Bifurcation of Limit Cycles and Center Conditions for Two Families of Kukles-Like Systems with Nilpotent Singularities

Abstract

We solve theoretically the center problem and the cyclicity of the Hopf bifurcation for two families of Kukles-like systems with their origins being nilpotent and monodromic isolated singular points.

1. Introduction

One of the main open problems in the qualitative theory of planar analytic differential systems is the case when the singular point is monodromic (the orbits move around the singular point). A monodromic point of an analytic system is either a center (i.e., a singular point with a punctures neighborhood filled with periodic orbits) or a focus (i.e., a singular point with a neighborhood where all the orbits are spirals which arrive at the equilibrium point in forward or backward time). The problem of distinguishing when a monodromic singular point is either a center or a focus is called the center problem.

The so-called center problem for planar vector fields has been intensively and extensively studied over the last century and is also closely related to Hilbert’s 16th problem. The singular point can be elementary or not in the sense whether the corresponding Jacobian matrix has zero eigenvalues. If the eigenvalues of the quoted matrix are imaginary with real part null, the singular point may be a focus or a center, which is known as the celebrated Poincaré-Lyapunov center problem and has been theoretically solved by Poincaré [1] and Liapunov [2]. If the matrix of the linear part at the singular point has its two eigenvalues equal to zero, but it is not identically null, by Andreev [3] we know what is the behavior of the solutions in a neighborhood of the singular point, except if it is a center of a focus (nilpotent center problem).

Yet for the bifurcation of limit cycles and center problem of nilpotent singular points in a planar vector field, its intrinsic dynamics is still far away from understanding due to the complexity and technical difficulties. Therefore, it is natural to restrict our study to nilpotent singularities. An analytic system of differential equations in the plane having an isolated nilpotent singularity, in some suitable coordinates, can be written as ̇,̇,𝑥=𝑦+𝑋𝑥,𝑦𝑦=𝑌𝑥,𝑦 (1) with 𝑋(𝑥,𝑦) and 𝑌(𝑥,𝑦) real analytic functions without constant nor linear terms defined in a certain neighborhood of the singularity.

The study of nilpotent singularities and their unfolding for vector fields is important not only for mathematical interest but also for practical reasons. To solve the finite cyclicity problem in the second part of Hilbert’s 16th problem, Dumortier et al. [4, 5] presented an impressive list of 121 graphics that occur in quadratic systems, among which many are nilpotent cases, studied in [6–11]. On the other hand, the center problem for nilpotent singularities is another interesting issue and has been partially solved; see [3, 12–19], and so forth.

Liu and Li [20] originated a method in studying the center problem of third-order nilpotent singularities by integral factor method, where a new definition of the focal values, quasi-Lyapunov constants, is provided. At the same time, the equivalence of quasi-Lyapunov constants and focal values is proved. A linear recursive formula for computing quasi-Lyapunov constants is also derived. Afterward, they proved that if the third-order nilpotent origin is a 𝑚-order weakened focus, then, by a small perturbation for the unperturbed system, there exist 𝑚 limit cycles in a neighborhood of the origin. Meanwhile, the origin becomes an elementary critical point and two complex singular points.

In the present paper, we are interested in the study of two families of Kukles-like systems with third-order nilpotent part of the form (𝑦,−2𝑥³) and, respectively, nonlinear part given by quartic and quintic homogeneous polynomial with an isolated singular point at the origin; that is, 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=−2𝑥³+𝑏₄₀𝑥⁴+𝑏₃₁𝑥³𝑦+𝑏₂₂𝑥²𝑦²+𝑏₁₃𝑥𝑦³+𝑏₀₄𝑦⁴,𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=−2𝑥³+𝑏₅₀𝑥⁵+𝑏₄₁𝑥⁴𝑦+𝑏₃₂𝑥³𝑦²+𝑏₂₃𝑥²𝑦³+𝑏₁₄𝑥𝑦⁴+𝑏₀₅𝑦⁵. (2)(3)

This paper will be organized as follows. In Section 2 we summarize some general definitions and results about third-order nilpotent singularities. In Section 3 we present the conditions for third-order nilpotent singularities to be a center and give the proof of the cyclicity for system (2). In the last section, we present the parallel results for system (3).

