Critical Periods of Perturbations of Reversible Rigidly Isochronous Centers

Lemma 1 (see [18].)For system (5),

where moreover, if n is odd, S₀(θ) ≡ 0, and

Lemma 2 (see [18].)Let n = m = 2 and λ = 0 in (5). Then

where h ∈ (0, 1/2) and D_ij are constants depending on the coefficients appearing in F_n, K_n, P_m, and Q_m. Furthermore, for sufficiently small ε,

• (i)

  the center O preserves the isochronicity when D₂₀ = D₀₂ = 0; period function T(h, ε, 0) is increasing (resp., decreasing) for h ∈ (0, 1/2) when D₂₀ = 0 and D₀₂ > 0 (resp., < 0);

• (ii)

  there is at most one critical period in (0, 1/2) when D₂₀ ≠ 0. Moreover, there is exactly one critical period in (0, 1/2) if and only if

  $$ ()

Lemma 3 (see [18].)Let n = m = 3 and λ = 0 in (5). Then

where h ∈ (0,1) and B_ij, D_ij are constants depending on the coefficients appearing in F_n, K_n, P_m, and Q_m. Furthermore, for sufficiently small ε, one has the following results.

• (i)

  The center O preserves the isochronicity when

  $$ ()

• (ii)

  If (20) do not hold, there are at most two critical periods in (0,1) and the maximum is achievable.

Theorem 4. Let (8) and (9) hold. Then, for system (5), one has

where

Proof. We follow the idea of proving Lemma 1 which is Theorem 2.1 given in [18].

From (21), we have

Let r(θ, ε, λ, h) be the solution of the above equation satisfying the initial condition r(0, ε, λ, h) = h. It can be written as a series of ε

where r₀(0, λ, h) = h, r_i(0, λ, h) = 0, i ≥ 1.

Substituting (30) into differential equation (29) and comparing the coefficients of ε⁰, we have

We can write r₀(θ, λ, h) = r₀₀(θ, h) + r₀₁(θ, h)λ + ⋯, where r₀₀(0, h) = h, r_0i(0, h) = 0, i ≥ 1. Then, substituting r₀(θ, λ, h) into (31), then comparing the coefficients of λ⁰ and λ, we can obtain Solving the above ODEs associated with the initial values r₀₀(0, h) = h and r₀₁(0, h) = 0, we can get (27) and (28). By (21) we have Therefore, It implies that Thus, (26) is proved. This ends the proof of Theorem 4.

Theorem 5. Let T(h, ε, λ) be defined as before with h ∈ (0, K). Then for 0<|ε | ≪|λ | ≪ 1, one has the following.

•  (i)

  The period function T(h, ε, λ) of system (5) is increasing (resp., decreasing) in h ∈ (0, K) if M₀(h) > 0 (resp., < 0) for all h ∈ (0, K).

•  (ii)

  If M₀(h) is not identically zero, the number of critical periods of  T(h, ε, λ) in (0, K) is not more than the number of zeros (take the multiplicity into consideration) of M₀(h) in (0, K). And there are exactly k critical periods if M₀(h) has exactly k simple positive zeros.

•  (iii)

  If M₀(h) ≡ 0, the number of critical periods of T(h, ε, λ) in (0, K) is not larger than the number of zeros of M₁(h) in (0, K). And k critical periods can appear if M₁(h) has k simple zeros. Similarly, the period function T(h, ε, λ) of (5) is increasing (resp., decreasing) in h ∈ (0, K) if M₁(h) > 0 (resp., < 0).

Proof. From (8) and (9)

If M₀(h) > 0, h ∈ (0, K), then there exists a λ₀ such that for 0<|λ | < λ₀, ∂T₁(h, λ)/∂h has the same sign with dT₁₀(h)/dh. For 0 < ε ≪ λ, the sign of the function ∂T(h, ε, λ)/∂h is the same as ∂T₁(h, λ)/∂h. Then the conclusion (i) is proved.

