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Limit Cycles and Invariant Curves in a Class of Switching Systems with Degree Four.

1. Introduction

It is well known that the 16th problem stated in 1900 by D. Hilbert is considered to be the most difficult problem in the 23 problems; it is far from being solved. Over past three decades, there have been many good results about this problem. As far as the maximal number of small-amplitude limit cycles which are bifurcated from an elementary center or focus is concerned, the best known result obtained by Bautin in 1952 [1] is M(2) = 3, where M(n) denotes the maximal number of small-amplitude limit cycles around a singular point with n being the degree of polynomials in the system. For cubic-degree system, many good results have also been obtained. For example, a cubic system was constructed by Lloyd and Pearson [2] to show 9 limit cycles with the aid of purely symbolic computation. Moreover, Yu and Tian [3] proved that there can exist 12 limit cycles around an elementary center in a planar cubic-degree polynomial system. As far as we know this is the best result obtained so far for cubic-degree polynomial systems with all limit cycles around a single singular point. For n [greater than or equal to] 4, because of the difficulty of computation of focal values, there are very few results. An example of a quartic system with 8 limit cycles bifurcating from a fine focus [4] was given by Huang et al. Theory of rotated equations and applications to a population model can be found in [5]; they gave a new method to solve the center problem.

As far as the maximal number of limit cycles of polynomial systems is concerned, the best results published were given as follows. Articles [6, 7] proved that H(2) [greater than or equal to] 4, then [8-10] gave H(3) [greater than or equal to] 12 and [11,12] obtained H(4) [greater than or equal to] 16 etc. Here, H(n) denotes the maximal number of limit cycles of polynomial systems. Furthermore, 13 limit cycles bifurcated from [Z.sub.2]-equivariant systems with degree 3 were proved in [13-15], respectively. An improvement on the number of limit cycles bifurcating from a nondegenerate center of homogeneous polynomial systems was given in [16].

Center and the coexistence of large and small-amplitude limit cycles problems are two closely related questions of the 16th problem. Algebraic trajectories play an important role in the dynamical behavior of polynomial systems, so it has been an interesting problem in planar polynomial systems. Over the past twenty years, many interesting results were got for quadratic systems; the authors in [17,18] proved that quadratic systems with a pair of straight lines or an invariant hyperbola, ellipse, can have no limit cycles other than the possible ellipse itself. Furthermore, if there is an invariant line, there can be no more than one limit cycle. The case of parabola was considered in [19]. For cubic systems, there exist different classes of cubic systems in which there may coexist an invariant hyperbola or straight lines with limit cycles (see [20-28]). For a given family of real planar polynomial systems of ordinary differential equations depending on parameters, the problem of how to find the systems in the family which become time-reversible was solved in [29].

In modelling many practical problems in science and engineering, switching systems have been widely used recently. The richness of dynamical behavior found in switching systems covers almost all the phenomena discussed in general continuous systems. For example, the maximum number of limit cycles bifurcating from the periodic orbits of the quadratic isochronous centers of switching system was studied in [30]. In [31], limit cycles in a class of continuous and switching cubic polynomial systems were investigated. Bifurcation of limit cycles in switching quadratic systems with two zones was considered in [30] . In [32, 33], the authors considered nonsmooth Hopf bifurcation in switching systems. Limit cycles bifurcating from centers of discontinuous quadratic systems were studied by Chen and Du [34]. Switching Bautin system was also investigated in [35]; they got 10 limit cycles for this system. [Z.sub.2]-equivariant cubic systems were also investigated, and 14 limit cycles were obtained in [36]. Bifurcation of periodic orbits by perturbing high-dimensional piecewise smooth integrable systems was investigated in [37]. Bifurcation theory for finitely smooth planar autonomous differential systems was considered in [38]. All results obtained show that the dynamical behavior of switching systems is more complex than continuous system.

About algebraic invariant curves, as far as we know, there are few papers to consider switching system with algebraic invariant curves. In this paper we are concerned with the limit cycle problem and the center problem for a class of degree four polynomial differential systems

[mathematical expression not reproducible], (1)

which have an invariant conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, and we prove the coexistence of large elliptic limit cycle that contains at least four small-amplitude limit cycles generated by Hopf bifurcations.

