1 Introduction

The double Hopf bifurcation is a codimension-2 bifurcation and has two Hopf bifurcation curves intersecting transversely at the singular point. Early studies on the Hopf bifurcation and its applications have been done by Marsden et al.^[1]. The non-resonant double Hopf bifurcation was studied by many researchers. Nayfeh and Chin^[2] studied the double Hopf bifurcation. With the aid of the perturbation technique, Yu^[3] computed the normal form of a general nonlinear system associated with the double Hopf bifurcation. Chamara and Coller^[4] studied the double Hopf interaction in a system of two airfoils using the theories of normal forms and center manifolds. Yu and Bi^[5] considered the double Hopf bifurcation in a nonlinear system including resonant and non-resonant cases using the normal form theory. Xie and Ding^[6] analyzed the double Hopf bifurcation of a vibro-impact system by using the method of Poincare map. They found that the vibro-impact system presents different types of complicated invariant tori when two parameters vary near the double Hopf bifurcation points. Revel et al.^[7] investigated the double Hopf bifurcation of a coupled electric circuit system and obtained several bifurcation diagrams associated using the truncated normal form.

The resonant double Hopf bifurcation will lead to more complex phenomena, such as tori and periodic orbits, period doubling, homoclinic and heteroclinic connections, and chaos. Using the methods of Melnikov and Petrov, Zhang and Yu^[8] studied the codimension-3 degenerate bifurcations of the generalized Lenard oscillator. Zhang and Yu^[9] investigated the homoclinic and heteroclinic bifurcations and the local bifurcations of a general parametrically and externally excited mechanical system. In particular, a degenerate bifurcation of codimension-3 was studied with the aid of multiple scales and the normal form theory. Using the normal form theory, Yu^[10] analyzed the double Hopf bifurcation and divided the parameter space into several regions. Govaerts et al.^[11] investigated the singularity in the unfolding of a 1:1 resonant double Hopf bifurcation. Namachchivaya et al.^[12] gave an explicit formula of Poincar\'{e}-Birkhoff normal form for the generalized Hopf bifurcation in the case of non-semisimple 1:1 resonance. Zhang and Ye^[13] dealt with the global and local bifurcations of nonlinear oscillations of a valve mechanism at an internal combustion engine system using the normal form theory. Using the Lyapunov direct method, Ge et al.^[14] obtained the Hopf bifurcation and other dynamic behaviors of a rotational machine system.

Various problems with resonant double Hopf bifurcations in higher dimensional nonlinear dynamical systems were studied. Gattulli et al.^[15-16] studied the double Hopf bifurcation of non-resonant and 1:1 resonant types with the multiple scale method. Based on the method of multiple scales, Zhang and Huo^[17] analyzed the bifurcations in the nonlinear oscillation system which is under the combined parametric and forcing excitations for 1/2 subharmonic resonance-primary parametric resonance, and got eight different diagrams of bifurcating response curves.

Some applications in mechanical systems for resonant double Hopf bifurcation were considered by many researchers. Luongo et al.^[18] used the multiple scale method to consider the double Hopf bifurcation of a general higher dimensional system. Le Blanc^[19] studied some cases of resonance double Hopf bifurcations in a nonlinear dynamical system.

Composite piezoelectric plates have been widely used in aerospace and aerospace engineering, with the advantages of good control ability, high specific strength, high specific stiffness, good fatigue resistance, and so on. The domestic and foreign scholars have done a lot of work in the field of piezoelectric shell dynamics and vibration control. Tzou^[20] obtained the governing equations of piezoelectric laminated shell and analyzed the governing equations using the Lyapunov stability theory and direct feedback method. Subsequently, Tzou^[21] established the dynamic model of the thin shell with distributed piezoelectric actuators. Tzou and Zhong^[22] derived the governing equations of the piezoelectric shell using the first-order shear theory. Pratt and Nayfeh^[23] used a new biaxial vibration control system to study the self excited vibration of flexible rod in the process of forming, and the curve diagram of the variation of the amplitude of the work piece with the cutting depth was given.

