Sage Journals: Discover world-class research

Abstract

We study the bifurcation of periodic solutions for viscoelastic belt with integral constitutive law in 1: 1 internal resonance. At the beginning, by applying the nonsingular linear transformation, the system is transformed into another system whose unperturbed system is composed of two planar systems: one is a Hamiltonian system and the other has a focus. Furthermore, according to the Melnikov function, we can obtain the sufficient condition for the existence of periodic solutions and make preparations for studying the stability of the periodic solution and the invariant torus. Eventually, we need to give the phase diagrams of the solutions under different parameters to verify the analytical results and obtain which parameters the existence and the stability of the solution are based on. The conclusions not only enrich the behaviors of nonlinear dynamics about viscoelastic belt but also have important theoretical significance and application value on noise weakening and energy loss.

1. Introduction

The research about the nonlinear dynamics is one of the frontier application problems of the international dynamics area. Bifurcation of periodic solutions about nonlinear dynamics system is a powerful tool to investigate these problems. In recent years, there are also some progresses about periodic solutions of three-dimensional systems. In 1990, Chow et al. [1] studied a bifurcation of homoclinic orbits, which is an analogue of period doubling in the limit of infinite period. In 1991, Perdios et al. [2] investigated the families of three-dimensional periodic orbits which branch off the families of planar periodic around the triangular equilibrium point. In 1998, Mehri and Mahdavi-Amiri [3] showed that the reduced spatial three-body problem with one small mass is to the first approximation the product of the spatial restricted three-body problem and a harmonic oscillator by using the methods of symplectic scaling and reduction. In 2005, Liu and Han [4] considered a 3-dimensional system having an invariant surface, derived new formula of Melnikov function, and obtained sufficient conditions for the existence of periodic orbits. In 2009, Liu and Han [5] investigated the bifurcation of periodic solutions of a 4-dimensional system depending on a small parameter and gave a new method to use the Melnikov function. From 2009 to 2012, Llibre et al. [6 –9] studied the bifurcation of periodic solutions from a class of systems which has a 4-dimensional center in 1: n resonance, p: q resonance, a 4-dimensional center in Rⁿ in resonance 1: n as well as the 2n-dimensional center, respectively. In 2013, Liu et al. [10] investigated the double Hopf bifurcation at zero equilibrium point and simulated the periodic solutions and 3-dimensional torus near the double Hopf bifurcation.

Viscoelastic belt system is an irreplaceable transmission device in the mechanical system and the belt is the core link part of the mechanical equipment. From the vehicles’ original machinery to the modern automatic equipment, products go through several changes and have abundant uses, so scholars around the home and abroad pay more attention to the features of nonlinear dynamics. In 1985, Ulsoy et al. [11] studied the stability of parametric vibration of a moving belt with a tensioner. In 1988, Wickert and Mote [12] studied the vibration and stability of axially moving materials. In 1998, Zhang and Zu [13, 14] studied the nonlinear vibration and stability of a parametrically excited viscoelastic belt by using the multiscale method. In 2004, Zhang et al. [15] investigated the bifurcation of periodic solutions and chaotic dynamics for a parametrically excited viscoelastic moving belt with 1: 3 internal resonance and obtained that there exist periodic, 2-periodic, 3-periodic, 5-periodic, and quasiperiodic responses and chaotic motions in viscoelastic moving belt. In 2008, Liu et al. [16] investigated the problem of the transverse nonlinear nonplanar oscillations of an axially moving viscoelastic belt with the integral constitutive law in the case of 1: 1 internal resonance.

In this paper, we use the bifurcation theories of high-dimensional system to investigate the existence and stability of the periodic solution of the viscoelastic belt with two degrees of freedom. In Section 2, we introduce the viscoelastic belt system and make the nonsingular linear transformation. In Section 3, we present the methods to study the bifurcation of periodic solutions; what is more, the conditions of the existence and stability of the periodic solution are obtained. In Section 4, we give the phase diagrams of the solutions under different parameters to verify the analytical results in Section 3. In Section 5, we summarize our results.