The rest of the paper is organized as follows. In the next section, we prove that the switching system (1) has an invariant conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, and there exists a large limit cycle in switching system (1); half attracting invariant conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, is found in switching systems. In Section 3, the first eight Lyapunov constants will be computed; bifurcation of limit cycles and center conditions of (1) are investigated. Section 4 is devoted to discuss the number of limit cycles with different parameter c of (1). At last, coexistence of invariant curve and limit cycles of (1) is drawn in Section 5.

2. Invariant Curve and Large Limit Cycle of (1)

In this section, we will prove that the switching system (1) has an invariant conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, and there exists a large limit cycle in switching system (1).

Lemma 1. The conic h(x, y) = [x.sup.2] + [cy.sup.2] -1 = 0, c [member of] R, is an invariant algebraic curve of system (1). In particular, if c > 0 and [dd.sub.1] = 0, this conic is an elliptic hyperbolic limit cycle, attracting if A > 0, A1 > 0, a repelling if X [lambda] 0, p[lambda].sub.1] [greater than or equal to] 0, and half attracting if [[lambda][lambda].sub.1] < 0.

Proof. It is easy to know that the conic h(x, y) = [x.sup.2] + [cy.sup.2] - 1, c [member of] R, is an invariant algebraic curve of systems

[mathematical expression not reproducible], (2)

and

[mathematical expression not reproducible], (3)

because

dh (x, y)/dt = h (x, y) [k.sub.i] (x, y), i = 1, 2, (4)

respectively, where

[mathematical expression not reproducible]. (5)

In particular, according to Lemma 1 in [39], if c > 0 and d [not equal to] 0, this conic is an elliptic hyperbolic limit cycle of system (2), attracting if [lambda] > 0 and a repelling if [lambda] < 0. Similarly, if c > 0 and [d.sub.1] [not equal to] 0, this conic is an elliptic hyperbolic limit cycle of system (3), attracting if [[lambda].sub.1] > 0 and a repelling if [[lambda].sub.1] < 0. Especially, if c > 0 and [dd.sub.1] = 0 and [[lambda][lambda].sub.1] < 0, the stability of the conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, is contradict for the upper half system and lower half system.

So, for switching system (1), the conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, is an invariant algebraic curve. Furthermore, if c > 0 and [dd.sub.1] [not equal to] 0, this conic is an elliptic hyperbolic limit cycle, and

attracting [lambda] > 0, [[lambda].sub.1] > 0;

repelling [lambda] < 0, [[lambda].sub.1] < 0;

half attraction [[lambda][lambda].sub.1] < 0. (6)

Remark 2. For planar continuous system, if c > 0 and d [not equal to] 0, the conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, is an elliptic hyperbolic limit cycle, attracting if [lambda] > 0, a repelling if [lambda] [less than or equal to] 0. For switching system, half attracting cases which are different from continuous systems appear. Namely, for the conic [x.sup.2] + [cy.sup.2] = 1, c [member of] R, it is attracting (repelling) for y > 0 and repelling (attracting) for y < 0. It is an interesting phenomena; see Figure 1.

3. Bifurcation of Limit Cycle and Center Conditions of (1)

First of all, it is easy to know that the origin of upper half system and lower half system is a fine focus if [[lambda][lambda].sub.1] = 0, so we let [lambda] = [[lambda].sub.1] = 0 in order to consider the center conditions and the number of small limit cycles. With the aid of symbolic computation, we obtain the following result.

Theorem 3. For system (1), the first eight Lyapunov constants at the origin are given by

[mathematical expression not reproducible], (7)

with two cases: (I) c = [+ or -] [square root of 2]/2.

[mathematical expression not reproducible], (8)

[mathematical expression not reproducible], (9)

where

[mathematical expression not reproducible]. (10)

Note that in computing the above expressions, [[mu].sub.k] = 2, ..., 8, and [[mu].sub.1] = [[mu].sub.2] = ... = [[mu].sub.k-1] = 0 have been used.

The following proposition follows directly from Theorem 3.

Proposition 4. The first eight Lyapunov constants at the origin of system (1) become zero if and only if one of the following conditions is satisfied:

[mathematical expression not reproducible], (11)

[mathematical expression not reproducible], (12)

[mathematical expression not reproducible]. (13)

They are also the center conditions of system (1).

