Bifurcations of periodic orbits, subharmonic solutions and invariant Tori of high-dimensional systems

Nonlinear Analysis 36 (1999) 319 – 329

Bifurcations of periodic orbits, subharmonic solutions and invariant Tori of high-dimensional systems Maoan Hana; 1 , Katie Jiangb; ∗ , David Green, Jr.c; 1 aDepartment

of Mathematics, Shanghai Jiao Tong University, Shanghai, 200030, People’s Republic of China bDepartment of Science & Mathematics, Kettering University, Flint, MI 48504-4898, USA cDepartment of Science & Mathematics, Kettering University, Flint, MI 48504-4898, USA

Received 5 June 1996; received in revised form 10 March 1997; accepted 14 April 1997

Keywords: Periodic orbit; Subharmonic solution; Invariant torus

1. Introduction and Preliminary Lemmas As we know, by means of the Melnikov functions or Lyapunov–Schmidt reduction, we can give sucient conditions for a periodic orbit of an unperturbed system to generate periodic or subharmonic solutions under autonomous or periodic perturbations. See [1–10, 12] for details. A periodic orbit of a plane Hamiltonian system may also generate invariant tori. This problem has been discussed extensively [7–9]. In 1987, Wiggins and Holmes [13] studied the bifurcations of periodic orbits, subharmonic solutions and small invariant tori near periodic orbits of the following three-dimensional system: x˙ = Hy (x; y; z);

y˙ = − Hx (x; y; z);

z˙ = 0

under autonomous or periodic perturbations, where H is a C r function, r ≥ 4. Note that the above system has two dierent rst integrals H (x; y; z) and z. ∗ 1

Corresponding author. Supported by a grant of NNSFC. Supported by Tian Yuan Fund, The National Natural Science Fund of China.

0362-546X/98/$19.00 ? 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 7 ) 0 0 6 6 9 - X

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In this paper, we consider the n-dimensional system (n ≥ 2) x˙ = f(x)

(1.1)

and its perturbation x˙ = f(x) + g(t; x; );

(1.2)

where x ∈ Rn ;  ∈ R and is small, f and g are C r functions, r ≥ 3 and g is periodic in t of period T . For the unperturbed system (1.1), we make the following assumptions: (A1) Eq. (1.1) has n − 1 dierent C r rst integrals Hi (x); i = 1; : : : ; n − 1, such that for each x in an open set U ⊂ Rn , the gradients DH1 (x); : : : ; DHn−1 (x) are linearly independent. (A2) There exists an open set V ⊂ Rn−1 such that for h ≡ (h1 ; : : : ; hn−1 )T ∈ V , Lh ≡ {x ∈ U | H (x) = h} is a compact subset of U not containing a critical point of Eq. (1.1). Here, we have put H (x) = (H1 (x); : : : ; Hn−1 (x))T . By (A1) and (A2), the family {Lh : h ∈ V } gives periodic orbits of Eq. (1.1) in U with n − 1 independent parameters h. Our goal is to investigate bifurcation phenomena of system (1.2) by perturbating the family Lh with  suciently small. We separate the paper into the autonomous and nonautonomous cases presented in Sections 2 and 3, respectively. As a preliminary, we rst prove two fundamental lemmas. We will introduce new coordinates around Lh by using its time-parameter representation. Note that for each h ∈ V; Lh is periodic. We suppose that Lh has a representation Lh : x = q(t; h);

0 ≤ t ≤ T (h);

where T (h) denotes the period of Lh for h ∈ V . We rescale the rst variables of q by introducing    ; h ; 0 ≤  ≤ 2; (1.3) G(; h) = q

(h) where (h) = 2=T (h). Then G is C r ; 2-periodic in  and satis es H (G(; h)) ≡ h:

(1.4)

