Bifurcation of Limit Cycles of a Class of Piecewise Linear Differential Systems in ℜ^𝟒 with Three Zones

Abstract

We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method.

1. Introduction and Statement of the Main Result

Piecewise linear systems are used extensively to model many physical phenomena, such as switching circuits in power electronics [1, 2] and impact and dry frictions in mechanical systems [3]. These systems exhibit not only standard bifurcations but also complicated dynamical phenomena not existing in smooth systems. The study and classification of various kinds of bifurcation phenomena for piecewise linear systems have attracted great attentions since the last century, see, for example, [4, 5] and the references therein.

In recent years, many papers studied the bifurcation of limit cycles and the number and distribution of these limit cycles. Most of them studied the planar piecewise linear system, see for example, [6–9] and the references quoted there. There are also some papers which studied bifurcation of limit cycles of 3D piecewise linear systems [10, 11]. For high-dimensional cases, there are a few papers [12–16]. Especially in [12] the authors studied the bifurcation of limit cycles of a class of piecewise linear systems in ℜ⁴. They showed that three is an upper bound for the number of limit cycles that bifurcate from periodic orbits.

In this paper, we study the limit cycles bifurcated from periodic orbits of a linear differential system in  ℜ⁴  when the perturbation is piecewise linear with two switching boundaries. We consider two classes of asymmetric perturbation. With the first class of asymmetric perturbation, six is the upper bound for the number of limit cycles bifurcated from periodic orbits, and there are perturbed systems having six limit cycles. With the second class of asymmetric perturbation, three is the upper bound for the number of limit cycles bifurcated from periodic orbits, which generalizes the result of the paper [12].

More precisely, we study the maximum number of limit cycles of the 4-dimensional continuous piecewise linear vector fields with three zones of the form 𝑥̇=𝐴₀𝑥+𝜀𝐹(𝑥), (1) for 𝜀≠0 sufficiently small real parameter, where 𝐴₀=,0−1001000000−10010 (2) and 𝐹∶ℜ⁴→ℜ⁴  is given by 𝐹(𝑥)=𝐴𝑥+𝜑𝑘^𝑇𝑥𝑏, (3) with  𝐴∈𝑀₄(ℜ),  𝑘,𝑏∈ℜ⁴⧵{0},  and  𝜑∶ℜ→ℜ  the piecewise linear function  if  𝑚₁<0<𝑚₂, 𝜑(𝑥)=ℎ𝑚₁,for𝑥∈−∞,𝑚₁,ℎ𝑥,for𝑥∈𝑚₁,𝑚₂,ℎ𝑚₂,for𝑥∈𝑚₂,+∞; (4) if  𝑚₁<𝑚₂<0, 𝜑(𝑥)=ℎ𝑚₁−ℎ𝑚₂,for𝑥∈−∞,𝑚₁,ℎ𝑥−ℎ𝑚₂,for𝑥∈𝑚₁,𝑚₂,0,for𝑥∈𝑚₂,+∞; (5) if  0<𝑚₁<𝑚₂, 𝜑(𝑥)=0,for𝑥∈−∞,𝑚₁,ℎ𝑥−ℎ𝑚₁,for𝑥∈𝑚₁,𝑚₂,ℎ𝑚₂−ℎ𝑚₁,for𝑥∈𝑚₂,+∞, (6)

where  ℎ∈ℜ⧵{0}. The independent variable is denoted by  𝑡; vectors of  ℜ⁴  are column vectors, and  𝑘^𝑇  denotes a transposed vector.

For  𝜀=0,  system (1) becomes 𝑥̇₁=−𝑥₂,𝑥̇₂=𝑥₁,𝑥̇₃=−𝑥₄,𝑥̇₄=𝑥₃. (7)

Our main results are the following.

Theorem 1. If  𝑚₁𝑚₂>0, six is the upper bound for the number of limit cycles of system (1) which bifurcate from the periodic orbits of system (7) with  𝜀  sufficiently small. Moreover, there are systems of form (1) having six limit cycles.

Theorem 2. If  𝑚₁𝑚₂<0, three is the upper bound for the number of limit cycles of system (1) which bifurcate from the periodic orbits of system (7) with  𝜀  sufficiently small. Moreover, there are systems of form (1) having three limit cycles.

It is worth to note that Theorem 2 generalizes the result of paper [12]. The method for computing the number of limit cycles bifurcated from periodic orbits is the averaging method, which is obtained by Buică and Llibre [17]. By means of the result of paper [18], we can study the stability of the limit cycles of Theorem 1; for more details see Remark 10.

Theorems 1 and 2 will be proved in Section 3. In Section 2, we review the results from the averaging theory necessary for proving these two theorems. Further discussions on the number of limit cycles of the perturbed system are present in Section 4. There is a conclusion given in the last section.

2. First-Order Averaging Method

The aim of this section is to review the first-order averaging method which is obtained by Buică and Llibre [17]. The advantage of this method is that the smoothness assumptions for the vector field of the differential system are minimal.

Theorem 3 (see [17]). Consider the following differential system: 𝑥̇(𝑡)=𝜀𝐻(𝑡,𝜀)+𝜀²𝑅(𝑡,𝑥,𝜀), (8) where  𝐻∶ℜ×𝐷→ℜ^𝑛,  𝑅∶ℜ×𝐷×(−𝜀₀,𝜀₀)→ℜ^𝑛  are continuous functions. T-periodic in the first variable, and  𝐷  is an open subset of  ℜ^𝑛. We define  ℎ∶𝐷→ℜ^𝑛  as ℎ(𝑧)=^𝑇₀𝐻(𝑠,𝑧)𝑑𝑠, (9) and assume that(i)𝐻 and  𝑅  are locally Lipschitz with respect to  𝑥;(ii)for  𝑎∈𝐷  with  ℎ(𝑎)=0, there exists a neighborhood  𝑉  of  𝑎  such that  ℎ(𝑧)≠0  for all  𝑧∈𝑉⧵{𝑎}  and  𝑑_𝐵(ℎ,𝑉,0)≠0.
Then, for  |𝜀|>0  sufficiently small, there exists an isolated T-periodic solution  𝜙(⋅,𝜀)  of system (8) such that  𝜙(⋅,𝜀)→𝑎  as 𝜀→0.