2. The System and Nonsingular Linear Transformation

We get the nondimensional nonlinear dynamical formulations of the viscoelastic belt by using the method in [16]:

\begin{matrix} v_{t t}^{'} + 2 γ v_{t x}^{'} + (γ^{2} - 1 - a \cos ω t) v_{x x}^{'} + 2 μ v_{t}^{'} - N_{1} (w, v) = 0, \\ w_{t t}^{'} + 2 γ w_{t x}^{'} + (γ^{2} - 1 - a \cos ω t) w_{x x}^{'} + 2 μ v w_{t}^{'} - N_{1} (w, v) = 0, \end{matrix}

(1)

where

\begin{matrix} N_{1} (w, v) = \frac{3}{2} E_{e} {(v_{x}^{'})}^{2} v_{x x}^{'} + E_{e} w_{x}^{'} w_{x x}^{'} v_{x}^{'} + v_{x x}^{'} \int_{0}^{t} (\frac{1}{2} {(v_{x}^{'})}^{2} + \frac{1}{2} {(w_{x}^{'})}^{2}) \frac{\partial E}{\partial τ} d τ + v_{x}^{'} \int_{0}^{t} \frac{\partial E}{\partial τ} (v_{x}^{'} v_{x x}^{'} + w_{x}^{'} w_{x x}^{'}) d τ + \frac{1}{2} E_{e} {({w,}_{x})}^{2} v_{x x}^{'}, \\ N_{2} (w, v) = \frac{3}{2} E_{e} {({w,}_{x})}^{2} w_{x x}^{'} + E_{e} v_{x}^{'} v_{x x}^{'} w_{x}^{'} + w_{x x}^{'} \int_{0}^{t} (\frac{1}{2} {(v_{x}^{'})}^{2} + \frac{1}{2} {(w_{x}^{'})}^{2}) \frac{\partial E}{\partial τ} d τ + w_{x}^{'} \int_{0}^{t} \frac{\partial E}{\partial τ} (v_{x}^{'} v_{x x}^{'} + w_{x}^{'} w_{x x}^{'}) d τ + \frac{1}{2} E_{e} {(v_{x}^{'})}^{2} w_{x x}^{'} . \end{matrix}

(2)

By using the multiscale method, we obtain the average equations of viscoelastic belt in 1: 2 internal resonances as follows:

\begin{matrix} {\dot{x}}_{1} = - μ x_{1} + (σ_{1} + α) x_{2} + α_{1} x_{1} (x_{1}^{2} + x_{2}^{2}) + α_{2} x_{2} (x_{1}^{2} + x_{2}^{2}) + α_{4} x_{2} (x_{3}^{2} + x_{4}^{2}) + α_{5} x_{1} (x_{3}^{2} - x_{4}^{2}) + α_{3} x_{1} (x_{3}^{2} + x_{4}^{2}) - α_{6} x_{1} x_{3} x_{4} + α_{6} x_{2} (x_{3}^{2} - x_{4}^{2}) + α_{5} x_{2} x_{3} x_{4}, \\ {\dot{x}}_{2} = (- σ_{1} + α) x_{1} - μ x_{2} - α_{2} x_{1} (x_{1}^{2} + x_{2}^{2}) + α_{1} x_{2} (x_{1}^{2} + x_{2}^{2}) - α_{4} x_{1} (x_{3}^{2} + x_{4}^{2}) + α_{3} x_{2} (x_{3}^{2} + x_{4}^{2}) - α_{6} x_{1} (x_{3}^{2} - x_{4}^{2}) + α_{5} x_{1} x_{2} x_{3} - α_{5} x_{2} (x_{3}^{2} - x_{4}^{2}) + α_{6} x_{2} x_{3} x_{4}, \\ {\dot{x}}_{3} = - μ x_{3} + (σ_{2} + β) x_{4} + β_{1} x_{3} (x_{3}^{2} + x_{4}^{2}) + β_{2} x_{4} (x_{3}^{2} + x_{4}^{2}) + β_{3} x_{3} (x_{1}^{2} + x_{2}^{2}) + β_{4} x_{4} (x_{1}^{2} + x_{2}^{2}) + β_{5} x_{3} (x_{1}^{2} - x_{2}^{2}) - β_{6} x_{1} x_{2} x_{3} + β_{6} x_{4} (x_{1}^{2} - x_{2}^{2}) + β_{5} x_{1} x_{2} x_{4}, \\ {\dot{x}}_{4} = (- σ_{2} + β) x_{3} - μ x_{4} - β_{2} x_{3} (x_{3}^{2} + x_{4}^{2}) + β_{1} x_{4} (x_{3}^{2} + x_{4}^{2}) - β_{4} x_{3} (x_{1}^{2} + x_{2}^{2}) + β_{3} x_{4} (x_{1}^{2} + x_{2}^{2}) - β_{6} x_{3} (x_{1}^{2} - x_{2}^{2}) + β_{5} x_{1} x_{2} x_{3} - β_{5} x_{4} (x_{1}^{2} - x_{2}^{2}) + β_{6} x_{1} x_{2} x_{4}, \end{matrix}

