Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.1996pt" id="M1" height="16.0296pt" version="1.1" viewBox="-0.0657574 -11.83 18.494 16.0296" width="18.494pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-91" d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"/></g><g transform="matrix(.012,0,0,-0.012,11.789,4.134)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg> Equivariant Quartic System

Yang, Jian; Liu, Jukun; Lu, Jingping

doi:https://doi.org/10.1155/2024/4786384

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Conclusion and Discussion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 4786384 | https://doi.org/10.1155/2024/4786384

Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching Equivariant Quartic System

Jian Yang,¹Jukun Liu,²and Jingping Lu^1,3

Academic Editor: Chunrui Zhang

Received11 Aug 2023

Revised11 Mar 2024

Accepted22 Mar 2024

Published02 May 2024

Abstract

In this paper, the limit cycles and local bifurcation of critical periods for a class of switching equivariant quartic system with two symmetric singularities are investigated. First, through the computation of Lyapunov constants, the conditions of the two singularities to become the centers are determined. Then, we prove that there are at most 18 limit cycles with a distribution pattern of 9-9 around the two symmetric singular points of the system. Numerical simulation is conducted to validate the obtained results. Furthermore, by calculating the period constants, we determine the conditions for the critical point to be a weak center of finite order. Finally, the number of local critical periods that bifurcate from the equilibrium point under the center conditions is discussed. This study presents the first example of a quartic switching smooth system with 18 limit cycles and 4 local critical periods bifurcating from two symmetric singular points.

1. Introduction

In the fields of automation, machinery, and engineering, there exist many discontinuous or nondifferentiable systems, which are called discontinuous or nonsmooth mechanical systems [1, 2]. One type of planar nonsmooth dynamical systems is known as a switching system, which is characterized by differential continuous vector fields in at least two different zones that are divided by at least one straight line or curve. The simple switching system can be described by the following equations:where and are analytic functions in and . It is evident that there are two systems in system (1). As referenced in [3, 4], the system is divided into the upper system, defined in the upper half plane for , and the lower system, defined in the lower half plane for . Filippov [5] was the first to establish the fundamental qualitative theory for piecewise smooth systems. Subsequently, there has been a significant interest in the qualitative analysis of such systems. Coll et al. [6] derived the explicit expressions for the first three Lyapunov constants of pseudofocus singular points. Recently, there has been a growing interest among mathematicians in extending the second part of Hilbert’s 16th problem to planar piecewise polynomial vector fields, aiming to determine an upper bound for the maximum number of limit cycles in these systems. While it is a well-known fact that smooth linear systems cannot generate limit cycles, this is not the case for piecewise smooth systems (see [7, 8]). In the realm of piecewise smooth nonlinear systems, significant progress has been made in exploring limit cycles. Tian and Yu [9] proposed a new approach for calculating the Lyapunov constant of a planar switched system, which was subsequently utilized to analyze the bifurcation of limit cycles in the switched Bautin system. Moreover, a specific example of switching systems was presented to demonstrate the emergence of 10 small-amplitude limit cycles bifurcating from a singular point. Liu and Romanovski [10] studied a class of piecewise quadratic system with quartic perturbation, showing that 11 limit cycles could bifurcate around the double homoclinic loop. Li et al. [11] modified an existing method for computing the focal values and period constants of switching systems associated with elementary singular points. They presented a cubic switching system and demonstrated that 15 small limit cycles can bifurcate from the singular point. Li et al. [12] investigated a class of quartic piecewise smooth system and showed that 8 limit cycles can bifurcate from a weak focus of finite order.

If the above equations in (1) admitthen the switching system (1) is -equivariant. Li and his collaborators have made a series of achievements in the study of dynamics such as limit cycles and integrability for -equivariant planar system, as evidenced by references [13–16]. In addition, there have been notable advancements in the application research of bifurcation of limit cycles in symmetric systems. For instance, Song and Xu introduced the delayed half-center CPG oscillator and conducted an analysis of the system’s dynamical behavior with various spatiotemporal patterns [17–19]. Building upon these patterns, they developed the delayed-CPG neural system and successfully implemented locomotion control for both a snake-like robot [20] and a quadruped robot [21]. For the switching -equivariant system, Guo et al. [22] constructed a class of switching -equivariant cubic system to obtain 18 limit cycles from centers. Li and Yu [23] added one more switching line and used the same equations provided in [11] to construct a correct -equivariant cubic system. Their analysis demonstrated that this new system can exhibit 15 limit cycles. Furthermore, Yu et al. [4] considered the bifurcation of limit cycles in a cubic switching -equivariant system, revealing the emergence of 18 limit cycles bifurcating from two symmetric singular foci.