Proof. When the conditions in (11) hold, system (1) can be brought to

[mathematical expression not reproducible]. (14)

Obviously, the system is symmetric with the y-axis, and so the origin is a center of system (14).

When the conditions in (12) hold, system (1) can be rewritten as

[mathematical expression not reproducible], (15)

which is symmetric with the x-axis, and so the origin is a center of system (15).

When the conditions in (13) hold, system (1) becomes a continuous system

dx/dy = y (1 + bx) (1- + [x.sup.2] + [cy.sup.2]),

dy/dt = x (1 + cy) (1- + [x.sup.2] + [cy.sup.2]). (16)

By elementary integration, the above system in [omega] = (x, y) | [x.sup.2] + [cy.sup.2] < 1 is topologically equivalent to the system

dx/dt = y (1 + bx),

dy/dt = x (1 + cy). (16)

which has the first analytic integral

H (x, y) = bc (by - cx) + [b.sup.2] ln (1 - cy)

+ [c.sup.2] ln (1 + bx). (18)

Remark 5. The phase plane of system (16) can be drawn by Maple; see Figure 2.

As far as limit cycles are concerned, it follows from Theorem 3 that at most 8 limit cycles can bifurcate from the origin of system (1). We have the following theorem.

Theorem 6. If the origin of system (l)isa 8th-order weak focus, then for 0 < [[delta].sub.1], [[delta].sub.2] [much less than] 1, 8 small-amplitude limit cycles can bifurcate from the origin of the perturbed system (l).

Proof. When the origin of system (1) is a 8th-order weak focus, the conditions

[mathematical expression not reproducible] (19)

should be satisfied. Furthermore, one has the following: When c = -2.09067,

[mathematical expression not reproducible] (20) When c = -1.9427,

[mathematical expression not reproducible] (21)

When c = -0.763201,

[mathematical expression not reproducible]. (22)

When c = 0.581824,

[mathematical expression not reproducible]. (23)

So it implies that 8 small-amplitude limit cycles can bifurcate from the origin of the perturbed system (1).

4. Number of Limit Cycles with Different Parameter c of (1)

In this section, we devote to discuss the number of limit cycles with different parameter c of (1). The following theorem could be concluded from Theorem 3.

Theorem 7. The number of limit cycles with different parameter c of(l) can be shown in the Table l.

Proof. Let [[mu].sub.1] = [[mu].sub.2] = [[mu].sub.3] = 0, it is easy to obtain that

[mathematical expression not reproducible]. (24)

If c = 0, it is easy to check that the origin is a three-order weak focus. Furthermore, if c [not equal to] 0, when c = [square root of 2]/2, the Lyapunov constants in case 1 yield that the origin is a seventh order weak focus.

When c [square root of 2]/2, the Lyapunov constants [[mu].sub.4] = [[mu].sub.5] = [[mu].sub.6] = [[mu].sub.7] = 0, [[mu].sub.8] = 0 yield that the origin is an eighth-order weak focus if

[mathematical expression not reproducible], (25)

where

[mathematical expression not reproducible]. (26)

[mathematical expression not reproducible]. (27)

It is easy to conclude that if [F.sub.3](c) > 0, [F.sub.4](c) > 0 and [F.sub.1](c) = 0, [F.sub.2](c) = 0, there exist 8 limit cycles; namely,

c [approximately equal to] -2.09067,-1.9427, -0.763201,0.581824. (28)

If [F.sub.3](c) > 0, [F.sub.4](c) > 0, and [F.sub.1](c) = 0, there exist 7 small limit cycles.

If [F.sub.3](c)[F.sub.4](c) < 0, there exist 6 limit cycles.

When c [not equal to] [square root of 2]/2, f(c) = 0, the Lyapunov constants ([[mu].sub.4] = [[mu].sub.5] = 0, [[mu].sub.6] [not equal to] 0 yield that the origin is a sixth-order weak focus. The conclusion can be given in Table 1 for simplify.

5. Coexistence of Invariant Curve and Limit Cycles of (1)

From above discussion, we study the coexistence of invariant curve and limit cycles of (1), by perturbation method of small parameters, the following conclusions could be got easily; for example, when c = -2.09067, there exist eight small limit cycles at least and [x.sup.2] - 2.09067 [y.sup.2] = 1 is an invariant algebraic curve. The distribution of limit cycle can be drawn in Figure 3.