We remind here that  𝑑_𝐵(ℎ,𝑉,𝑎)  denotes the Brouwer degree of the function  ℎ  with respect to the set  𝑉  and the point  𝑎, as is defined in [19]. The following fact is useful for the proof of Theorems 1 and 2.

Fact 1. Let ℎ∶𝐷→ℜ^𝑛 be a 𝐶¹ function, with ℎ(𝑎)=0, where 𝐷 is an open subset of ℜ^𝑛 and 𝑎∈𝐷. Whenever 𝑎 is a simple zero of  ℎ (i.e., 𝐽(𝑎)≠0), there exists a neighborhood 𝑉 of 𝑎 such that ℎ(𝑧)≠0 for all 𝑧∈𝑉⧵{𝑎}. Then, 𝑑_𝐵(ℎ,𝑉,0)∈{−1,1}.

3. Proof of Main Theorems

The proof of Theorems 1 and 2 is based on the first-order averaging method presented in the previous section. In order to apply this method, we will first reduce the four parameters of the vector  𝑘  in the definition of the function  𝐹(𝑥)  to one, and then we will change the variables in order to transform the system into the standard form for the averaging method. After that, we will calculate the number of its isolated zeros.

Lemma 4. By a linear change of variables, system (1) can be transformed into the system 𝑥̇=𝐴₀𝑥+𝜀𝐴𝑥+𝜀𝜑𝑥₁𝑏, (10) where  𝐴∈𝑀₄(ℜ)  is an arbitrary matrix and  𝑏=(𝑏₁,𝑏₂,0,0)^𝑇  or  𝑏=𝑒₃.

Proof. A linear change of variables  𝑥=𝑃𝑦, with  𝑃  invertible, transforms system (1) into 𝑦̇=𝑃⁻¹𝐴₀𝑃𝑦+𝜀𝑃⁻¹𝐴𝑃𝑦+𝜀𝜑𝑘^𝑇𝑃𝑦𝑃⁻¹𝑏, (11) where  𝑏=(𝑏₁,𝑏₂,𝑏₃,𝑏₄)^𝑇,  𝑘=(𝑘₁,𝑘₂,𝑘₃,𝑘₄)^𝑇.
We have to find  𝑃  invertible which satisfies 𝑃⁻¹𝐴₀𝑃=𝐴₀,𝑘^𝑇𝑃=𝑒^𝑇₁. (12) It is easy to obtain that  𝑃⁻¹  has the following form: 𝑃⁻¹=𝑘₁𝑘₂𝑘₃𝑘₄−𝑘₂𝑘₁−𝑘₄𝑘₃𝑝₃₁𝑝₃₂𝑝₃₃𝑝₃₄−𝑝₃₂𝑝₃₁−𝑝₃₄𝑝₃₃. (13) Thus, we have 𝑃⁻¹𝑏=𝑏, (14) where 𝑏₁=⁴_𝑖=1𝑘_𝑖𝑏_𝑖,𝑏₂=−𝑘₂𝑏₁+𝑘₁𝑏₂−𝑘₄𝑏₃+𝑘₃𝑏₄,𝑏₃=⁴_𝑖=1𝑝_3𝑖𝑏_𝑖,𝑏₄=−𝑝₃₂𝑏₁+𝑝₃₁𝑏₂−𝑝₃₄𝑏₃+𝑝₃₃𝑏₄. (15) If  𝑏²₁+𝑏²₂≠0, it is easy to find  𝑃⁻¹  invertible with  𝑝₃₁,  𝑝₃₂,  𝑝₃₃,  𝑝₃₄  satisfying 𝑏₃=⁴_𝑖=1𝑝_3𝑖𝑏_𝑖=0,𝑏₄=−𝑝₃₂𝑏₁+𝑝₃₁𝑏₂−𝑝₃₄𝑏₃+𝑝₃₃𝑏₄=0. (16) If  𝑏²₁+𝑏²₂=0, it is easy to find  𝑃⁻¹  invertible with  𝑝₃₁,  𝑝₃₂,  𝑝₃₃,  𝑝₃₄  satisfying 𝑏₃=⁴_𝑖=1𝑝_3𝑖𝑏_𝑖=1,𝑏₄=−𝑝₃₂𝑏₁+𝑝₃₁𝑏₂−𝑝₃₄𝑏₃+𝑝₃₃𝑏₄=0. (17) Changing  variables  𝑦  to  𝑥  with  𝑥=𝑦, then we obtain system (10).

The standard form of the averaging method is obtained by changing variables (𝑥₁,𝑥₂,𝑥₃,𝑥₄) to (𝑟,𝜃,𝜌,𝑠) with 𝑥₁=𝑟cos𝜃,𝑥₂𝑥=𝑟sin𝜃,₃=𝜌cos(𝜃+𝑠),𝑥₄=𝜌sin(𝜃+𝑠). (18) Thus, system (10) is transformed into the following system: 𝑑𝑟𝑑𝜃=𝜀𝐻₁𝜃,𝑟,𝜌,𝑠+𝜀²𝑂(1),𝑑𝜌𝑑𝜃=𝜀𝐻₂𝜃,𝑟,𝜌,𝑠+𝜀²𝑂(1),𝑑𝑠𝑑𝜃=𝜀𝐻₃𝜃,𝑟,𝜌,𝑠+𝜀²𝑂(1), (19) where  𝐻₁,  𝐻₂,  and  𝐻₃  are given by 𝐻₁=cos𝜃𝐹₁+sin𝜃𝐹₂,𝐻₂=cos(𝜃+𝑠)𝐹₃+sin(𝜃+𝑠)𝐹₄,𝐻₃=1𝑟sin𝜃𝐹₁−1𝑟cos𝜃𝐹₂−1𝜌sin(𝜃+𝑠)𝐹₃+1𝜌cos(𝜃+𝑠)𝐹₄ (20) and for every  𝑖=1,2,…,4, 𝐹_𝑖=𝑎_𝑖1𝑟cos𝜃+𝑎_𝑖2𝑟sin𝜃+𝑎_𝑖3𝜌cos(𝜃+𝑠)+𝑎_𝑖4()()𝜌sin𝜃+𝑠+𝜑𝑟cos𝜃𝑏_𝑖, (21) where  𝑎_𝑖𝑗  are elements of the matrix  𝐴  of Lemma 4.