In the qualitative theory of differential equations, a crucial focus is on identifying isochronous centers and the bifurcation of critical periods within a period annulus. Similar to smooth systems, this study also aims to analyze whether the orbits near the center of system (1) have the same period. If these orbits share the same period, the origin of the system (1) is classified as an isochronous center. Alternatively, if the orbits near the center do not share the same period, it becomes essential to determine the order of the weak center and the maximum number of local critical periods that bifurcate from a finite order weak center. Chicone and Jacobs [24] introduced the theory of local bifurcation of critical periods and the definition of weak center of k-order for smooth systems. Additional relevant results for smooth systems can be found in [25–27] and references therein. However, limited research has been conducted on the investigation of isochronous centers and the bifurcation of critical periods for piecewise smooth systems. Chen et al. [28] proposed a new method for calculating the periodic constant near the center of a piecewise smooth system and discussed the problem of bifurcation of local critical periods for a switching equivariant cubic system. It was found that there are at most 5 local critical periods bifurcating from a single singular point. Huang et al. [29] investigated small-amplitude crossing limit cycles and local critical periods for some classes of piecewise smooth Kukles systems of degrees 3 and 4 and acquired new lower bounds for the local cyclicity and the criticality.

Inspired by the abovementioned work, we aim to determine a new lower bound on the maximum number of limit cycles and local critical periods for a class of quartic switching smooth systems. In this paper, we investigate the limit cycles and local bifurcation of critical periods for the following quartic switching smooth system:where and are real parameters.

The rest of the paper is organized as follows. In the next section, we present preliminary information on the algorithm for calculating the Lyapunov constants and period constants for piecewise smooth systems, which can be easily implemented in computer algebraic software. In Section 3, we determine the necessary and sufficient condition for the equilibrium point to be a center by computing the Lyapunov constants for system (3) at the origin. In Section 4, the bifurcation of limit cycles for system (3) is investigated, demonstrating that there can be at most 18 limit cycles surrounding two symmetric equilibrium points under a suitable perturbation. In Section 5, the corresponding period constants are computed under the obtained center conditions, revealing that the equilibrium point of system (3) is not an isochronous center and that the maximum number of local critical period bifurcations from two symmetric equilibrium points is 4. Finally, conclusions are given in Section 6.

2. Preliminaries

In order to investigate the limit cycles and local bifurcation of critical periods for piecewise smooth systems, this section introduces fundamental concepts and techniques that will be utilized in subsequent sections. The approach outlined in [28, 30] is employed in this study to calculate Lyapunov constants and period constants for piecewise smooth systems, following the procedure described in the following.

The general differential systemunder the polar coordinate transformationcan be rewritten aswhere are polynomials of and , given by

Letbe the solutions of system (6) with the initial of .

Define the positive half-return map and the negative half-return map , respectively, bywhere are the Taylor’s coefficients. We obtain the successor function of system (4),where are the Taylor’s coefficients, as illustrated in Figure 1.

The computation of Lyapunov constants using (11) is complex because it involves the composition of two maps and in the displacement map. Another method to calculate the Lyapunov constants was introduced by the authors of [11]. This method involves performing a change of variables to transform system (4) into the following form.

According to Lemma 3 in [31], the displacement map can be calculated aswhere represents the inverse map of (see Figure 2) and is the positive half-return map of system (12) (see Figure 3). The map of system (4) is equivalent to the positive half-return map of system (12). The positive half-return map of system (12) is defined as follows:where are the Taylor’s coefficients. Consequently, a new function is derived aswhere is called the th-order Lyapunov constants at the origin of system (4).

Definition 1 (see [28, 29]). The origin of system (3) is a th-order weak focus if and only if , and the origin is called a rough focus if the order of weak focus is 0, i.e., ; the origin is a center if for all .