2. The Formula to Compute the Quasi-Lyapunov Constants

The origin is a third-order monodromic critical point if and only if the system is of the following form: 𝑑𝑥𝑑𝑡=𝑦+𝜇𝑥²+^∞_𝑖+2𝑗=3𝑎_𝑖𝑗𝑥^𝑖𝑦^𝑗=𝑋𝑥,𝑦,𝑑𝑦𝑑𝑡=−2𝑥³+2𝜇𝑥𝑦+^∞_𝑖+2𝑗=4𝑏_𝑖𝑗𝑥^𝑖𝑦^𝑗=𝑌𝑥,𝑦. (4)

The quasi-Lyapunov constant is defined and a way of computing them is provided in [20].

Theorem 1. For any positive integer 𝑠 and a given number sequence 𝑐_0𝛽,𝛽≥3, (5) one can construct successively the terms with the coefficients 𝑐_𝛼𝛽 satisfying 𝛼≠0 of the formal series 𝑀𝑥,𝑦=𝑦²+^∞_𝛼+𝛽=3𝑐_𝛼𝛽𝑥^𝛼𝑦^𝛽=^∞_𝑘=2𝑀_𝑘𝑥,𝑦, (6) such that 𝜕𝑋+𝜕𝑥𝜕𝑌()𝜕𝑦𝑀−𝑠+1𝜕𝑀𝜕𝑥𝑋+𝜕𝑀𝑌=𝜕𝑦_∞𝑚=3𝜔_𝑚𝑠,𝜇𝑥^𝑚, (7) where for all 𝑘, 𝑀_𝑘(𝑥,𝑦) is a 𝑘-homogeneous polynomial of 𝑥,𝑦 and 𝑠𝜇=0.

It is easy to see that (7) is linear with respect to the function 𝑀, so that we can easily find in [20] the following recursive formulae for the calculation of 𝑐_𝛼𝛽 and 𝜔_𝑚(𝑠,𝜇).

Theorem 2. For 𝛼≥1, 𝛼+𝛽≥3 in (6) and (7), 𝑐_𝛼𝛽 can be uniquely determined by the recursive formula 𝑐_𝛼𝛽=1()𝛼𝐴𝑠+1^{𝛼−1,𝛽+1}+𝐵_{𝛼−1,𝛽+1}. (8) For 𝑚≥1, 𝜔_𝑚(𝑠,𝜇) can be uniquely determined by the recursive formula 𝜔_𝑚𝑠,𝜇=𝐴_𝑚,0+𝐵_𝑚,0, (9) where 𝐴_𝛼𝛽=^{𝛼+𝛽−1}_𝑘+𝑗=2[𝑘−(𝑠+1)(𝛼−𝑘+1)]×𝑎_𝑘𝑗𝑐_{𝛼−𝑘+1,𝛽−𝑗},𝐵_𝛼𝛽=^{𝛼+𝛽−1}_𝑘+𝑗=2𝑗−(𝑠+1)𝛽−𝑗+1×𝑏_𝑘𝑗𝑐_{𝛼−𝑘,𝛽−𝑗+1}. (10) Notice that in (10), one sets 𝑐₀₀=𝑐₁₀=𝑐₀₁𝑐=0,₂₀=𝑐₁₁=0,𝑐₀₂𝑐=1,_𝛼𝛽=0,𝑖𝑓𝛼<0𝑜𝑟𝛽<0. (11)

We see from Theorem 1 that if the origin of system (4) is 𝑠-class or ∞-class, then, by choosing {𝑐_𝛼𝛽}, such that 𝜔_2𝑘+1𝑠,𝜇=0,𝑘=1,2,…, (12) we can obtain a solution group of {𝑐_𝛼𝛽} of (12); thus, we have 𝜆_𝑚=𝜔_2𝑚+4𝑠,𝜇.2𝑚−4𝑠−1 (13)

Clearly, the recursive formulas presented by Theorem 2 are linear with respect to all 𝑐_𝛼𝛽. Therefore, it is convenient to realize the computations of quasi-Lyapunov constants by using computer algebraic system like MATHEMATICA.

3. Center Conditions and Limit Cycles of System (2)

By Theorems 1 and 2, we directly have the following.