When c = -[square root of 2]/2, there exist seven small limit cycles at least and [x.sup.2] - ([square root of 2]/2) [y.sup.2] = 1 is an invariant algebraic curve. The distribution of limit cycle can be drawn in Figure 4.

When c = 0, there exist three small limit cycles at least and x = 1 and x = -1 are two invariant lines. The distribution of limit cycle can be drawn in Figure 5.

When c = 0.581824, there exist eight small limit cycles and a large limit cycle [x.sup.2] + 0.581824 [y.sup.2] = 1 at the same time, namely, nine limit cycles in total for this system. The distribution of limit cycle can be drawn in Figure 6.

When c = [square root of 2]/2, there exist seven small limit cycles at least and there is a large limit cycle [x.sup.2] + ([square root of 2]/2) [y.sup.2] = 1 at the same time. The distribution of limit cycles can be drawn in Figure 7.

When c = 1, there exist six small limit cycles at least and there is a large limit cycle [x.sup.2] + [y.sup.2] = 1 at the same time. The distribution of limit cycles can be drawn in Figure 8.

6. Conclusion

In this paper, a class of switching systems is investigated; the coexistence of small limit cycles and algebraic an invariant curve is proves. An interesting phenomenon that the algebraic invariant curve [x.sup.2] +[cy.sup.2] = 1, c > 0, can be half attracting is found.

https://doi.org/10.1155/2018/4716047

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors' Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

The authors would like to thank the support of Shandong Provincial Natural Science Foundation (no. ZR2014FL017) and the National Natural Science Foundation of China (nos. 71874172, 11275186, and 91024026).

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Xinli Li (iD), (1,2) Huijie Yang, (1) and Binghong Wang (1,3)

(1) School of Business, University of Shanghai for Science and Technology, Shanghai 200093, China

(2) Logistics School, Linyi University, Linyi 276000, China

(3) Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China

Correspondence should be addressed to Xinli Li; lixinli0539@sina.com

Received 10 September 2018; Accepted 31 October 2018; Published 25 November 2018

Academic Editor: Liguang Wang

Caption: Figure 1: The half attracting conic when A > 0, [[lambda].sub.1] < 0 or [lambda] < 0, [[lambda].sub.1] > 0.

Caption: Figure 2: The phase plane of system (16) when b = 1, c = 1 or b = 1, c = -1.

Caption: Figure 3: When c = -2.09067, an invariant algebraic curve [x.sup.2] 2.09067 [y.sup.2] = 1 and eight small limit cycles.

Caption: Figure 4: When c = -[[square root of 2]/2, an invariant algebraic curve [x.sup.2] - ([square root of 2/2]) [y.sup.2] = 1 and seven small limit cycles.

Caption: Figure 5: When c = 0, two invariant lines x = [+ or -]1 and three small limit cycles.

Caption: Figure 6: When c = 0.581824, an invariant algebraic curve [x.sup.2] + 0.581824 [y.sup.2] = 1 and eight small limit cycles.

Caption: Figure 7: When c = [square root of 2]/2, an invariant algebraic curve [x.sup.2] + ([[square root of 2]/2) [y.sup.2] = 1 and seven small limit cycles.

Caption: Figure 8: When c = 1, an invariant algebraic curve [x.sup.2] + [y.sup.2] = 1 and six small limit cycles.
Table 1: The maximum number of small limit cycles around the
origin for different parameter c.

                               The maximum number of
Parameter c                  small limit cycle around
                                    the origin

(-[infinity], -6.14829)                  6
(-6.14829, -2.09067)                     7
-2.09067                                 8
(-2.09067, -1.94278)                     7
-1.94278                                 8
(-1.94278, -0.7632018)                   7
-0.7632018                               8
(-0.7632018, -0.707107]                  7
(-0.707107, -0.588861]                   6
(-0.588861, -0.44638)                    7
[-0.44638, -0.031805]                    6
(-0.031805, 0)                           7
0                                        3
(0,0.031805)                             7
[0.031805, 0.129745]                     6
(0.129745, 0.581824)                     7
0.581824                                 8
(0.581824, 0.707107]                     7
(0.707107, +[infinity])                  6
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Title Annotation:Research Article
Author:Li, Xinli; Yang, Huijie; Wang, Binghong
Publication:Journal of Function Spaces
Date:Jan 1, 2018
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