Further, suppose that h₁, h₂, …, h_k are the k zeros of M₀(h) with the multiplicity m₁, m₂, …, m_k, respectively. We only need to prove that the number of critical periods of T(h, ε, λ) is less than or equal to $$. For the purpose, it suffices to prove that ∂T(h, ε, λ)/∂h has at most $$ roots. Thus, we only need to prove that ∂T(h, ε, λ)/∂h has at most m_j zeros near h_j, j = 1,2, …, k. At first, we prove that ∂T₁(h, λ)/∂h has at most m_j zeros near h_j for λ small. If it is not the case, then ∂T₁(h, λ)/∂h has at least m_j + 1 zeros near h_j; that is, there exists λ_n → 0 such that ∂T₁(h, λ_n)/∂h has at least m_j + 1 zeros h_i,n  (i = 1,2, …, m_j + 1) near h_j, where h_i,n → h_j  (n → ∞). By Rolls theorem, ∂²T₁(h, λ_n)/∂h² has at least m_j zeros near h_j,  …, and  $$ has at least one zero $$ near h_j. Thus, $$. Letting n → ∞, we have $$. This is a contradiction. Thus, ∂T₁(h, λ)/∂h has at most m_j zeros near h_j. Using the same method, for sufficiently small ε, Γ(h, ε, λ) or ∂T/∂h has at most m_j zeros near h_j. Thus, T(h, ε, λ) has at most m_j critical periods for each h_j. Thus, we have proved the first part of (ii).

For the second part of (ii), assume that h₁, h₂, …, h_k are the k simple zeros of M₀(h); that is, M₀(h_j) = 0 and $$, j = 1,2, …, k. Then

By the Implicit Function Theorem, there is a unique function h_j(λ) such that Γ₁(h_j(λ), λ) = 0 and h_j(0) = h_j, j = 1,2, …, k. Therefore h₁(λ), h₂(λ), …, h_k(λ) are k simple zeros of ∂T₁(h, λ)/∂h. By the same method, there is a unique function $$ such that $$, and $$. The second part of (ii) is proved.

When M₀ ≡ 0, then

Then, we can prove conclusion (iii) in the same way as proving conclusion (ii). The proof is completed.

Case 1 (n = m = 2,  l = k = 3). In this case, system (5) becomes

where a₁₁ ≠ 0. For system (40), O is a center because of the symmetry.

We only need to consider the case of a₁₁ > 0 in (40); otherwise, we can use the transformation x → −x, y → −y to change (40) into the same form with opposite signs to the coefficients. Further, suppose a₁₁ = 1, otherwise, system (40) can be simplified as the form of system (41) by the transformation u = a₁₁x, v = a₁₁y. In this case, system (40) becomes

It is easy to conclude that system (41) ∣_ε=0 has a first integral of the form

Theorem 6. For system (41), one has

•  (i)
  $$ ()

•  

  where h ∈ (0,1/2). For sufficiently small ε and λ, there is at most one critical period if M₀(h) is not identically zero and c₂₀ ≠ 0; there is exactly one critical period if and only if ρ≔(c₀₂ − b₁₁)/c₂₀ < −3. Otherwise, the period function T(h, ε, λ) is increasing for h ∈ (0,1/2) (resp., decreasing) if c₂₀ > 0 (resp., < 0);

•  (ii)

  if $$, then for 0 < h ≪ 1,

  $$ ()

•  

  where

  $$ ()

There can appear three critical periods in h ∈ (0, 1/2) if T₁₁(h) is not identically zero.

Proof. For system (41), from (13), (14), (22)–(25), we have

From (27), we have From (10), we have Then, from [18], we have where $$ and Support that c₂₀ ≠ 0. By (50) we have four zeros of F(δ) as follows: where ρ is given by ρ = (c₀₂ − b₁₁)/c₂₀. It is not difficult to examine that δ_± is not in (0,1) and $$ is not in (0,1). In addition, $$ if and only if ρ < −3. Thus, there is at most one critical period in (0, 1/2). If ρ ≥ −3, F(δ) has no zero in h ∈ (0,1/2). From the expression of dT₁₀(h)/dh, we can easily check that dT₁₀(h)/dh > 0 (resp., < 0) when c₂₀ > 0 (resp., < 0). Therefore, for sufficiently small ε and λ, ∂T(h, ε, λ)/∂h > 0 (resp., < 0) if c₂₀ > 0 (resp., < 0). The proof of conclusion (i) is completed.