We take  𝜀₀  sufficiently small,  𝑚  arbitrarily large and 𝐷_𝑚=∣∈1𝑟,𝜌,𝑠𝑟,𝜌,𝑠𝑚,𝑚²×𝑆. (22) Then, the vector of system (19) is well defined and continuous on  𝑆×𝐷_𝑚×(−𝜀₀,𝜀₀). Moreover, the system is  2𝜋-periodic with respect to variable  𝜃  and locally Lipschitz with respect to variables  (𝑟,𝜌,𝑠). Our next step is to find the corresponding function  ℎ∶𝐷→ℜ³,  ℎ=(ℎ₁,ℎ₂,ℎ₃), where ℎ_𝑖𝑟,𝜌,𝑠=₀^2𝜋𝐻_𝑖𝑟,𝜃,𝜌,𝑠𝑑𝜃, (23) for  𝑖=1,2,3.

In order to calculate the exact expression of  ℎ, we denote 𝐼₁(𝑟)=₀^2𝜋𝜑(𝑟cos𝜃)cos𝜃𝑑𝜃,𝐼₂(𝑟)=₀^2𝜋𝜑(𝑟cos𝜃)sin𝜃𝑑𝜃, (24)(25) for  each  𝑟>0, where  𝜑  is the piecewise linear function given by (4)–(6). Without loss of generality, we assume that the slope  ℎ  of  𝜑  is positive.

Lemma 5. The integrals  𝐼₁  and  𝐼₂  given by (24)-(25), respectively, have the following expressions: 𝐼₂(𝑟)=0,∀𝑟>0,𝑚₁,𝑚₂, (26) and(1) if  0<𝑚₁<𝑚₂, 𝐼₁(𝑟)=0,𝑖𝑓0<𝑟≤𝑚₁,𝐽𝑟,𝑚₁,𝑖𝑓𝑚₁<𝑟<𝑚₂,𝐾𝑟,𝑚₁,𝑚₂,𝑖𝑓𝑟≥𝑚₂; (27) (2) if  𝑚₁<𝑚₂<0, 𝐼₁(𝑟)=0,𝑖𝑓0<𝑟≤𝑚₂,−𝐽𝑟,𝑚₂,𝑖𝑓𝑚₂<𝑟<𝑚₁,𝐾𝑟,𝑚₁,𝑚₂,𝑖𝑓𝑟≥𝑚₁; (28) (3) if  𝑚₁<0<𝑚₂  and  |𝑚₁|<|𝑚₂|, 𝐼₁(𝑟)=𝜋ℎ𝑟,𝑖𝑓0<𝑟≤𝑚₁,𝜋ℎ𝑟+𝐽𝑟,𝑚₁,𝑖𝑓𝑚₁<𝑟<𝑚₂,𝜋ℎ𝑟+𝐾𝑟,𝑚₁,𝑚₂,𝑖𝑓𝑟≥𝑚₂; (29) (4) if  𝑚₁<0<𝑚₂  and  |𝑚₁|>|𝑚₂|, 𝐼₁(𝑟)=𝜋ℎ𝑟,𝑖𝑓0<𝑟≤𝑚₂,𝜋ℎ𝑟−𝐽𝑟,𝑚₂,𝑖𝑓𝑚₂<𝑟<𝑚₁,𝜋ℎ𝑟+𝐾𝑟,𝑚₁,𝑚₂,𝑖𝑓𝑟≥𝑚₁; (30) (5) if  𝑚₁<0<𝑚₂  and  |𝑚₁|=|𝑚₂|=𝑚, 𝐼₁(𝑟)=()𝜋ℎ𝑟,𝑖𝑓0<𝑟≤𝑚,𝜋ℎ𝑟−2𝐽𝑟,𝑚,𝑖𝑓𝑟>𝑚, (31) where 𝐽𝑟,𝑚_𝑖=ℎ𝑟arctan𝑟²−𝑚²_𝑖𝑚_𝑖−𝑚_𝑖𝑟²−𝑚²_𝑖𝑟², (32) for 𝑖=1,2, and 𝐾𝑟,𝑚₁,𝑚₂=ℎ𝑟arctan𝑟²−𝑚²₁𝑚₁−arctan𝑟²−𝑚²₂𝑚₂−𝑚₁𝑟²−𝑚²₁𝑟²+𝑚₂𝑟²−𝑚²₂𝑟². (33)

The proof of this lemma is given in the appendix.

Remark 6. If  𝑚₁<0<𝑚₂  and  |𝑚₁|=|𝑚₂|, system (1) can be transformed into the system which is studied in the paper [12].

Lemma 7. If  𝑚₁𝑚₂>0, one defines  𝑚=max(|𝑚₁|,|𝑚₂|),  𝑚=min(|𝑚₁|,|𝑚₂|)  and consider the equation  𝐼₁(𝑟)=𝑐𝑟, 𝑟>0  with  𝐼₁  given by (24), and  𝑐 is  a real parameter. Then, (1)if  𝑐<0  or  𝑐>ℎ(arctan(𝑚/𝑚)−arctan(𝑚/𝑚)), the equation has no solutions;(2)if  𝑐=0, then the interval  (0,𝑚]  is a continuum of solutions;(3)if 𝑐=ℎ(arctan(𝑚/𝑚)−arctan(𝑚/𝑚)), there is an unique solution  𝑟^∗=𝑚²+𝑚²;(4)if  𝑐∈(0,ℎ(arctan(𝑚/𝑚)−arctan(𝑚/𝑚))), there are two solutions  𝑟^∗₁<𝑚²+𝑚²  and  𝑟^∗₂>𝑚²+𝑚².

