k. Then LFh(x) = Cl + CAx + Cf(x) = Cl + kh(x) + g(h(x)). Again, taking x 2 N1 we can conclude that all periodic orbits are contained in the set defined by the equation Cl + kh(x) + g(h(x)) = 0. We obtain that if the equation Cl + kt + g(t) = 0 has no real roots then our system has no periodic orbits. Otherwise, let f1, . . ., fl be all real distinct roots of this equation. If l = 1 then all periodic orbits are contained in the plane Cx = f1. Now if l > 1 and fmax = max{fs; s = 1, . . ., l}; fmin = min{fs; s = 1, . . ., l} then N1 ¼ [ls¼1 h1 ðfs Þ and all periodic orbits are contained in the frustum {fmin 6 h(x) 6 fmax}. Obviously, it is followed from Proposition 1 and Lemma 2. We remark that if the system (1) is given in the observer form, see e.g. in [5], with the observation function h(x) = Cx, then the condition (8) holds. Let us apply our results for the general quadratic system. In this case f½2 ðxÞ ¼ ðxT S 1 x; . . . ; xT S n xÞT ; where matrices S i ¼ ðsijk Þj;k¼1;...;n; i ¼ 1; . . . ; n, are symmetric. Corollary 5. Suppose that there are a positive semidefinite (n · n)-matrix P and a nonzero vector-row C 2 Rn such that P and C satisfy the system of equations n X ci sijk þ bcj ck ¼ pjk ; j; k ¼ 1; . . . ; n ð9Þ i¼1
and C is an eigenvector of some (n · n) matrix A0 with a real nonzero eigenvalue k. Then each periodic orbit C is contained in the set (6) with d =1 for some real n. Proof. By (9) we get: LF h(x) = Cl + CAx + Cf[2](x) = Cl + kh(x) + C(A A0)x bh(x)2 + xTPx = Cl + kh(x) bh(x)2 + (x x*)TP(x x*) + n for some vector x* and real scalar n. Therefore if x 2 N1 then Cl + n + kh 6 bh2. It remains to apply Proposition 1. h Let P = 0 in (9). If the quadratic equation bh2 kh Cl ¼ 0;
ð10Þ
has no real roots respecting h considered as an indeterminate then the general quadratic system has no periodic orbits. If (10) has the unique real root f (of multiplicity 2) then each periodic orbit C is contained in the set {h(x) = f}. If (10) has two real roots f1 < f2 then each periodic orbit C is contained in the frustum {f1 6 h(x) 6 f2}.
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Remark 1. It is clear that we get the localization of periodic orbits as well if we replace positive definiteness (semidefiniteness) properties by the negative definiteness (semidefiniteness) correspondingly in assertions formulated above. Remark 2. Conditions for the positive definiteness (semidefiniteness) of a quadratic form are well-known, see e.g. in [4, Chapter 7.2]. Necessary and sufficient conditions for the positive definiteness of a homogeneous polynomial form of an even degree (P4) are described in the paper [21] for dimensions 2, 3 and 4. One algorithm for checking positive semidefiniteness of a homogeneous polynomial form of an even degree (P4) is contained in [11]. From now up to the end of this section we replace the condition that C in (4) is an eigenvector of A by the weaker condition. Namely, we have Proposition 6. Assume that there are a (n · n)-matrix A0 and a vector-row C 2 Rn such that (1) CA0 = kC, with real k 5 0; (2) there is a polynomial w(x) such that Cf(x) = w(C(A A0)x). Then hinf ðN1 ðv; hÞÞ ¼ inf ðk1 ðCl þ y þ wðyÞÞÞ; y
ð11Þ
hsup ðN1 ðv; hÞÞ ¼ supðk1 ðCl þ y þ wðyÞÞÞ: y
Proof. Indeed, let us take h(x) = Cx. We compute LF hðxÞ ¼ Cl þ CAx þ Cf ðxÞ ¼ Cl þ khðxÞ þ CðA A0 Þx þ wðCðA A0 ÞxÞ: By using y :¼ C(A A0)x we get that the set N1 can be defined by the equation Cl + kh + y + w(y) = 0. Now it is clear that the last formula leads to localization bounds (11). h Finally, we provide computations for two special cases. Case 1. By A(i) we denote the matrix A in which the ith column is replaced by the zero column. Further, let D :¼ A A(i). Let us assume that CA(i) = kC, with real k 5 0, and Cf(x) = w(xi) for some polynomial w. Therefore we establish that n X cj aji xi þ wðxi Þ; LF hðxÞ ¼ Cl þ CAðiÞ x þ CDðxÞ þ Cf ðxÞ ¼ Cl þ khðxÞ þ j¼1
and we get localization bounds hinf ¼ inf k1 Cl þ y
n X
!! cj aji y þ wðyÞ
;
j¼1
hsup ¼ sup k
1
Cl þ
y
n X
!! cj aji y þ wðyÞ
:
j¼1
Case 2. Suppose that there is a (n · n)-matrix A0 and a vector-row C 2 Rn such that (1) CA0 = kC, with real k 5 0; (2) X cj xT S j x ¼ cxT ðA A0 ÞT C T CðA A0 Þx; j
for some real c 5 0. Then hinf = k1Cl + c1k1/4; hsup = 1 provided ck1 < 0 and hsup = k1Cl + c1k1/4; hinf = 1 provided ck1 > 0. Indeed, let us take h(x) = Cx. We compute LF hðxÞ ¼ Cl þ CAx þ Cf ðxÞ ¼ Cl þ khðxÞ þ CðA A0 Þx þ cxT ðA A0 ÞT C T CðA A0 Þx: By using y :¼ C(A A0)x we get that N1 can be defined by Cl þ kh þ y þ cy 2 ¼ 0: The last formula leads to localization bounds (11) where we take w(y) = cy2.