Zhang et al.^[24] studied the nonlinear dynamic responses of the composite laminated piezoelectric plate under the combined action of lateral load, in-plane load, and the excitation pressure electric using the normal form theory and the energy phase method. Zhang et al.^[25] investigated the dynamic model of a composite laminated piezoelectric rectangular plate, and considered the periodic motion and chaotic dynamic response of the composite laminated piezoelectric plate in the case of 1:3 internal resonances. Furthermore, Zhang et al.^[26] studied the bifurcation and chaotic dynamics of a composite laminated piezoelectric rectangular plate in the case of 1:2 internal resonances.

In this paper, our work aims at investigating the double Hopf bifurcation of such a composite laminated piezoelectric plate and showing how the singular points bifurcate to many forms of the solutions. We first transform a two-degree-of-freedom nonlinear system into autonomous equations in two different coordinates using the method of multiple scales. The bifurcation response equations are obtained for the composite laminated piezoelectric plate with the primary parameter resonance, i.e., 1:3 internal resonance. The bifurcation diagram is achieved on the parameter plane σ₂-μ₂ using the singularity theory. Furthermore, the different steady state solutions of the average equations are analyzed. By numerical simulation, the composite laminated piezoelectric plate can present the periodic vibration and quasi-periodic vibrations.

2 Double Hopf bifurcation of composite laminated piezoelectric plate with primary parameter resonance, i.e., 1:3 internal resonance

We consider a composite laminated piezoelectric plate with four edges simply supported, whose length, width, and height are a, b, and h, respectively. The piezoelectric plate is considered as regular symmetric cross-ply laminates with n layers. Some of these layers are made of the piezoelectric polyvinylidene fluoride (PVDF) materials as actuators, while the others are made of fiber-reinforced composite materials. The coordinate system is constructed on the mid-plane, as shown in Fig. 1.

Assume that the displacements of any point in the piezoelectric plate are u, v, and w along the directions of x, y, and z, respectively. The displacements of any point on the middle surface of the piezoelectric plate are u₀, v₀, and w₀ along the directions of x, y, and z, respectively. It is assumed that the in-plane excitations of the piezoelectric plate are loaded along the y-direction at x=0 and the x-direction at y=0 with the form of q₀ +q_x cos (Ω₁t) and q₁ +q_y cos (Ω₂t), respectively. The transverse excitation, which loads to the composite laminated piezoelectric plate, is represented by q=q₃ cos (Ω₃t). The dynamic electrical loading is expressed as E₃=E_z cos (Ω₄t). According to Ref. [26], we obtain a two-degree-of-freedom nonlinear system for the composite laminated piezoelectric plate,

(1a)

(1b)

We use the singularity theory to study the double Hopf bifurcation for the piezoelectric plates under the combined effects of external and internal excitations. Assume that the nonlinear system (1) is a weakly nonlinear system, and add a small perturbation ε to the damping terms, parametric excitation terms, external excitation terms, and nonlinear terms. Then, we obtain the following equations:

(2a)

(2b)

Consider the primary parametric resonance, i.e., 1:3 internal resonance of the composite laminated piezoelectric plate. The resonance relationship is

(3)

where σ₁ and σ₂ are two detuning parameters.

For convenience of analysis, let Ω =3. Using the method of multiple scales^[27], the averaged equation of the composite laminated piezoelectric plate in the polar form is obtained as follows:

(4a)

(4b)

The averaged equation in the Cartesian form can be expressed as

(5a)

(5b)

(5c)

(5d)

Let

(6)

Put (6) into (4) . Then, the averaged equation of the composite laminated piezoelectric plate in the polar form is obtained as

(7a)

(7b)

(7c)

(7d)

In order to study the steady state solutions to (7) , namely, the periodic solutions or quasi-periodic solutions to (2) , let the left-hand side of (7) equal zero. Then, the bifurcation response equation of (2) can be written as follows:

(8a)

(8b)

Let and . Then, (8) can be represented as

(9a)

(9b)

Since (9a) has one zero solution, it can be further simplified as

(10a)

(10b)

Next, we will consider the various different steady state solutions to (9b), (10) and their changes along with the variation of the parameters μ₁, μ₂, σ₁, σ₂, α_i (i=2, 3, …, 7), and β_j (j=6, 7, 8). For convenience of analysis, we take a₂ =1 and a₁ =1 in (10a) and (9b), respectively. Then, (10a) and (9b) can be written as

(11a)

(11b)

After simplification, the bifurcation response equations of the composite laminated piezoelectric plate can be represented as

(12a)

(12b)

Let

(13)

Using the condition equation (13) , the different steady state solutions to the bifurcation response equation (12a) are obtained.