Proof. If  0<𝑚₁<𝑚₂, we have  𝑚=𝑚₂  and  𝑚=𝑚₁. It is easy to see that all  𝑟∈(0,𝑚]  are a solution if  𝑐=0. If  𝑐≠0  changing the variable  𝑢=𝑟²−𝑚²₁/𝑚₁  and defining  𝑔(𝑢)=𝐼₁(𝑟)/ℎ𝑟, we obtain the equivalent equation 𝑔(𝑢)=𝑢arctan𝑢−1+𝑢²,if𝑢∈0,𝑚²₂−𝑚²₁𝑚₁,𝑢arctan𝑢−1+𝑢²−arctan1+𝑢²𝑚²₁−𝑚²₂𝑚₂+𝑚₂1+𝑢²𝑚²₁−𝑚²₂𝑚²₁1+𝑢²,if𝑢∈𝑚²₂−𝑚²₁𝑚₁,+∞ (34) with simple computation; we find that the function  𝑔  is strictly monotonically increasing of variable  𝑢  when  𝑢∈(0,𝑚₂/𝑚₁)  and strictly monotonically decreasing when  𝑢∈(𝑚₂/𝑚₁,+∞). The function  𝑔  gets to the maximal value  𝑔_max=arctan(𝑚₂/𝑚₁)−arctan(𝑚₁/𝑚₂)  when  𝑢=𝑚₂/𝑚₁. Also we have  𝑓(𝑢)→0  as  𝑢→0  and  𝑢→+∞. The proof is similar if  𝑚₁<𝑚₂<0.

Lemma 8. If  𝑚₁𝑚₂<0, one defines  𝑚=min(|𝑚₁|,|𝑚₂|)  and  consider the equation  𝐼₁(𝑟)=𝑐𝑟,  𝑟>0  with  𝐼₁  given by (24), and  𝑐 is  a real parameter. Then,(1)if  𝑐<0  or  𝑐>𝜋ℎ, the equation has no solutions;(2)if  𝑐=𝜋ℎ, then the interval  (0,𝑚]  is a continuum of solutions;(3)if  𝑐∈(0,𝜋ℎ), there is an unique solution  𝑟^∗.

Proof. We only consider the case when  |𝑚₁|<|𝑚₂|: the proof is similar when  |𝑚₁|>|𝑚₂|  and  |𝑚₁|=|𝑚₂|. It is easy to see that all  𝑟∈(0,|𝑚₁|]  are a solution if  𝑐=𝜋ℎ. If  𝑐≠0  changing the variable  𝑢=𝑟²−𝑚²₁/𝑚₁  and defining  𝑔(𝑢)=𝐼₁(𝑟)/ℎ𝑟, we obtain the equivalent equation 𝑔(𝑢)=𝑢𝜋+arctan𝑢−1+𝑢²,if𝑚𝑢∈²²−𝑚²₁𝑚₁,𝑢,0𝜋+arctan𝑢−1+𝑢²−arctan1+𝑢²𝑚²₁−𝑚²₂𝑚₂+𝑚₂1+𝑢²𝑚²₁−𝑚²₂𝑚²₁1+𝑢²,if𝑢∈−∞,𝑚²₂−𝑚²₁𝑚₁. (35) With simple computation, we find that the function  𝑔  is strictly monotonically increasing of variable  𝑢. It is easy to know  𝑓(𝑢)→𝜋  as  𝑢→0  and  𝑓(𝑢)→0  as  𝑢→−∞.

With Lemma 5, we obtain the expressions for the components of function  ℎ, ℎ₁𝑟,𝜌,𝑠=𝑐₁𝑟+𝑐₂𝜌cos𝑠+𝑐₃𝜌sin𝑠+𝑏₁𝐼₁(𝑟),ℎ₂𝑟,𝜌,𝑠=𝑐₇𝜌+𝑐₅𝑟cos𝑠+𝑐₆𝑟sin𝑠+𝑏₃cos𝑠𝐼₁(𝑟),ℎ₃𝑟,𝜌,𝑠=𝑐₄+𝑐₃cos𝑠−𝑐₂𝜌sin𝑠𝑟+𝑐₆cos𝑠−𝑐₅𝑟sin𝑠𝜌−𝑏₃𝐼sin𝑠¹(𝑟)𝜌−𝑏₂𝐼₁(𝑟)𝑟, (36) where  𝑐_𝑖  are constants that depend linearly on  𝑎_𝑖𝑗𝑐₁=𝑎₁₁+𝑎₂₂𝜋,𝑐₂=𝑎₁₃+𝑎₂₄𝑐𝜋,₃=𝑎₁₄−𝑎₂₃𝜋,𝑐₄=𝑎₄₃+𝑎₁₂−𝑎₃₄−𝑎₂₁𝑐𝜋,₅=𝑎₃₁+𝑎₄₂𝜋,𝑐₆=𝑎₄₁−𝑎₃₂𝑐𝜋,₇=𝑎₃₃+𝑎₄₄𝜋. (37) According to Theorem 3 and Fact 1, for each simple zero  (𝑟^∗,𝜌^∗,𝑠^∗)  of (36) there is an isolated  2𝜋-periodic solution    𝜙(⋅,𝜀)  of system (19) with  |𝜀|≠0  sufficiently small such that  𝜙(⋅,𝜀)→(𝑟^∗,𝜌^∗,𝑠^∗)  as 𝜀→0. Any isolated  2𝜋-periodic solution of system (19) with  |𝜀|≠0  sufficiently small corresponds to a limit cycle of system (10). Thus, the most important task is to calculate the number of the simple zeros of function  ℎ. We solve the two first equations of (36), then, we get 𝐼₁(𝑟)=𝑘₂(𝑠)𝑑(𝑠)𝑘𝜌,𝑟=¹(𝑠)𝑑(𝑠)𝜌, (38) where 𝑑(𝑠)=𝑏₁𝑐₅−𝑏₃𝑐₁cos𝑠+𝑏₁𝑐₆𝑘sin𝑠,₁(𝑠)=𝑏₃cos𝑠𝑐₂cos𝑠+𝑐₃sin𝑠+𝑏₁𝑐₇,𝑘₂(𝑠)=𝑐₁𝑐₇−𝑐₂𝑐₅cos²𝑠−𝑐₃𝑐₆sin²𝑠−𝑐₂𝑐₆+𝑐₃𝑐₅sin𝑠cos𝑠. (39) Substituting (38) into the third equation, we obtain ℎ₃=𝑓(𝑠)𝑟,𝜌,𝑠𝑑(𝑠)𝑘₁(𝑠)=0, (40) where 𝑓(𝑠)=𝑐₄𝑑(𝑠)𝑘₁(𝑠)+𝑐₃cos−𝑐₂sin𝑠𝑑²(𝑠)+𝑐₆cos𝑠−𝑐₅sin𝑠𝑘²₁(𝑠)−𝑏₃sin𝑠𝑘₁(𝑠)𝑘₂(𝑠)−𝑏₂𝑑(𝑠)𝑘₂(𝑠). (41) It is necessary to study the zeros of  𝑓  instead of the zeros of  ℎ.