For conclusion (ii), let M₀(h) ≡ 0, which gives c₂₀ = c₀₂ − b₁₁ = 0 or S₁(θ) = 0. From (26), we have that

Then we expand T₁₁(h) at h = 0 by letting

By (52), we have Note that and that Therefore, A₂, A₄, A₅, A₆ can be taken as free parameters. Hence, we can change the sign of A₂, A₄, A₅, A₆, satisfying which ensures that $$ has three positive zeros in h near h = 0. Thus, conclusion (ii) is proved.

Case 2 (n = m = 3, l = k = 4). In this case, system (40) becomes

where we suppose a₂₁ ≠ 0. This is a new system which is not studied in [18]. For the case a₂₁ > 0, by the rescaling system (58) can be written as where If a₂₁ < 0, system (58) can be simplified as (60) similarly by the change $$, $$.

Theorem 7. For system (60), one has

•  (i)
  $$ ()

•  

  where h ∈ (0,1). If M₀(h) is not identically zero, there are at most two critical periods in h ∈ (0,1) for sufficiently small ε and λ, and the maximum can be achievable;

•  (ii)

  if M₀(h) ≡ 0, one has

  $$ ()

•  

  and there are at most two critical periods in h ∈ (0,1) if T₁₁(h) is not identically zero.

Proof. From (13), (14), (22)–(25), we have that for system (60)

From (27) and (28), we have Hence, from (10),

Thus, $$, where $$, $$(D₁₂ − B₂₁)δ² − 2B₀₃δ − B₀₃. If D₃₀ = 0, $$, we can control the coefficients D₁₂ − B₂₁, −2B₀₃, −B₀₃, which gives two zeros of $$ in (0,1). If D₃₀ ≠ 0, according to the expression of $$, we suppose that δ₁, δ₂, δ₃, δ₄ are four zeros of $$. From the relationship of root and coefficients, we have δ₁ + δ₂ + δ₃ + δ₄ = −2; thus not all of the zeros are in (0,1). Then from [18], and it is impossible for $$ to have 3 zeros in (0,1). Thus, the first part of conclusion (i) is proved.

Next, we give an example to show that the zeros in (0,1) can be achievable. We can choose D₃₀ = 1300, D₁₂ − B₂₁ = −2553, B₀₃ = 116, then, $$(2δ − 1)(5δ − 2)(130δ² + 377δ + 58), so, $$ has two simple zeros 2/5 and 1/2 in (0,1). Thus, T(h, ε, λ) has exactly two critical periods in (0,1). The proof of conclusion (i) is completed.

For the conclusion (ii), if M₀(h) ≡ 0, we get D₃₀ = B₀₃ = D₁₂ − B₂₁ = 0 or S₂(θ) = 0.

Hence, from (26), we have

So, $$, where $$, $$$$. From the proof above, we can get result (ii). This ends the proof of Theorem 7.

Corollary 8. For system (58) with a₂₁ > 0, one obtains

where $$. For sufficiently small ε and λ, there are at most two critical periods for T(h, ε, λ) in h ∈ (0,1) and the maximum number can be achievable.

Proof. Let T(h, ε, λ) denote the period function of (60) and $$ denote the period function of (58). Then we have $$, and $$. Hence by Theorem 7 and (61), we have for system (58),

which gives the formula of T₁₀(h). Other conclusions in the corollary can also be gotten from results in Theorem 7 similarly. This ends the proof of Corollary 8.

Case 3 (n = l = 2, m = k = 3). In this case, system (5) becomes

Without loss of generality, we suppose a₁₁ = 1, otherwise, system (70) can be simplified as the form of system (71) by the transformation u = a₁₁x, v = a₁₁y. In this case, system (70) becomes

Theorem 9. For system (71) and sufficiently small ε and λ, one has

• (i)
  $$ ()
  where
  $$ ()
  There can exist three critical periods for system (71) if T₁₀(h) is not identically zero;

• (ii)

  if $$, one has

  $$ ()
  where
  $$ ()

There can appear three critical periods for system (71) if T₁₁(h) is not identically zero.

Proof. From (13), (14), (22)–(25), we have

From (10), we can get where By the same method as the proof of Theorem 6, we have conclusion (i).

For conclusion (ii), if $$, giving S₁(θ) = S₂(θ) = 0, thus from (26),

where We can get conclusion (ii) with the same method as the proof of Theorem 6. This ends the proof of Theorem 9.