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4. Examples Example 7. Let us consider the Lanford system examined by Hopf in [3] x_ 1 ¼ ðv 1Þx1 x2 þ x1 x3 ; x_ 2 ¼ x1 þ ðv 1Þx2 þ x2 x3 ; x_ 3 ¼ vx3 x21 þ x22 x23 ;
v > 0:
We choose h(x) = x3. Then f1 = 0; f2 = v and all periodic orbits are located in the semispace {v 6 x3 6 0}. Remark 3. This type of localization which is called the semispace localization can be easily established if there is a number j, 1 6 j 6 n, such that Fj(x) = lj + axj + xTPx, with a 5 0 and, P be a positive or negative semidefinite quadratic form. There are many three-dimensional quadratic right-side systems satisfying this condition which are considered in the existing literature on this subject. For example, one can mention the system described by Rucklidge in [13], the modified Ro¨ssler system with a quadratic nonlinearity [20], the system described by Liu et al. in [9], some of quadratic Sprott systems, see Table 1, Examples E; F; L; M; P in [13] and others. Example 8. Let us take the jerky system with a quadratic monomial x_ 1 ¼ x2 ; x_ 2 ¼ x3 ; 3 X x_ 3 ¼ a3s xs þ bx2j : s¼1
Here j = 1 or 2 or 3; a3s; s = 1, 2, 3; b are parameters; b 5 0. As an example of this system, we recall the system proposed in [19] by Tesi, Abed, Genesio and Wang, with a31 = 1; a32 = a; a33 = l; b = 1; j = 1. Below for the sake of brevity let us impose the condition a3s 5 0; s = 1, 2, 3. We take h(x) = Cx and compute LF hðxÞ ¼ c3 a31 x1 þ ðc1 þ c3 a32 Þ þ ðc2 þ c3 a33 Þx3 þ c3 bx2j . Let c3 a31 ¼ kc1 ;
c1 þ c3 a32 ¼ kc2 ;
c2 þ c3 a33 ¼ kc3 :
By using (12) we have: LF hðxÞ ¼ khðxÞ þ 1
hjN1 ¼ k
c3 bx2j .
ð12Þ
Now let j = 1 or 2. Then we obtain that
c3 bx2j :
By taking c1 = 1 in (12) we get c33 a231 c23 a33 a31 c3 a32 ¼ 1;
c2 ¼ c33 a231 c3 a33 :
ð13Þ
Let H be a set of real roots (c2*, c3*) of (13). Then all periodic orbits are located in the semispace AðC Þ ¼ fxjC x P 0; k1 c3 b < 0; C ¼ ð1; c2 ; c3 Þg; with (c2*, c3*) 2 H, or in the semispace AðC Þ ¼ fxjC x 6 0; k1 c3 b > 0; C ¼ ð1; c2 ; c3 Þg; with (c2*, c3*) 2 H, by the remark in the end of Section 2, they are contained in the set \ðc2 ;c3 Þ 2 HAðC Þ: If j = 3 then by successive applying LF to the formula LFh(x) = h(k + c3bh) we get that {h = 0} and {k + c3bh = 0} are invariant planes and by Proposition 2.2 in [16] all periodic orbits are located in these planes. Example 9. We take the quartically perturbed harmonic oscillator, [10] x_ 1 ¼ x2 þ ðax1 þ bx1 x2 þ dx31 þ ex41 Þ; x_ 2 ¼ x1 þ gx41 ; here ; a; b; d; e; g are some nonzero parameters. Let e 5 1. We take h(x) = x1 and form the function GðxÞ ¼ bðc1 x1 þ c2 x2 Þ4 þ c1 ex41 þ c2 gx41 . By taking b = 1 and c1e + c2g = 1 we get G as a positive definite.