Note that the partial derivative Dh G of the vertical vector G in h is an n × (n − 1) matrix with n − 1 columns. There exists a 1 × n vector orthogonal to the n − 1 vectors given by the n − 1 columns. The following lemma tells us that the 1 × n vector is uniquely determined under an extra condition. Lemma 1.1. There exists a unique 1 × n vector (; h) which is C r in (; h) and 2-periodic in  for h ∈ V and 0 ≤  ≤ 2; such that (; h)Dh G(; h) = 0;

(; h)D G(; h) = 1:

Proof. We rst prove that the n − 1 vertical vector of the n × (n − 1) matrix Dh G and the vector D G together are linearly independent. Otherwise, the vector D G could be

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a linear combination of the n − 1 columns of Dh G. Equivalently, there would exist an (n − 1) × 1 vertical vectors such that D G = Dh G · :

(1.5)

On the other hand, dierentiating (1.4) in  and h separately we have DH (G)D G = 0;

(1.6)

DH (G)Dh G = In−1 ;

(1.7)

and from Eq. (1.3) and the fact that q is a solution of Eq. (1.1), D G = f(G)= (h):

(1.8)

From Eqs. (1.5)–(1.7) it follows that = 0, and therefore from Eqs. (1.5) and (1.8) we have f(G) = 0. This is a contradiction. Denote by S1 the superplane spanned by the n−1 horizontal vectors of the (n−1) × n matrix DH (G), and by S2 the superplane spanned by the n − 1 vertical vectors of Dh G. From the assumption (A1) and (1.7), S1 and S2 are both of dimension n − 1. From Eq. (1.6) and the above proof we see that D G is normal to S1 , and not parallel to S2 . Then the angle  between S1 and S2 satis es 0 ≤ ¡ and  6= =2. We can assume 0 ≤ ¡=2. (Otherwise, use  −  instead of .) If  = 0; we have the desired vector  given by  = (D G)T =|D G|2 : Let 0¡¡=2 and set L = S1 ∩ S2 . Now we rotate S1 around the straight line L with the angle  such that the new position of S1 after rotation is the same as the original position of S2 . Let D G become after rotation. The vector satis es

· D G = |D G|2 cos ¿0: Then the following vector =

|D G|2 cos 

will satisfy our requirements. This ends the proof. Lemma 1.2. The periodic transformation x = G(; h);

0 ≤  ≤ 2;

h∈V

(1.9)

transforms Eq. (1.2) into the following C r−1 system: ˙ = (h) + (; h)g(t; G(; h); ); h˙ = DH (G(; h)g(t; G(; h); ):

(1.10)

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Proof. Dierentiating the both sides of Eq. (1.9) with respect to t, and using Eq. (1.2) we have D G ˙ + Dh G h˙ = f(G) + g(t; G; ):

(1.11)

Multiplying both sides of Eq. (1.11) on the left by DH(G) and using Eqs. (1.6) – (1.8), we obtain h˙ = DH (G)g(t; G; ): Similarly, multiplying both sides of Eq. (1.11) on the left by (; h) and using Lemma 1.1 and Eq. (1.8), we have ˙ = (; h)f(G) + (; h)g(t; G; ) = (h) + (; h)g(t; G; ): This completes the proof. 2. Periodic orbits and invariant tori in the autonomous case In this section we suppose that g is independent of the time t. Then Eq. (1.2) becomes the autonomous system x˙ = f(x) + g(x; ):

(2.1)

In this case, from Eq. (1.10) we have the 2-periodic system dh = M (; h) + 2 P(; h; ); d

(2.2)

where M (; h) =

1 DH (G(; h))g(G(; h); 0);

(h)

and P is a C r−1 vector function being 2-periodic in . Note that the parameter  is small. The averaging theorem [5,6,12] implies that there is a 2-periodic transformation such that Eq. (2.2) can be transformed into dh = M (h) + 2 p(;  h; ); d

(2.3)

for  small enough, where p is 2-periodic in , and Z 2 1  M (h) = M (; h) d 2 0 I 1 DH (x)g(x; 0) dt: = 2 Lk

(2.4)