Since k₁₁≥0 and k₁₃≥0, if the following condition is satisfied, namely,

(14)

there are three solutions to (12a), which can be written as

(15a)

(15b)

(15c)

In other conditions, there is only one zero solution to (12a).

Put (13) into (12) . Then, the different steady state solutions to the bifurcation response equation (12b) are obtained.

If the following conditions are satisfied, namely,

(16)

there are three solutions to (12b), which can be written as

(17)

If △₂＞0, namely,

(18)

and

(19)

there are two solutions to (12b), which can be written as

(20)

with

(21)

If △₂ =0 and B≠0, namely,

(22)

and furthermore, the following conditions are also satisfied:

(23)

then there is only one solution to (12b), which can be written as

(24)

If we have the following conditions:

(25)

then we obtain three solutions to (12b), which can be written as

(26)

If the following conditions are satisfied:

(27)

there will be two solutions to (12b), which can be written as

(28)

If △₂<0, namely,

(29)

and

(30)

there will be only one solution to (12b), which can be written as

(31)

If the following condition is satisfied:

(32)

then there will be two solutions to (12b), which can be written as

(33)

If the following conditions are satisfied:

(34)

there will be three solutions to (12b), which can be written as

(35)

where

(36)

In order to examine the stability of the steady state solutions to (7) , we should consider the Jacobi matrix at the singular points of (5) . At the zero point, the Jacobi matrix of (5) is

(37)

The characteristic polynomial of (5) is

(38)

Four eigenvalues of (5) are expressed as

(39)

The Jacobi matrix of (5) at the nontrivial point is obtained as

(40)

The characteristic equation is

(41)

In the convenient analysis, (41) can be expressed as

(42)

in which the coefficients in (42) can be obtained in Appendix A.

Here, the eigenvalues of (5) can be written as follows:

(43)

in which the coefficients in (43) can be obtained in Appendix A.

Then, we obtain the steady non-zero solutions which can be written as follows:

(44)

It is clearly seen that the inequality (44) can be satisfied by the solutions given in (15) , (18) , (24) , and (31) , while the inequality (44) cannot be satisfied by the solutions given in (28) and (33) .

Based on the above analysis, the parameter plane σ₂-μ₂ can be divided into different zones by the steady critical curves given by the steady state solutions. The local bifurcation set is constituted by all the curves, as shown in Fig. 2.

3 Numerical simulations

Using the MAPLE program, we study different bifurcations of the composite laminated piezoelectric plate subjected to the combined effects of incentives and lateral excitations.

According to the aforementioned analysis, we choose the following parameters and initial conditions for (5) :

The bifurcation diagram is depicted for the composite laminated piezoelectric plate in the parameter plane σ₂-μ₂, as shown in Fig. 2. Next, we analyze the bifurcations for different steady state solutions to (5) .

In the areas A₂ and A₃, the zero solution is a unique singular point. From (43) , we know that the singular point is a stable focus so that the zero solution is stable. When crossing the lines L₁ and L₂ into the regions C₁ and C₂, two non-zero solutions fork from the above zero solution, which can be expressed by (20) , while the two non-zero solutions are unstable. The system presents pitchfork bifurcation at these points.

When crossing the line L₄ from the areas C₁ and C₂ into the areas B₁ and B₂, the steady solution changes its stability, a pair of complex conjugate eigenvalues of (5) cross the imaginary axis and become a pair of simple pure imaginary eigenvalues, the system has Hopf bifurcation at the singular points, and limit cycles occur from the initial points. The corresponding solutions can be written as (33) .

When crossing the line L₃ from the areas C₁ and C₂ into the regions D₁ and D₂, the steady solution also changes its stability, the system has Hopf bifurcation at the singular points, and limit cycles occur from the initial points. The corresponding solutions can also be written as (33) .

When crossing the line L₄ from the area A₂ into the area A₁, the stability of the singular point is sensitively dependent on the changes of the parameters, the system has pitchfork bifurcation on the line L₄, and two non-zero solutions occur from the zero solution which can be expressed by (28) .