(3)

where α_i, β_i are the combined coefficients.

We transform the system (3) by applying a nonsingular linear transformation.

Let

\begin{matrix} α \neq σ_{1}, β \neq σ_{2}, \\ \tilde{m} = \sqrt{μ^{2} - α^{2} + σ_{1}^{2}} > 0, \\ \tilde{n} = \sqrt{μ^{2} - β^{2} + σ_{2}^{2}} > 0 . \end{matrix}

(4)

Note the following linear transformation (TF) as

\begin{matrix} u_{1} = \frac{1}{\tilde{m}} ((α - σ_{1}) x_{1} + μ x_{2}), u_{2} = x_{2}, \\ v_{1} = \frac{1}{\tilde{n}} ((β - σ_{2}) x_{3} + μ x_{4}), v_{2} = x_{4}, \\ τ = \tilde{m} t . \end{matrix}

(5)

Then, by using the transformation (TF), system (3) has the form

\begin{matrix} \frac{d u_{1}}{d τ} = - u_{2} + M_{1 a} (u_{1}, u_{2}, v_{1}, v_{2}), \\ \frac{d u_{2}}{d τ} = u_{1} - b_{0100} u_{2} + M_{1 b} (u_{1}, u_{2}, v_{1}, v_{2}), \end{matrix}

(6a)

\begin{matrix} \frac{d v_{1}}{d τ} = - c_{0001} v_{2} + M_{2 c} (u_{1}, u_{2}, v_{1}, v_{2}), \\ \frac{d v_{2}}{d τ} = d_{0010} v_{1} - d_{0001} v_{2} + M_{2 d} (u_{1}, u_{2}, v_{1}, v_{2}), \end{matrix}

(6b)

where the expression of M_ji(u₁,u₂,v₁,v₂), j = 1, 2, 3, 4, i = a,b,c,d, is shown in Appendix B, and the coefficients are shown in Appendix A.

The systems are topologically equivalent after the nonsingular linear transformation, so we can study the existence of periodic solution of system (3) through studying it of system ((6a) and (6b)).

3. Bifurcation of Periodic Solution

In this section, we investigate the existence and stability of periodic solution of system ((6a) and (6b)).

3.1. Lemmas

In this part, we use the method in [14] to study the bifurcation of periodic solution of certain 4-dimensional system.

Consider the following C^r (r≥3) 4-dimensional system:

\frac{d x}{d τ} = f (x) + ε P (x, y, ε),

(7a)

\frac{d y}{d τ} = g (y) + ε Q (x, y, ε) .