Lemma 9. The function  𝑓∶[0,2𝜋)→ℜ  given by formula (41) can have at most six isolated zeros, and they appear in pairs  {𝑠^∗,𝑠^∗+𝜋(mod2𝜋)}.

Proof. Substituting  cos𝑠=𝑥  and  sin𝑠=√1−𝑥²  in  𝑓(𝑠)=0  we get 𝐷₁𝑥+𝐷₃𝑥³+𝐷₀+𝐷₂𝑥²√1−𝑥²=0, (42) where 𝐷₀=𝑏²₁𝑐₂𝑐²₆−𝑐₄𝑐₆𝑐₇+𝑐₅𝑐²₇+𝑏₁𝑏₂𝑐₃𝑐²₆−𝑐₁𝑐₆𝑐₇,𝐷₁=𝑏²₁2𝑐₂𝑐₅𝑐₆−𝑐₃𝑐²₆−𝑐₄𝑐₅𝑐₇−𝑐₆𝑐²₇+𝑏²₃𝑐₁𝑐₃𝑐₇−𝑐₆𝑐²₃+𝑏₁𝑏₂2𝑐₂𝑐₅𝑐₆+𝑐₃𝑐²₅−𝑐₁𝑐₅𝑐₇,𝐷₂=𝑏²₁𝑐₂𝑐²₅−𝑐₂𝑐²₆−2𝑐₃𝑐₅𝑐₆+𝑏²₃𝑐₁𝑐₂𝑐₇+𝑐₂𝑐²₁−𝑐₁𝑐₃𝑐₄−2𝑐₂𝑐₃𝑐₆+𝑏₁𝑏₂2𝑐₂𝑐₅𝑐₆+𝑐₃𝑐²₅−𝑐₃𝑐²₆,𝐷₃=𝑏²₁𝑐₃𝑐²₆−𝑐₃𝑐²₅−2𝑐₂𝑐₅𝑐₆+𝑏²₃𝑐²₃𝑐₆−𝑐₃𝑐²₁−𝑐₁𝑐₂𝑐₄−𝑐₁𝑐₃𝑐₇−𝑐₆𝑐²₂+𝑏₁𝑏₂𝑐₂𝑐²₅−2𝑐₃𝑐₅𝑐₆−𝑐₂𝑐²₆. (43) When we consider the case  cos𝑠=𝑥  and  sin𝑠=−√1−𝑥²,  𝑓(𝑠)=0  becomes 𝐷₁𝑥+𝐷₃𝑥³−𝐷₀+𝐷₂𝑥²√1−𝑥²=0. (44) It follows that we have to find solutions of (42) or (44) in the interval  [−1,1]. This is equivalent to 𝐷₁𝑥+𝐷₃𝑥³−𝐷₀+𝐷₂𝑥²²1−𝑥²=0 (45) which is the polynomial equation 𝐷²₃+𝐷²₂𝑥⁶+2𝐷₁𝐷₃+2𝐷₀𝐷₂−𝐷²₂𝑥⁴+𝐷²₁+𝐷²₀−2𝐷₀𝐷₂𝑥²−𝐷²₀=0. (46) This equation can have at most six roots in the interval  [−1,1]. Then,  𝑓(𝑠)=0  has at most six solutions  𝑠∈[0,2𝜋).   Since  𝑓(𝑠+𝜋)=−𝑓(𝑠)  for all  𝑠∈[0,2𝜋), it is clear that if  𝑠^∗  is a zero of  𝑓  then  𝑠^∗+𝜋(mod2𝜋)  is also a zero.

The functions  𝑓(𝑠),  𝑑(𝑠),  𝑘₁(𝑠),  and  𝑘₂(𝑠)  have the properties  𝑓(𝑠+𝜋)=−𝑓(𝑠),  𝑑(𝑠+𝜋)=−𝑑(𝑠),  𝑘₁(𝑠+𝜋)=𝑘₁(𝑠),  and  𝑘₂(𝑠+𝜋)=𝑘₂(𝑠). So, we have 𝑘₁(𝑠)𝑑(𝑠)𝑘>0⟹¹(𝑠+𝜋)𝑑(𝑠+𝜋)<0. (47) Thus, the equation ℎ₃=0  at most three zeros that satisfy  𝑘₁(𝑠)/𝑑(𝑠)>0. With Lemma 7 for a fixed  𝑠^∗, we at most find two isolated value of  𝑟^∗  from  𝐼₁(𝑟)/𝑟=𝑘₂(𝑠^∗)/𝑑(𝑠^∗). With Lemma 8 for a fixed  𝑠^∗, we at most find one isolated value of  𝑟^∗  from  𝐼₁(𝑟)/𝑟=𝑘₂(𝑠^∗)/𝑑(𝑠^∗). For fixed  𝑠^∗  and fixed  𝑟^∗,  𝜌/𝑟=𝑘₁(𝑠^∗)/𝑑(𝑠^∗)  gives at most one isolated value for  𝜌^∗. Thus, we conclude that if  𝑚₁𝑚₂>0  the maximum number of limit cycles for system (1) is six, and if  𝑚₁𝑚₂<0  the maximum number of limit cycles for system (1) is three.