The main issue at hand is how the order of the weak focus determines the maximum number of limit cycles that bifurcate from a finite order weak focus. In 2015, Chen, Romanovski, and Zhang provided the following result to address this question.

Proposition 2 (see [32]). Assuming that the origin is a weak focus of order in system (3) associated with the parameter value , it can be concluded that at most limit cycles bifurcate from this weak focus at the parameter value .

To determine if there are perturbations that result in the bifurcation of exactly k limit cycles, one can refer to the following lemma as outlined in reference, Lemma 4 [9].

Lemma 3 (see [9]). If there exists such that and , system (3) just has limit cycles from the origin (where ).

Using the computer algebra system, system (7) becomeswhere are the Taylor’s coefficients.

We obtainwhere is the Taylor’s coefficients.

Therefore, the periodic functions can expand as

By means of transformation , the half-periodic functions of system (4) changes into positive half-periodic functions .

We have,where is a th-order periodic constant.

If , the origin of system (4) is a th-order weak center. If , the origin is a strong center. If for all , the origin is an isochronous center.

Similar to smooth systems, a local critical period is a period that corresponds to a critical point of the period function resulting from a bifurcation from a weak center. More precisely, the definition of local critical periods is as follows.

Definition 4 (see [28]). Suppose that the origin is a weak center corresponding to the parameter . If there exists a such that has solutions in every sufficiently small neighborhood of , then system (3) has local critical periods bifurcating from the center.

In what follows, the key issue is how the order of the weak focus decides the maximum number of local critical periods that bifurcate from a finite order weak center. In 2022, Huang, He, and Cai presented the following result to answer this question.

Proposition 5 (see [29]). If the origin is a weak center of order in system (3) associated with the parameter value , then no more than local critical periods emerge from this weak center at the parameter value .

To determine if there exist perturbations such that exactly k local critical periods are created, one can refer to the following lemma as outlined in reference [28, 33].

Lemma 6 (see [28, 33]). If there exists such that and , system (3) just has critical periods bifurcating from the origin (where ).

In addition, we also want to know whether the origin of switching systems is a center if the first th-order Lyapunov constants vanish. Some criteria for determining whether the equilibrium point of switching systems is a center are provided in [11].

Lemma 7 (see [11]). The necessary and sufficient conditions for the origin of system (2) to be a center if one of the following conditions are satisfied:(a)System (3) is symmetric with the -axis, namely, .(b)System (3) is symmetric with the -axis, namely, .

3. Lyapunov Constants and Center Conditions

In this section, we compute the Lyapunov constants of the corresponding equilibria and find center conditions of system (3) at the origin. First of all, it is easy to check that system (3) always has two singular point and . Due to the symmetry, we choose only to analyze in detail.

By applying the transformation and introducing linear perturbations to system (3), we obtain the following system (for simplicity, we still denote by , respectively):where the singular point (1, 0) of system (3) changes into the origin of system (20).

The system (20) can be transformed into the form of system (6) under the polar coordinate transformation (5). Considering the systemthe expansion of the system (21) around can be expressed aswhere is a polynomial in and . Note that

Combining the above equation with (21) and (22) yields

Substituting the solution into (22) results in andwhere are polynomials in . Then, according to the proposition [28], Proposition 2, and (15) and using the software Mathematica (a code snippet is provided in the Appendix), we obtain the following theorems.

Theorem 8. The first ten Lyapunov constants at the origin of system (20) are given by

Case 9.

Case 10. (i.e. )whereIn the above expressions of , we have already set .

Next, we discuss the center conditions of system (20). From Theorem 8, we obtain the following results.

Theorem 11. The first ten Lyapunov constants at the origin of system (20) are zero if and only if the following conditions are satisfied:

Proof. From the expressions of the first ten Lyapunov constants given in Theorem 8, the sufficiency of the conditions in Theorem 12 is obvious. Next, we prove that the conditions in Theorem 12 are also necessary.
Letting the second Lyapunov constant , we have andTaking yields and . If , we get and . Setting , we obtain andThen, solving yields or . If , we have . This leads to the conditions . On the other hand, under the condition , taking (in particular ), we getTaking , we obtain (i.e., ) andNext, we need to judge whether the equations and have common solutions, that is, to discuss whether or not the polynomials and have common zeros. For this purpose, we compute the Grbner basis of the ideal and getThis means that the polynomials and have no common zeros. The proof of Theorem 12 is complete.