When crossing the line L₅ from the area A₃ into the area A₄, the system has pitchfork bifurcation on the line L₅, and other two non-zero solutions occur from the zero solution which can also be expressed by (28) .

When crossing the lines L₆ and L₇ from the areas D₁ and D₂ into the regions E₁ and E₂, the number of the steady state solutions remains unchanged, but its stability changes.

When crossing the line L₅ from the areas E₁ and E₂ into the areas F₁ and F₂, two pairs of complex conjugate eigenvalues of (5) cross the imaginary axis and become two pairs of simple pure imaginary eigenvalues, and the system has double Hopf bifurcation on the line L₅. In the areas F₁ and F₂, (26) and (31) are satisfied at the same time, and the system has four non-zero solutions which can be expressed by (28) and (33) .

When crossing the lines L₆ and L₇ from the area A₄ into the regions G₁ and G₂, one eigenvalue of the system becomes zero on the lines L₆ and L₇, and the system generates saddle-node bifurcation. In the regions G₁ and G₂, the system has three non-zero solutions which can be expressed by (35) .

Figures 3-7 represent different forms of nonlinear vibration characteristics of the composite laminated piezoelectric plate. In the following figures, diagrams (a) and (b) represent the two-dimensional phase diagrams on the planes (x₁, x₂) and (x₃, x₄ ), respectively. Graphs (c) and (d) are the waveform diagrams on the planes (t, x₁) and (t, x₃), respectively. Diagrams (e) and (f) are the three-dimensional phase diagrams in the spaces (x₁, x₂, x₃) and (x₃, x₄, x₁), respectively.

Select the damping coefficient and tuning parameters of the piezoelectric composite laminates to be μ₂ =0.11 and σ₂=0.12, respectively. The other parameters and initial value of the piezoelectric composite laminates are the same as those in Fig. 2. We get the nonlinear vibration near the equilibrium solutions of the piezoelectric composite laminates, as shown in Fig. 3. From Figs. 3(a) and 3(b), we know that the solutions of the system tend to zero solution. Then, it is asymptotically stable.

In Fig. 4, the damping coefficient and tuning parameters of the composite laminated piezoelectric plate are μ₂ =0.11 and σ₂=0.23, respectively. The other parameters and initial values of the piezoelectric composite laminates are the same as those in Fig. 3. The equilibrium solution of the piezoelectric composite laminates generates Hopf bifurcation. The nonlinear vibration of the piezoelectric composite laminates is periodic motion.

When the damping coefficient and tuning parameters of composite laminated piezoelectric plate are μ₂ =0.26 and σ₂=0.26, respectively, the equilibrium solutions also generate Hopf bifurcation, and the nonlinear vibration of the piezoelectric plates is also periodic motion, as shown in Fig. 5.

Change the parameters of the composite laminated piezoelectric plate. Take the damping coefficient and tuning parameters of the composite laminated piezoelectric plate as μ₂ =0.57, σ₂=0.29 and μ₂ =0.68, σ₂=0.49, respectively. The equilibrium solutions generate double Hopf bifurcation, and the nonlinear vibration of the piezoelectric plates is quasi-periodic motion, as shown in Figs. 6 and 7.

4 Conclusions

The double Hopf bifurcation of a composite laminated piezoelectric plate is studied under the combined parametric excitation and external excitation in the primary parametric resonance, i.e., 1:3 internal resonance. Using a multiple scale method, the average equations of the composite laminated piezoelectric plate are obtained. Furthermore, the bifurcation equations of the composite laminated piezoelectric plate are achieved. Through the analysis of the bifurcation equations of the system, the nonlinear dynamic responses of the composite laminated piezoelectric plate are analyzed on the parameter plane based on the bifurcation diagram shown in Fig. 2. The theoretical and numerical results indicate that the Hopf bifurcation and double Hopf bifurcation occur for the composite laminated piezoelectric plate.

Further research will be carried out based on this paper, such as detailed analysis of the local and global bifurcations of the composite laminated thin plate, and the investigation of the classification and unfolding of the composite laminated thin plate system.

Appendix A

The coefficients given in (42) are

where

The parameters given in (43) can be expressed as

Acknowledgements The authors gratefully acknowledge the support of the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the jurisdiction of Beijing.