Remark 10. Using the main result of [18], the stability of the limit cycles associated with the solution  (𝑟^∗,𝜌^∗,𝑠^∗)  is given by the eigenvalues of the matrix 𝜕ℎ₁,ℎ₂,ℎ₃𝜕𝑟,𝜌,𝑠_{(𝑟,𝜌,𝑠)=(𝑟∗,𝜌∗,𝑠∗)}. (48) In order to show that there exist examples with exactly six limit cycles, we consider the following values of the coefficients: 𝑐₁=−1,𝑐₂=−4√33,𝑐₃𝑐=0,₄=−4√33,𝑐₅=0,𝑐₆=1,𝑐₇=−1,𝑏₁=1,𝑏₂=𝑏₃=𝑏₄𝑚=0,ℎ=6,₁=1,𝑚₂=√3. (49) More precisely, the system has the following form: 𝑥̇=𝐴₀𝑥+𝜀𝐴𝑥+𝜀𝜑𝑥₁𝑏, (50) where 𝜑𝑥₁=0,for𝑥₁∈(−∞,1),6𝑥₁−6,for𝑥₁∈1,√3,6√3−6,for𝑥₁∈√,𝐴3,+∞₀=,𝑎0−1001000000−10010𝐴=¹¹𝑎₁₂𝑎₁₃𝑎₁₄𝑎₂₁𝑎₂₂𝑎₁₄𝑎₂₄𝑎₃₁𝑎₃₂𝑎₃₃𝑎₃₄𝑎₄₁−𝑎₃₁𝑎₄₃𝑎₄₄,1000𝑏=, (51) and  𝑎_𝑖𝑗∈ℜ  satisfy 𝑎₁₁+𝑎₂₂𝜋=−1,𝑎₁₃+𝑎₂₄√𝜋=−433,𝑎₄₃+𝑎₁₂−𝑎₃₄−𝑎₂₁√𝜋=−433,𝑎₄₁−𝑎₃₂𝜋=1,𝑎₃₃+𝑎₄₄𝜋=−1. (52)
It is easy to know  6(arctan√3−arctan(√3/3))=𝜋. Computing the six solutions of  𝑓(𝑠)=0, we get  {𝜋/6,𝜋/3,𝜋/2,7𝜋/6,4𝜋/3,3𝜋/2}. The values of  𝑑(𝑠^∗),  𝑘₁(𝑠^∗)  and  𝑘₂(𝑠^∗)  are given in Table 1.

Table 1 

The values of 𝑠^∗, 𝑑(𝑠^∗), 𝑘₁(𝑠^∗), and 𝑘₂(𝑠^∗).

There are three values of  𝑠^∗  that satisfy  𝑘₁(𝑠^∗)/𝑑(𝑠^∗)>0  and  0<𝑘₂(𝑠^∗)/𝑘₁(𝑠^∗)<𝜋. These three solutions are  {𝜋/6,𝜋/3,𝜋/2}.

The six values of solution  𝑠^∗,  𝑟^∗,  𝜌^∗  and the value of the Jacobian at the solution  (𝑟^∗,𝜌^∗,𝑠^∗)  are given in Table 2.

Table 2 

The values of solution 𝑠^∗,𝑟^∗,𝜌^∗ and the Jacobian 𝐽ℎ(𝑟^∗,𝜌^∗,𝑠^∗).

4. Conclusion

In this paper, we have studied the limit cycles bifurcated from periodic orbits of a linear differential system in  ℜ⁴  when the perturbation is piecewise linear with two switching boundaries. We considered two classes of asymmetric perturbation. We have found that the perturbed system could have at most six limit cycles with one class of the asymmetric perturbation and three limit cycles with the other class of asymmetric perturbation, which generalized the result of paper [12].

Appendix

The Proof of Lemma 5

Case 1 (0<𝑚₁<𝑚₂). We have  |𝑟sin𝜃|≤𝑚₁  and  |𝑟cos𝜃|≤𝑚₁  for all  𝜃∈[0,2𝜋)  if  0<𝑟≤𝑚₁. Then,  𝜑(𝑟cos𝜃)=0  for every  𝜃. Thus, 𝐼₁(𝑟)=𝐼₂(𝑟)=0. (A.1) We now fix  𝑚₁<𝑟<𝑚₂  and consider  𝜃_𝑐∈(0,𝜋/2)  which satisfies  𝑟cos𝜃_𝑐=𝑚₁. Then, we have 𝐼₁(𝑟)=^𝜃𝑐₀ℎ𝑟cos𝜃−ℎ𝑚₁+cos𝜃𝑑𝜃_{2𝜋2𝜋−𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁cos𝜃𝑑𝜃=ℎ𝑟𝜃_𝑐−ℎ𝑚₁sin𝜃_𝑐,𝐼₂(𝑟)=^𝜃𝑐₀ℎ𝑟cos𝜃−ℎ𝑚₁+sin𝜃𝑑𝜃_{2𝜋2𝜋−𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁sin𝜃𝑑𝜃=0. (A.2) We now fix  𝑟≥𝑚₂  and consider  0<𝜃^∼_𝑐<𝜃_𝑐<𝜋/2  which satisfies  𝑟cos𝜃^∼_𝑐=𝑚₂. Then, we have 𝐼₁(𝑟)=^∼𝜃𝑐_{−∼𝜃𝑐}ℎ𝑚₂−ℎ𝑚₁+cos𝜃𝑑𝜃_{𝜃𝑐∼𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁+cos𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐2𝜋−𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁𝐼cos𝜃𝑑𝜃,₂(𝑟)=^∼𝜃𝑐_{−∼𝜃𝑐}ℎ𝑚₂−ℎ𝑚₁+sin𝜃𝑑𝜃_{𝜃𝑐∼𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁+sin𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐2𝜋−𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₁sin𝜃𝑑𝜃. (A.3) With simple computation, we get 𝐼₁(𝑟)=ℎ𝑚₂sin𝜃^∼_𝑐−ℎ𝑚₁sin𝜃_𝑐+ℎ𝑟𝜃_𝑐−ℎ𝑟𝜃^∼_𝑐,𝐼₂(𝑟)=0, (A.4) where sin𝜃_𝑐=𝑟²−𝑚²₁𝑟,sin𝜃^∼_𝑐=𝑟²−𝑚²₂𝑟,𝜃_𝑐=arctan𝑟²−𝑚²₁𝑚₁,𝜃^∼_𝑐=arctan𝑟²−𝑚²₂𝑚₂. (A.5)(A.6)