Theorem 12. The origin of system (3) is a center if and only if condition (30) is true.

Proof. When the condition (30) holds, the system (20) can be rewritten in the formObviously, the upper system and the lower system of system (36) are symmetric with the -axis. Thus, the origin of system (3) is a center.

4. Bifurcation of Limit Cycles

The discussion now shifts to the maximum number of limit cycles bifurcating from the origin of system (20). To determine the highest order of the origin as a weak focus, we refer to the Proof of Theorem 12. It is established that the polynomials and have no common root but and should have. Therefore, we need to solve to find the solutions of and . It can be demonstrated that the equations have 6 groups of real solutions, and we take one of them as follows:

According to the above analysis and Theorem 12, we obtain the following theorem.

Theorem 13. The origin of system (20) is a th-order weak focus if and only if the following conditions are satisfied:

Moreover, one can directly verify that the Jacobian evaluated at the critical point equals

According to Lemma 3, the following theorem is concluded.

Theorem 14. Assume that the conditions of Theorem 14 hold, then 9 limit cycles can bifurcate from the equilibrium point of system (20) after a small perturbation. Accordingly, 18 limit cycles can exist in a small neighborhood of two symmetric equilibria (1.0) and of system (3).

Theorem 15 ensures that there are 9 large-amplitude limit cycles in system (20) under small perturbations. In order to obtain 9 small-amplitude limit cycles bifurcating from the equilibrium point, it is necessary to find the exact 9 positive roots derived from solving the polynomial equation

Finding explicit parameter values to achieve a numerical realization is generally challenging. However, for our case, perturbations can be implemented sequentially, allowing for the identification of parameter values that result in 9 positive solutions for (40). It is essential to ensure that the perturbations satisfy the condition

We take perturbations in the backward order as follows: on for , on for , on for , on for , on for , on for , on for , on for , and on for . More precisely, we chooseUsing the abovementioned perturbed parameter values, the resulting Lyapunov constants arefor which (40) has the following 9 positive roots:as expected.

5. Local Bifurcation of Critical Periods

In this section, we investigate the order of a finite weak center and the number of local critical periods when the origin is a center for system (20).

When condition (30) is satisfied, using the computer algebra system Mathematica, the first three period constants of system (20) at the origin are given as follows:

For every that was calculated, we already imposed that . Through analysis, it was found that the equation has no real root, that is, the equations , and have no common zeros. Therefore, the origin of system (20) is not an isochronous center.

Moreover, it is evident that while and do not share any common zeros, and should. By solving , we can find the solutions for and . Subsequently, the Jacobian determinant at the critical point is determined to be

According to the above analysis and Lemma 6, we have the following theorem.

Theorem 15. The highest order of the weak center for system (20) at the origin is 2 if the condition (30) holds, i.e., at most 2 critical periods are generated from the center of system (20). Accordingly, at most 4 critical periods can bifurcate from two symmetric equilibria (1, 0) and of system (3).

6. Conclusion and Discussion

Based on an exact symbolic computation of Lyapunov constants and period constants, we have studied the center problem and determined the maximum number of limit cycles and local critical periods for a class of switching equivariant quartic system. Our findings reveal that at most 18 limit cycles and 4 local critical periods can bifurcate from two symmetric singular points. Up to our knowledge, the result of 18 limit cycles and 4 local critical periods are two new lower bounds on the maximum number of small-amplitude limit cycles and local critical periods for quartic switching smooth systems. The complexity of calculating Lyapunov constants and period constants in piecewise smooth systems increases with the degree of system (1). Compared to smooth systems, it is more challenging to improving the maximum number of limit cycles and local critical periods in these systems. Therefore, developing a more efficient method for calculating Lyapunov constants and period constants in piecewise smooth systems remains a challenging task.

Appendix

Data Availability

A code snippet is provided in the Appendix.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (12061016) and the Basic Ability Enhancement Program for Young and Middle-Aged Teachers of Guangxi (2022KY0198 and 2022KY0254).

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Copyright © 2024 Jian Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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