Lemma 3. Assume that 𝑠 is a natural number. One can derive a power series (6) for system (2) under which (7) is satisfied, where 𝑐[][][]𝑐[][][]0,0=0,𝑐1,0=0,𝑐0,1=0,2,0=0,𝑐1,1=0,𝑐0,2=1. (14) In addition, for any natural numbers 𝛼,𝛽, 𝑐[𝛼,𝛽] is given by the following recursive formula: 𝑐𝛼,𝛽=−𝑏₄₀(1+𝑠)2+𝛽𝑐−5+𝛼,2+𝛽+𝑏₃₁()𝑐()𝑐1−1+𝑠1+𝛽−4+𝛼,1+𝛽+21+𝑠2+𝛽−4+𝛼,2+𝛽+𝑏₂₂2−(1+𝑠)𝛽𝑐−3+𝛼,𝛽+𝑏₁₃3−(1+𝑠)−1+𝛽×𝑐−2+𝛼,−1+𝛽+𝑏₀₄()×(𝛼(4−1+𝑠−2+𝛽×𝑐−1+𝛼,−2+𝛽𝑠+1))⁻¹, (15) and, for any natural number 𝑚, 𝜔_𝑚 is given by the following recursive formula: 𝜔_𝑚=−𝑏₄₀(1+𝑠)𝑐[−4+𝑚,1]+𝑏₃₁𝑐[−3+𝑚,0]+2(1+𝑠)𝑐[−3+𝑚,1]+𝑏₂₂(3+𝑠)𝑐[−2+𝑚,−1]+𝑏₁₃(3+2(1+𝑠))𝑐[−1+𝑚,−2]+𝑏₀₄(4+3(1+𝑠))𝑐[𝑚,−3]. (16)

The straightforward computing according to Lemma 3 with MATHEMATICA gives 𝜔₆𝜔=0,₇=12𝑏₃₁−2𝑏₃₁𝑠+6𝑐₀₃+6𝑠𝑐₀₃,𝜔₈1∼−𝑏10⁴⁰𝑏₃₁(),𝜔−3+4𝑠₉∼−2𝑏₁₃(−1+𝑠). (17)

To make 𝜔₉=0, we will consider two different cases: 𝑏₁₃=0 and 𝑠=1.

Case 1 (𝑏₁₃=0). Taking expression (13) into account, the first four quasi-Lyapunov constants of system (2) are given by 𝜆₁𝜆=0,₂∼1𝑏10⁴⁰𝑏₃₁,𝜆₃∼1𝑏21³¹𝑏₂₂,𝜆₄2∼−9𝑏₃₁𝑏₀₄. (18)

Now we can prove the following result.

Theorem 4. In this case, the origin of system (2) is a center if and only if the first four quasi-Lyapunov constants vanish; that is, one of the following two series is satisfied: 𝑏₃₁=𝑏₁₃𝑏=0,₄₀=𝑏₂₂=𝑏₁₃=𝑏₀₄=0. (19)(20)

Proof. It is direct to obtain conditions (19) and (20) by imposing 𝜆_𝑖=0, 𝑖=1,2,3,4.
On the other hand, when condition (19) holds, system (2) goes over to 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=−2𝑥³+𝑏₄₀𝑥⁴+𝑏₂₂𝑥²𝑦²+𝑏₀₄𝑦⁴, (21) whose vector field is symmetric with respect to the 𝑥-axis. When condition (20) holds, system (2) goes over to 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=𝑥³−2+𝑏₃₁𝑦, (22) whose vector field is symmetric with respect to the 𝑦-axis. This ends the proof.

Case 2 (𝑠=1). Taking expression (13) into account, the first five quasi-Lyapunov constants of system (2) are given by 𝜆₁𝜆=0,₂∼1𝑏10⁴⁰𝑏₃₁,𝜆₃∼11055𝑏₃₁𝑏₂₂+9𝑏₄₀𝑏₁₃,𝜆₄∼16300680𝑏₂₂𝑏₁₃−1400𝑏₃₁𝑏₀₄−117𝑏³₄₀𝑏₁₃,𝜆₅∼1𝑏8085000¹³×476000𝑏₀₄−286495𝑏²₄₀𝑏₂₂+18018𝑏⁵₄₀. (23)

For the same reason as before, we obtain the following.

Theorem 5. In this case, the origin of system (2) is a center if and only if the first five quasi-Lyapunov constants vanish; that is, the following series is satisfied: 𝑏₄₀=𝑏₂₂=𝑏₀₄=0. (24)

Proof. It is direct to obtain conditions (24) by imposing 𝜆_𝑖=0, 𝑖=1,2,3,4,5.
On the other hand, when condition (24) holds, system (2) goes over to 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=𝑥−2𝑥²+𝑏₃₁𝑥²𝑦+𝑏₁₃𝑦³, (25) whose vector field is symmetric with respect to the 𝑦-axis. Then, we have proved the theorem.