Case 2 (𝑚₁<𝑚₂<0). We have   |𝑟sin𝜃|≤|𝑚₂|  and  |𝑟cos𝜃|≤|𝑚₂|  for all  𝜃∈[0,2𝜋)  if  0<𝑟≤|𝑚₂|. Then,  𝜑(𝑟cos𝜃)=0  for every  𝜃. Thus, 𝐼₁(𝑟)=𝐼₂(𝑟)=0. (A.7) We fix now  |𝑚₂|<𝑟<|𝑚₁|  and consider  𝜃_𝑐∈(𝜋/2,𝜋)  which satisfies  𝑟cos𝜃_𝑐=𝑚₂. Then, we have 𝐼₁(𝑟)=^{2𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑟cos𝜃−ℎ𝑚₂cos𝜃𝑑𝜃=𝜋ℎ𝑟−ℎ𝑟𝜃_𝑐+ℎ𝑚₂sin𝜃_𝑐,𝐼₂(𝑟)=^{2𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑟cos𝜃−ℎ𝑚₂sin𝜃𝑑𝜃=0. (A.8) We now fix  𝑟≥|𝑚₁|  and consider  𝜃^∼_𝑐∈(𝜋/2,𝜋)  which satisfies  𝑟cos𝜃^∼_𝑐=𝑚₁. Obviously,  𝜃_𝑐<𝜃^∼_𝑐. Then, we have 𝐼₁(𝑟)=^∼𝜃𝑐_𝜃𝑐ℎ𝑟cos𝜃−ℎ𝑚₂+cos𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐∼𝜃𝑐}ℎ𝑚₁−ℎ𝑚₂+cos𝜃𝑑𝜃_{2𝜋−𝜃𝑐2𝜋−∼𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₂𝐼cos𝜃𝑑𝜃,₂(𝑟)=^∼𝜃𝑐_𝜃𝑐ℎ𝑟cos𝜃−ℎ𝑚₂+sin𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐∼𝜃𝑐}ℎ𝑚₁−ℎ𝑚₂+sin𝜃𝑑𝜃_{2𝜋−𝜃𝑐2𝜋−∼𝜃𝑐}ℎ𝑟cos𝜃−ℎ𝑚₂sin𝜃𝑑𝜃. (A.9) With simple computation, we get 𝐼₁(𝑟)=ℎ𝑚₂sin𝜃_𝑐−ℎ𝑚₁sin𝜃^∼_𝑐−ℎ𝑟𝜃_𝑐+ℎ𝑟𝜃^∼_𝑐,𝐼₂(𝑟)=0, (A.10) where sin𝜃_𝑐=𝑟²−𝑚²₂𝑟,sin𝜃^∼_𝑐=𝑟²−𝑚²₁𝑟,𝜃_𝑐=𝜋+arctan𝑟²−𝑚²₂𝑚₂,𝜃^∼_𝑐=𝜋+arctan𝑟²−𝑚²₁𝑚₁. (A.11)

Case 3 (𝑚₁<0<𝑚₂  and  |𝑚₁|<|𝑚₂|). We have  |𝑟sin𝜃|≤|𝑚₁|  and  |𝑟cos𝜃|≤|𝑚₁|  for all  𝜃∈[0,2𝜋)  if  0<𝑟≤|𝑚₁|.   Then,  𝜑(𝑟cos𝜃)=ℎ𝑟cos𝜃  for every  𝜃. Thus, 𝐼₁(𝑟)=₀^2𝜋ℎ𝑟cos²𝐼𝜃𝑑𝜃=𝜋ℎ𝑟,₂(𝑟)=₀^2𝜋ℎ𝑟cos𝜃sin𝜃𝑑𝜃=0. (A.12) We fix now  |𝑚₁|<𝑟<|𝑚₂|  and consider  𝜃_𝑐∈(𝜋/2,𝜋)  which satisfies  𝑟cos𝜃_𝑐=𝑚₁. Then, we have 𝐼₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑟cos²+𝜃𝑑𝜃_{2𝜋−𝜃𝑐𝜃𝑐}ℎ𝑚₁cos𝜃𝑑𝜃=ℎ𝑟𝜃_𝑐−ℎ𝑚₁sin𝜃_𝑐,𝐼₂(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑟sin𝜃cos𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑚₁sin𝜃𝑑𝜃=0. (A.13) We now fix  𝑟≥𝑚₂  and consider  𝜃^∼_𝑐∈(0,𝜋/2)  which satisfies  𝑟cos𝜃^∼_𝑐=𝑚₂. Then, we can write 𝐼₁(𝑟)=^∼𝜃𝑐_{−∼𝜃𝑐}ℎ𝑚₂cos𝜃𝑑𝜃+^𝜃𝑐_∼𝜃𝑐ℎ𝑟cos²+𝜃𝑑𝜃_{2𝜋−𝜃𝑐𝜃𝑐}ℎ𝑚₁cos𝜃𝑑𝜃+^{2𝜋−∼𝜃𝑐}_{2𝜋−𝜃𝑐}ℎ𝑟cos²𝜃𝑑𝜃,𝐼₂(𝑟)=^∼𝜃𝑐_{−∼𝜃𝑐}ℎ𝑚₂sin𝜃𝑑𝜃+^𝜃𝑐_∼𝜃𝑐+ℎ𝑟cos𝜃sin𝜃𝑑𝜃_{2𝜋−𝜃𝑐𝜃𝑐}ℎ𝑚₁sin𝜃𝑑𝜃+^{2𝜋−∼𝜃𝑐}_{2𝜋−𝜃𝑐}ℎ𝑟cos𝜃sin𝜃𝑑𝜃. (A.14)(A.15) With simple computation, we get 𝐼₁(𝑟)=ℎ𝑚₂sin𝜃^∼_𝑐−ℎ𝑚₁sin𝜃_𝑐+ℎ𝑟𝜃_𝑐−ℎ𝑟𝜃^∼_𝑐,𝐼₂(𝑟)=0, (A.16)(A.17) where sin𝜃_𝑐=𝑟²−𝑚²₁𝑟,sin𝜃^∼_𝑐=𝑟²−𝑚²₂𝑟,𝜃_𝑐=𝜋+arctan𝑟²−𝑚²₁𝑚₁,𝜃^∼_𝑐=arctan𝑟²−𝑚²₂𝑚₂. (A.18)(A.19)