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Obviously, M (h) is a vector of n − 1 components. Applying the averaging theorem to Eq. (2.3) we have the following theorem: Theorem 2.1. Suppose that h0 ∈ V . A necessary condition for Eq. (2.1) to have a periodic orbit near Lh0 for small  6= 0 is that M (h0 ) = 0:

(2.5)

Let Eq. (2.5) hold, and det DM (h0 ) 6= 0. Then Eq. (2.1) has a unique periodic orbit near Lh0 for small  6= 0. The periodic orbit is hyperbolic if DM (h0 ) has no eigenvalues having zero real part. For the bifurcation of invariant tori, we have from Theorem 4.4.2 [5] that Theorem 2.2. Suppose that the autonomous system h˙ = M (h)

(2.6)

has a hyperbolic periodic orbit. Then Eq. (2.1) has an invariant 2-torus for small  6= 0. Remark 2.1. If, apart from ; Eq. (2.1) contains another parameter, from Theorem 4.3.1 [5] we can use Eq. (2.6) to discuss the saddle-node bifurcation of periodic orbits or the bifurcation of invariant tori of Hopf type. As an example, consider the system x˙ 1 = x2 −  x1 x3 ; x˙ 2 = −x1 −  x2 x3 ; x˙ 3 = (2x12 − 1 − x33 + x3 ):

(2.7)

When  = 0, Eq. (2.7) has two rst integrals H1 ≡ x12 + x22 and H3 ≡ x3 , which satisfy (A1) and (A2). By taking V = {(h1 ; h2 )T : h1 ¿0}; Lh has a parameter representation of the form p p (2.8) Lh : x = ( h1 sin t; h1 cos t; h2 ); 0 ≤ t ≤ 2 = T (h): Note that for Eq. (2.7)  2x1 2x2 DH (x) = 0 0

0 1

 :

From Eqs. (2.4) and (2.7), we have M (h) = (−2h1 h2 ; h1 − 1 − h32 + h2 )T : Hence, Eq. (2.6) becomes h˙1 = −2h1 h2 ;

h˙2 = h1 − 1 − h32 + h2 ; h1 ¿0:

(2.9)

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The above system is equivalent to h˙2 = ey − 1 − h32 + h2 ;

y˙ = −2h2 ;

(2.10)

where y = ‘n h1 . Applying Theorem 1.3 [11] we can prove that Eq. (2.10) has a unique hyperbolic limit cycle surrounding a hyperbolic critical point. Thus, by Theorems 2.1 and 2.2, the system (2.7) has a unique invariant torus and a periodic orbit inside the torus for small  6≡ 0. Note that Eq. (2.9) has a unique saddle point on the line h1 = 0. It follows that Eq. (2.7) has a periodic orbit of saddle type outside the torus for small  6≡ 0.

3. Subharmonic solutions and invariant tori in the nonautonomous case We now consider the general periodic system (1.2). Suppose that T (h0 ) m = T k

(3.1)

is rational for h0 ∈ V , where m ≥ 1; k ≥ 1 and (m; k) = 1. The solution ((t; t0 ; r; ); h(t; t0 ; r; )) of Eq. (1.10) satisfying (t0 ; t0 ; r; ) = 0;

h(t0 ; t0 ; r; ) = r

has an expansion of the form (t; t0 ; r; ) = (r)(t − t0 ) + 1 (t; t0 ; r) + O(2 ); h(t; t0 ; r; ) = r + h1 (t; t0 ; r) + O(2 );

(3.2)

for  small enough, where 1 (t; t0 ; r) =  (t; t0 ; r; 0); h1 (t; t0 ; r) = h (t; t0 ; r; 0). Notice that

(h) = (r) + D (r)(h − r) + O(|h − r|2 ): Substituting the above and Eq. (3.2) into Eq. (1.10) and comparing the coecients of  on both sides of the equation, we can obtain ˙1 = D (r)h1 + ( (r)(t − t0 ); r)g(t; G( (r)(t − t0 ); r); 0); h˙1 = DH (G( (r)(t − t0 ); r))g(t; G( (r)(t − t0 ); r); 0); which gives Z h1 (t; t0 ; r) =

t

t0

DH (G( (r)(s − t0 ); r))g(s; G( (r)(s − t0 ); r); 0) ds:

(3.3)

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From Eqs. (1.3) and (3.3), we have Z t0 +mT DH (q(t − t0 ; r))g(t; q(t − t0 ; r); 0) dt h1 (mT + t0 ; t0 ; r) = t0

Z =

mT

0

DH (q(t; r))g(t + t0 ; q(t; r); 0) dt

≡ M m=k (t0 ; r):

(3.4)

Clearly, the function M m=k is T -periodic in t0 . Set N m=k (t0 ; r) = 1 (mT + t0 ; t0 ; r);

(3.5)

t0 = {(t; ; h): t = t0 ; 0 ≤  ≤ 2; h∈ V }: According to the de nition of the Poincare map given in Section 1.5 of [5], the Poincare map of Eq. (1.10) is given by P; t0 : t0 → t0 ; P; t0 (0 ; h0 ) = (∗ (t0 + T; t0 ; 0 ; h0 ); h∗ (t0 + T; t0 ; 0 ; h0 )); where (∗ (t; t0 0 ; h0 ); h∗ (t; t0 ; 0 ; h0 )) is the solution of Eq. (1.10) with initial value (0 ; h0 ) at t = t0 . Denote by P;mt0 the mth iteration of P; t0 . Take (0 ; h0 ) = (0; r) and note that ∗ (t; t0 ; 0; r) = (t; t0 ; r; ), h∗ (t; t0 ; 0; r) = h(t; t0 ; r; ). From Eqs. (3.2), (3.4) and (3.5) we have (P;mt0 − Id)(0; r) = ((t0 + mT; t0 ; r; ); h(t0 + mT; t0 ; r; ) − r) = ( (r)mT; 0) + (N m=k (t0 ; r); M m=k (t0 ; r)) + O(2 ):

(3.6)

From Eq. (3.1), we have

(h0 )mT = 0 (mod 2);

mT = 0 (mod T ):

Hence, from Eq. (3.6), for small  6= 0 and |r − h0 | small, (0; r) is a xed point of P;mt0 if and only if F1 (t0 ; r; ) ≡ D (h0 )(r − h0 ) + O(|r − h0 |2 ) + N m=k (t0 ; r) + O(2 ) = 0; F2 (t0 ; r; ) ≡ M m=k (t0 ; r) + O() = 0:

(3.7)

We can now prove the following theorem. Theorem 3.1. Suppose that Eq. (3:1) holds. (i) For small  6= 0 a necessary condition for the periodic orbit Lh0 to generate a subharmonic solution of order m of Eq. (1:2) is that there exists t0∗ ∈ [0; 2] such that M m=k (t0∗ ; h0 ) = 0:

(3.8)

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(ii) Suppose that Eq. (3:8) is satis ed. Let the n × n determinant 0 D (h0 ) J≡ 6= 0: Dt0 M m=k Dr M m=k ∗ (t ; h ) 0

(3.9)

0

Then for small  6= 0 Eq. (1:2) has a subharmonic solution x (t) of order m with the property lim x (t) = q(t − t0∗ ; h0 ):

(3.10)

→0

Proof. The conclusion (i) follows directly from the second equation of Eq. (3.7). Suppose now that Eq. (3.8) is satis ed. Then, by Eq. (3.7) we have F1 (t0∗ ; h0 ; 0) = 0; and

F2 (t0∗ ; h0 ; 0) = 0;

@(F1 ; F2 ) = @(t0 ; r) (t0 ; r; )=(t ∗ ; h0 ; 0) 0

0

D (r)

Dt0 M m=k

Dr M m=k

! : (t0∗ ; h0 )