It is not hard to see that condition (20) is a particular case of condition (24); in summary, the following result characterizes the center of system (2).

Theorem 6. System (2) has a center at the origin if and only if one of the following two sets of conditions holds:
(𝐼)𝑏₃₁=𝑏₁₃=0,
(𝐼𝐼)𝑏₄₀=𝑏₂₂=𝑏₀₄=0.

Remark 7. Indeed, in the light of the theory of [21], the origin is not only a center but also an analytic center under condition (I).

Referring to the expressions given in (18) and (23), we have the following.

Lemma 8. The origin of system (2) is a fine focus of maximum order five. Further, it is of order five if and only if 𝑏₄₀=𝑏₃₁=𝑏₂₂=0,𝑏₁₃𝑏₀₄≠0. (26)

Consider the perturbed system of (2) 𝑑𝑥(𝜀)𝑑𝑡=𝛿𝑥+𝑦,𝑑𝑦(𝜀)𝑑𝑡=2𝛿𝑦−2𝑥³+𝑏₂₁(𝜀)𝑥²𝑦+𝑏₄₀(𝜀)𝑥⁴+𝑏₃₁(𝜀)𝑥³𝑦+𝑏₂₂(𝜀)𝑥²𝑦²+𝑏₁₃𝑥𝑦³+𝑏₀₄𝑦⁴. (27)

Theorem 9. Under a small perturbation of system (2), for a small parameter 𝛿, in a neighborhood of the origin of system (27), there exist exactly 5 small amplitude limit cycles enclosing the elementary node 𝑂(0,0).

The next example shows this fact.

Example 10. Take 𝛿(𝜀)=−𝜀³⁰,𝑏₂₁(𝜀)=𝜀²⁰,𝑏₄₀(𝜀)=𝜀⁶,𝑏₃₁(𝜀)=−𝜀⁶,𝑏₂₂(𝜀)=−𝜀²,𝑏₁₃=𝑏₀₄=1. (28) Straightforward computations by using expression (23) give the first 5 quasi-Lyapunov constants of the origin of system (27): 𝜆₁=13𝜀²⁰+𝑜𝜀²⁰≈0.333333𝜀²⁰+𝑜𝜀²⁰,𝜆₂1∼−𝜀10¹²+𝑜𝜀¹²=−0.1𝜀¹²+𝑜𝜀¹²,𝜆₃∼3𝜀35⁶+𝑜𝜀⁶≈0.0857143𝜀⁶+𝑜𝜀⁶,𝜆₄∼−34𝜀315²+𝑜𝜀²≈−0.107937𝜀²+𝑜𝜀²,𝜆₅∼68(1)(1).1155+𝑜≈0.0588745+𝑜 (29) Thereby, for 0<𝜀≪1, system (27) has 5 limit cycles Γ_𝑘∶𝑟=𝑟(̃𝜃,ℎ_𝑘(𝜀)) in a small neighborhood of the origin, where ℎ_𝑘(𝜀)=𝑜(𝜀^𝑘),  𝑘=1,2,3,4,5.

4. Center Conditions and Limit Cycles of System (3)

Similar to Lemma 3, we have the following.

Lemma 11. Assume that 𝑠 is a natural number. One can derive a power series (6) for system (3) under which (7) is satisfied, where 𝑐[][][]𝑐[][][]0,0=0,𝑐1,0=0,𝑐0,1=0,2,0=0,𝑐1,1=0,𝑐0,2=1. (30) In addition, for any natural numbers 𝛼,𝛽, 𝑐[𝛼,𝛽] is given by the following recursive formula: 𝑐𝛼,𝛽=−𝑏₅₀(1+𝑠)2+𝛽𝑐−6+𝛼,2+𝛽+𝑏₄₁1−(1+𝑠)1+𝛽𝑐−5+𝛼,1+𝛽+𝑏₃₂()𝛽𝑐()𝑐2−1+𝑠−4+𝛼,𝛽+21+𝑠2+𝛽−4+𝛼,2+𝛽+𝑏₂₃3−(1+𝑠)−1+𝛽×𝑐−3+𝛼,−1+𝛽+𝑏₁₄4−(1+𝑠)−2+𝛽×𝑐−2+𝛼,−2+𝛽+𝑏₀₅()×(𝛼(5−1+𝑠−3+𝛽×𝑐−1+𝛼,−3+𝛽𝑠+1))⁻¹, (31) and, for any natural number 𝑚, 𝜔_𝑚 is given by the following recursive formula: 𝜔_𝑚=−𝑏₅₀(1+𝑠)𝑐[−5+𝑚,1]+𝑏₄₁𝑐[−4+𝑚,0]+𝑏₃₂()𝑐[]()𝑐[]3+𝑠−3+𝑚,−1+21+𝑠−3+𝑚,1+𝑏₂₃(3+2(1+𝑠))𝑐[−2+𝑚,−2]+𝑏₁₄(4+3(1+𝑠))𝑐[−1+𝑚,−3]+𝑏₀₅(5+4(1+𝑠))𝑐[𝑚,−4]. (32)