Case 4 (𝑚₁<0<𝑚₂  and  |𝑚₁|>|𝑚₂|). We have  |𝑟sin𝜃|≤𝑚₂  and  |𝑟cos𝜃|≤𝑚₂  for all  𝜃∈[0,2𝜋)if  0<𝑟≤𝑚₂. Then,  𝜑(𝑟cos𝜃)=ℎ𝑟cos𝜃  for every𝜃. Thus, 𝐼₁(𝑟)=₀^2𝜋ℎ𝑟cos²𝐼𝜃𝑑𝜃=𝜋ℎ𝑟,₂(𝑟)=₀^2𝜋ℎ𝑟cos𝜃sin𝜃𝑑𝜃=0. (A.20) We fix now  𝑚₂<𝑟<|𝑚₁|and consider𝜃_𝑐∈(0,𝜋/2)which satisfies𝑟cos𝜃_𝑐=𝑚₂. Then, we have 𝐼₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚₂cos𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑟cos²𝜃𝑑𝜃=𝜋ℎ𝑟−ℎ𝑟𝜃_𝑐+ℎ𝑚₂sin𝜃_𝑐,𝐼₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚₂sin𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑟cos𝜃sin𝜃𝑑𝜃=0. (A.21) We now fix𝑟≥|𝑚₁|and consider  𝜃^∼_𝑐∈(𝜋/2,𝜋)  which satisfies  𝑟cos𝜃^∼_𝑐=𝑚₁. Then, we have 𝐼₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚₂cos𝜃𝑑𝜃+^∼𝜃𝑐_𝜃𝑐ℎ𝑟cos²+𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐∼𝜃𝑐}ℎ𝑚₁cos𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_{2𝜋−∼𝜃𝑐}ℎ𝑟cos²𝐼𝜃𝑑𝜃,₂(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚₂sin𝜃𝑑𝜃+^∼𝜃𝑐_𝜃𝑐+ℎ𝑟cos𝜃sin𝜃𝑑𝜃_{2𝜋−∼𝜃𝑐∼𝜃𝑐}ℎ𝑚₁sin𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_{2𝜋−∼𝜃𝑐}ℎ𝑟cos𝜃sin𝜃𝑑𝜃. (A.22) With simple computation, we get 𝐼₁(𝑟)=ℎ𝑚₂sin𝜃_𝑐−ℎ𝑚₁sin𝜃^∼_𝑐−ℎ𝑟𝜃_𝑐+ℎ𝑟𝜃^∼_𝑐,𝐼₂(𝑟)=0, (A.23) where sin𝜃_𝑐=𝑟²−𝑚²₂𝑟,sin𝜃^∼_𝑐=𝑟²−𝑚²₁𝑟,𝜃_𝑐=arctan𝑟²−𝑚²₂𝑚₂,𝜃^∼_𝑐=𝜋+arctan𝑟²−𝑚²₁𝑚₁. (A.24)

Case 5 (𝑚₁<0<𝑚₂  and  |𝑚₁|=|𝑚₂|=𝑚). We have  |𝑟sin𝜃|≤𝑚  and  |𝑟cos𝜃|≤𝑚  for all  𝜃∈[0,2𝜋)  if  0<𝑟≤𝑚.   Then,  𝜑(𝑟cos𝜃)=ℎ𝑟cos𝜃  for every  𝜃. Thus, 𝐼₁(𝑟)=₀^2𝜋ℎ𝑟cos²𝐼𝜃𝑑𝜃=𝜋ℎ𝑟,₂(𝑟)=₀^2𝜋ℎ𝑟cos𝜃sin𝜃𝑑𝜃=0. (A.25) We fix now  𝑟>𝑚and consider  𝜃_𝑐∈(0,𝜋/2)  which satisfies  𝑟cos𝜃_𝑐=𝑚. Then, we have 𝐼₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚cos𝜃𝑑𝜃+^{𝜋−𝜃𝑐}_𝜃𝑐ℎ𝑟cos²−𝜃𝑑𝜃_{𝜋+𝜃𝑐𝜋−𝜃𝑐}ℎ𝑚cos𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_{𝜋+𝜃𝑐}ℎ𝑟cos²𝐼𝜃𝑑𝜃,₁(𝑟)=^𝜃𝑐_−𝜃𝑐ℎ𝑚sin𝜃𝑑𝜃+^{𝜋−𝜃𝑐}_𝜃𝑐−ℎ𝑟cos𝜃sin𝜃𝑑𝜃_{𝜋+𝜃𝑐𝜋−𝜃𝑐}ℎ𝑚sin𝜃𝑑𝜃+^{2𝜋−𝜃𝑐}_{𝜋+𝜃𝑐}ℎ𝑟cos𝜃sin𝜃𝑑𝜃. (A.26) With simple computation, we get 𝐼₁(𝑟)=𝜋ℎ𝑟+2ℎ𝑚sin𝜃_𝑐−2ℎ𝑟𝜃_𝑐,𝐼₂(𝑟)=0, (A.27) where sin𝜃_𝑐=√𝑟²−𝑚²𝑟,𝜃_𝑐√=arctan𝑟²−𝑚²𝑚. (A.28) This completes the proof of the lemma.

Acknowledgment

This work is supported in part by the National Natural Science Foundation of China (1097 2082).