The condition (3.9) implies that the determinant of the Jacobian of (F1 ; F2 ) with respect to (t0 ; r) at (t0∗ ; h0 ; 0) is not zero. Hence, we have from the implicit function theorem that there exist neighborhoods U0 of (t0∗ ; h0 ) and V0 of  = 0 such that for each  ∈ V0 , Eq. (3.7) have a unique solution (t0 ; r) = (t0 (); r()) = (t0∗ ; h0 ) + O(): By substituting the above into Eq. (3.2), we know that Eq. (1.10) has a subharmonic solution of order m of the form  (t) ≡ (t; t0 (); r()) = (h0 )(t − t0∗ ) + O(); h (t) ≡ h(t; t0 (); r()) = h0 + O(): Then inserting the above into Eq. (1.9) and using Eq. (1.3), we have that Eq. (1.2) has a subharmonic solution of order m of the form x (t) ≡ G( (t); h (t)) = G( (h0 )(t − t0∗ ); h0 ) + O() = g(t − t0∗ ; h0 ) + O(); which yields Eq. (3.10). This completes the proof. In the following, we discuss the existence of invariant tori of Eq. (1.2). For simplicity, we only consider a special case: the period of the family of periodic orbits Lh is constant. Suppose the constant is T0 . Then T (h) = T0

for all h ∈ V:

(3.11)

In this case, the C r−1 system (1.10) becomes 2 + (; h)g(t; G; ); ˙ = T0 h˙ = DH (G)g(t; G; ):

(3.12)

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For T0 =T being irrational, let Z T Z 2 1 dt DH (G(; h)g(t; G(; h); 0) d; P(h) = 2T 0 Z T Z0 2 1 dt (; h)g(t; G(; h); 0) d: Q(h) = 2T 0 0

327

(3.13)

Theorem 3.2. Suppose that Eq. (3:11) holds and T0 =T is irrational. If there exists h∗ ∈ V such that P(h∗ ) = 0; and the matrix DP(h∗ ) has no eigenvalues having zero real part; then for small  6= 0 Eq. (1:2) has an invariant torus in the phase space R n × S 1 ; which has Lh∗ × S 1 as its limit as  tends to zero. Proof. Since T0 =T is irrational, by applying the averaging theorem [6], there exists a near-identity periodic transformation which carries Eq. (3.12) into ˙ = 2=T0 + Q(h) + Q1 (t; ; h; ); (3.14) h˙ = P(h) + P1 (t; ; h; ); where P1 and Q1 are periodic in t and  with periods T and 2, respectively, and P1 = Q1 = 0 when  = 0. Then Theorem 3.2 follows by applying Theorem 7.7.3 and its Corollary of [6] to Eq. (3.14). Further, let T0 m = (3.15) T k be rational. Then 2 2k ≡ R: = T0 mT Let  =  − Rt: From Eq. (3.12), we have ˙ = ( + Rt; h)g(t; G( + Rt; h); ); h˙ = DH (G( + Rt; h))g(t; G( + Rt; h); ): Eq. (3.17) has the period mT in t. Put Z mT 1  P(; h) = DH (G( + Rt; h))g(t; G( + Rt; h); 0) dt; mT Z0 mT 1  Q(; h) = ( + Rt; h)g(t; G( + Rt; h); 0) dt: mT 0

(3.16)

(3.17)

(3.18)

By the method of averaging, Eq. (3.17) can be transformed into the form  ˙ = Q(; h) + 2 Q 1 (t; ; h; ); ˙h = P(;  h) + 2 P1 (t; ; h; ): Then similar to Theorems 2.1 and 2.2, we have the following theorem.