Taking expression (13) into account, the first four quasi-Lyapunov constants of system (3) are given by 𝜆₁𝜆=0,₂∼15𝑏₄₁,𝜆₃∼27𝑏₂₃,𝜆₄∼43𝑏₀₅. (33)

Hence, we arrive at the following.

Theorem 12. System (3) has a center at the origin if and only if the following set of conditions holds:
(𝐼𝐼𝐼)𝑏₄₁=𝑏₂₃=𝑏₀₅=0.

Proof. It is direct to obtain condition (III) by imposing 𝜆_𝑖=0, 𝑖=1,2,3,4.
On the other hand, when condition (III) is satisfied, system (3) reduces to 𝑑𝑥𝑑𝑡=𝑦,𝑑𝑦𝑑𝑡=𝑥−2𝑥²+𝑏₅₀𝑥⁴+𝑏₃₂𝑥²𝑦²+𝑏₁₄𝑦⁴, (34) whose vector field is symmetric with respect to the origin. The claim follows.

Remark 13. Indeed, in the light of the theory of [21], the origin is not only a center but also an analytic center under condition (III).

We obtain the following by expression (33).

Lemma 14. The origin of system (3) is a fine focus of maximum order four. Further, it is of order four if and only if 𝑏₄₁=𝑏₂₃=0,𝑏₀₅≠0. (35)

Consider the perturbed system of (3): 𝑑𝑥(𝜀)𝑑𝑡=𝛿𝑥+𝑦,𝑑𝑦(𝜀)𝑑𝑡=2𝛿𝑦−2𝑥³+𝑏₂₁(𝜀)𝑥²𝑦+𝑏₅₀𝑥⁵+𝑏₄₁(𝜀)𝑥⁴𝑦+𝑏₃₂𝑥³𝑦²+𝑏₂₃(𝜀)𝑥²𝑦³+𝑏₁₄𝑥𝑦⁴+𝑏₀₅𝑦⁵. (36)

Theorem 15. Under a small perturbation of system (3), for a small parameter 𝛿, in a neighborhood of the origin of system (36), there exist exactly 4 small amplitude limit cycles enclosing the elementary node 𝑂(0,0).

Example 16. Take 𝛿(𝜀)=𝜀²⁰,𝑏₂₁(𝜀)=−𝜀¹²,𝑏₄₁(𝜀)=𝜀⁶,𝑏₂₃(𝜀)=−𝜀²,𝑏₀₅=1. (37) Straightforward computations by using expression (33) give the first 4 quasi-Lyapunov constants of the origin of system (36): 𝜆₁1=−3𝜀¹²+𝑜𝜀¹²≈−0.333333𝜀¹²+𝑜𝜀¹²,𝜆₂∼15𝜀⁶+𝑜𝜀⁶=0.2𝜀⁶+𝑜𝜀⁶,𝜆₃2∼−7𝜀²+𝑜𝜀²≈−0.285714𝜀²+𝑜𝜀²,𝜆₄∼43+𝑜(1)≈1.33333+𝑜(1). (38) Thereby, for 0<𝜀≪1, system (36) has 4 limit cycles Γ_𝑘∶𝑟=𝑟(̃𝜃,ℎ_𝑘(𝜀)) in a small neighborhood of the origin, where ℎ_𝑘(𝜀)=𝑜(𝜀^𝑘),  𝑘=1,2,3,4.

Acknowledgments

This work is supported in part by the National Nature Science Foundation of China (11101126 and 11271110) and Scientific Research Start-up Fund for Doctoral Scholars of HAUST (09001524).