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K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627
Let us take A0 in the form ! ð0Þ a11 1 1 0 and C ¼ ð1; ð1 eÞg1 Þ:
ð14Þ
Then C is the eigenvector of A0 with the corresponded eigenvalue k ¼ ð1 eÞ1 g; ð0Þ a11
ð15Þ 1
1
as (1 e) g + (1 e)g . Therefore the corresponded matrix A A0 is ! 2 a ð1 eÞ1 g ð1 eÞg1 0 : 0 0
provided we assign
In order to compute the bound n we should form the polynomial dðxÞ ¼ ð2 a ð1 eÞ1 g ð1 eÞg1 Þx1 þ bx1 x2 þ 2 dx31 : Further, we apply the necessary extremum conditions to the function G + d. With this goal we form the system of equations g þ bx2 þ 32 dx21 þ 4ðx1 þ ð1 eÞg1 x2 Þ3 þ 4x31 ¼ 0; bx1 þ 4ð1 eÞg1 ðx1 þ ð1 eÞg1 x2 Þ3 ¼ 0;
ð16Þ
with g = 2a (1 e)1g (1 e)g1. The system (16) has the real solution(s). In order to find them we substitute the expression for (x1 + (1 e)g1x2)3 taken from the second equation of (16) into the first equation and after this we express x2 as a polynomial of x1. We substitute the expression obtained for x2 into the second equation of (16). As a result, we get the algebraic equation of the 9th degree respecting x1. Substituting real roots for x1 of this equation into the first equation of (16) we obtain corresponding roots for x2. Thus we have found the set X of real solutions of (16). Now we deduce that the bound n from (7) is given by n ¼ minðGðxÞ þ dðxÞÞ: x2X
ð17Þ
Hence all periodic orbits are contained in the set defined by {n + kCx 6 (Cx)4}, with C, k and n introduced in formulae (14), (15) and (17).
5. Conclusions In this paper we study the localization problem of periodic orbits of multi-dimensional continuous-time systems with a polynomial right side of even degree. We present new results concerning domains containing all periodic orbits enlarging the class of systems examined in [18]. Though we cannot provide the localization of all periodic orbits into a compact domain we localize all periodic orbits in some unbounded sets of a simple structure: semispaces, their finite intersections, etc. The principal idea of our approach consists in solving the conditional extremum problem described in [7] and applying homogeneous polynomial forms of even degree. Our results are used in the analysis of a location of periodic orbits of the Lanford system, the jerky system with one quadratic monomial and the quartically perturbed harmonic oscillator.
References [1] [2] [3] [4] [5] [6]
Babloyantz A, Krishchenko A, Nosov A. Analysis and stabilization of nonlinear systems. Comput Math Appl 1997;34:356–69. Hardy Y, Steeb WH. The Rikitake two-disk dynamo system and domains with periodic orbits. Int J Theoret Phys 1999;38:2413–7. Hopf E. A mathematical example displaying the features of turbulence. Comm Pure Appl Math 1948;1:303–22. Horn RA, Johnson CR. Matrix Analysis Cambridge: Cambridge University Press; 1986. Isidori A. Nonlinear Control Systems, third ed., Berlin: Springer; 1995. Kocarev L, Parlitz U, Hu B. Lie derivatives and dynamical systems. Chaos, Solitons & Fractals 1998;9:1359–66.
K.E. Starkov / Chaos, Solitons and Fractals 25 (2005) 621–627 [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20] [21]
627
Krishchenko AP. Localization of limit cycles. Differ Equ 1995;31:1826–33. Krishchenko AP. Estimation of domains with cycles. Comput Math Appl 1997;34:325–32. Liu C, Liu T, Liu L, Liu K. A new chaotic attractor. Chaos, Solitons & Fractals 2004;5:1031–8. Perko L. In: Differential Equations and Dynamical Systems. second ed., Texts in Applied Mathematics, New York, Berlin: Springer; 1996. Pogromsky A, Nijmeijer H, Rooda K. A negative criterion for a class of hybrid systems. In: Proceedings of 16th international symposium mathematical theory of networks and systems MTNSÕ2004, Leuven, Belgium, 2004. Powers V, Wo¨rmann T. An algorithm for sums of squares of real polynomials. J Pure Appl Algebra 1998;127:99–104. Rucklidge AM. Chaos in models of double convection. J Fluid Mech 1992;237:209–29. Sprott JC. Some simple chaotic flows. Phys Rev E 1994;50:647–50. Starkov K. Some results on periodic orbits of polynomial systems. Stab Control Theory Appl 2002;4:30–42. Starkov K. Localization of periodic orbits of autonomous systems based on high order extremum conditions. Math Probl Eng 2004;3:277–90. Starkov K, Krishchenko AP. Ellipsoidal estimates for domains containing all periodic orbits of general quadratic systems. In: CDROM with proceedings of the 16th international symposium on mathematical theory of networks and systems MTNSÕ2004, Leuven, Belgium, 2004. Starkov K, Krishchenko AP. Localization of periodic orbits of polynomial systems by ellipsoidal estimates. Chaos, Solitons & Fractals 2005;3:981–8. Tesi A, Abed EH, Genesio R, Wang HO. Harmonic balance analysis of period-doubling bifurcations with applications for control of nonlinear dynamics. Automatica 1996;32:1255–71. Thomas R. The Ro¨ssler equations revisited in terms of feedback control. J Biol Syst 1999;7:225–37. Uteshev AYu, Shulyak SG. HermiteÕ method of separation of solutions of algebraic equations and its applications. Linear Algebra Appl 1992;177:49–88.