(3.19)

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M. Han et al. / Nonlinear Analysis 36 (1999) 319 – 329

Theorem 3.3. Suppose that Eqs. (3:11) and (3:15) holds. (i) If the autonomous system  ˙ = Q(; h);

 h) h˙ = P(;

(3.20)

has an elementary critical point at (0 ; h0 ); then for small  6= 0; Eq. (1:2) has a subharmonic solution x (t) of order m and with the property lim x (t) = G(0 + Rt; h0 ):

→0

(ii) If Eq. (3:20) has a hyperbolic periodic orbit; then for small  6= 0 Eq. (1:2) has a two dimensional invariant torus. As a simple application of the above theorem, consider x˙ 1 = 12 x2 ; x˙ 2 = − 12 x1 + [x2 (x12 − 1) + 2x1 (cos t − 1)]:

(3.21)

We have that H (x) = x12 + x22 ; T0 = 4; T = 2; √ G(; h) = h(sin ; cos )T ; h¿0 1 (; h) = √ (cos ; − sin ): h Therefore,

√ DH (G) = 2 h(sin ; cos ); √ g(t; G) = h(0; 2 sin (cos t − 1) + cos (h sin2  − 1))T :

Hence, in this case, Eq. (3.12) becomes   1 1 2 2 ˙  = +  −2 sin (cos t − 1) + sin 2(h sin  − 1) ; 2 2 h˙ = 2h[sin 2(cost − 1) + cos2 (h sin2  − 1)]:

(3.22)

Note that we have m = 2; k = 1. Thus, from Eq. (3.18)   1  P(; h) = h − 1 + h sin 2; 4  Q(; h) = 2 + cos 2; and therefore Eq. (3.20) becomes the following cylinder system:   h ˙ ˙  = 2 + cos 2; h=h − 1 + sin 2 : 4

(3.23)

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Let y = 1=h. We get from Eq. (3.23) 1 − sin 2 1 dy = y− ; d 2 + cos 2 4(2 + cos 2) which is a linear periodic equation and has a unique positive hyperbolic -periodic solution y = y∗ (): It follows that Eq. (3.23) has a unique hyperbolic periodic orbit on the region h¿0. Hence, from Theorem 3.3, the system (3.21) has a unique invariant torus for small  6= 0. Also, from (3.23), we see that the trivial invariant torus h = 0 of Eq. (3.22) is asymptotically stable (unstable) for ¿0 (¡0). This means that the zero solution of Eq. (3.21) is asymptotically stable (unstable) for ¿0 (¡0). Acknowledgements The authors are very grateful to the referee for his careful investigations and valuable suggestions during the revision of the paper. References [1] C. Chicone, Bifurcation of nonlinear oscillations and frequency entrainment near resonance, SIAM J. Math. Anal. 23 (1992) 1577–1608. [2] C. Chicone, Lyapunov–Schmidt reduction and Melnikov integrals for bifurcation of periodic solutions in coupled oscillators, J. Di. Eqs. 112 (1994) 407– 447. [3] C. Chicone, Periodic solutions of a system of coupled oscillators near resonance, SIAM J. Math. Anal. 26 (1995) 1257–1283. [4] C. Chicone, A geometric approach to regular perturbation theory with an application to hydrodynamics, Trans. Amer. Math. Soc. 347 (1995) 4559 – 4598. [5] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. [6] J.K. Hale, Ordinary Dierential Equations, Robert E. Krieger Publishing Co., Malabar, Florida, 1980. [7] M. Han, Bifurcations of invariant Tori and subharmonic solutions for periodic perturbed systems, Sci. in China (Ser. A) 11 (1994) 1325 –1336. [8] M. Han, Bifurcation theory of invariant Tori of planar periodic perturbed systems, Sci. in China (Ser. A) 5 (1996) 509 – 519. [9] M. Han, K. Jiang, Bifurcation from a periodic orbit to a Torus, Dyn. of Continuous, Discrete Impulsive Systems 1 (1995) 267– 281. [10] M. Han, K. Jiang, Local bifurcations of periodic orbits in genear autonomous systems, Chin. Ann. Math. (in Chinese) 17A (2) (1996) 179 –188. [11] K. Jiang, M. Han, Boundedness of solutions and existence of limit cycles for a nonlinear system, Nonlin. Anal. TMA 26 (1996) 1995 – 2006. [12] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, 1990. [13] S. Wiggins, P. Holmes, Periodic orbits in slowly varying oscillations, SIAM J. Math. Anal. 18 (1987) 3, 592 – 611.