(7b)

Suppose that the following conditions hold.

(A) The planar autonomous system

\frac{d x}{d t} = f (x)

(8)

is a Hamiltonian system with C^{r + 1} Hamiltonian H(x), and there exists an open interval J⊂R, such that system (8) has a family of periodic orbits ${L_{h} : h \in J}$ , where $L_{h} = {x : H (x) = h}$ .

(B) y = 0 is a focus of planar autonomous system

\frac{d y}{d t} = g (y) .

(9)

Without loss of generality, we assume that

D g (0) = (\begin{pmatrix} 0 & ω \\ - ω & 0 \end{pmatrix}) = B .

(10)

Assume that L_h has a parameter representation x = q(t,h), 0 ≤ t ≤ T(n), where T(h) denotes the period of L_h for h∈J. Let

\begin{matrix} G (θ, h) = q (\frac{T (h)}{2 π} θ, h), 0 \leq θ \leq 2 π, \\ M (r) = \int_{0}^{2 π} f (G (θ, r)) \land P (G (θ, r), 0,0) d θ, \end{matrix}

(11)

where (a₁,a₂)^T∧(b₁,b₂)^T = a₁b₂-a₂b₁, and we have the following.

Lemma 1. Suppose that $0 < | ε | \leq 1$ .

If M(r)≠0 for any r∈J, the system ((7a) and (7b)) has no periodic orbits with period near T(r) in a neighborhood of ${\bar{L}}_{r}$ .

If there exists h₀∈J, such that M(h₀) = 0, M′(h₀)≠0, and ωT(h₀)≠2kπ, then system ((7a) and (7b)) has a unique periodic orbit with period near T(h₀) in a neighborhood of ${\bar{L}}_{r}$ .

Let

\begin{matrix} d_{1} = \frac{T (h_{0})}{2 π} \int_{0}^{2 π} (Q_{1 y_{1}} (G (θ, h_{0}), 0,0) + Q_{2 y_{2}} (G (θ, h_{0}), 0,0)) d θ, \\ d_{2} = - \frac{T (h_{0})}{2 π} \int_{0}^{2 π} [F (G (θ, h_{0})) \land (P_{x} (G (θ, h_{0}), 0,0) G (θ, h_{0})) + (F (G (θ, h_{0})) G_{θ} (θ, h_{0})) \land P (G (θ, h_{0}), 0,0)] d θ . \end{matrix}

(12)

Lemma 2. If there exists h₀∈J, such that M(h₀) = 0, M′(h₀)≠0, T′(h₀) = 0, and $T (h_{0}) ω / 2 π$ is an irrational number. For $0 < | ε | ≪ 1$ , in a neighborhood of L_{h
₀}, one has the following:

if 2d₂ε<d₁ε<0, the periodic orbit L_ε of system ((7a) and (7b)) is asymptotically stable, and there is a nontrivial invariant torus S_2ε of system ((7a) and (7b)), which is unstable;

if d₂ε<0<d₁ε, the periodic orbit L_ε of system ((7a) and (7b)) is unstable, and there is a nontrivial invariant torus S_2ε of system ((7a) and (7b)), which is stable;

if d₁d₂≠0, d₁ε<2d₂ε, or 0<2d₂ε<d₁ε, the periodic orbit L_ε of system ((7a) and (7b)) is unstable, and there is a nontrivial invariant torus S_2ε of system ((7a) and (7b)), which is also unstable.

3.2. Dynamic Analysis of the System

In this section, we investigate the existence and stability of periodic solutions of system ((6a) and (6b)) through studying the bifurcation of periodic solutions of system ((18a) and (18b)). In system ((6a) and (6b)), let

u = {(u_{1}, u_{2})}^{T}, v = {(v_{1}, v_{2})}^{T},

(13)

and note that

\begin{matrix} f (u) = {(- u_{2}, u_{1})}^{T}, \\ g (v) = {(- c_{0001} v_{2} + M_{3 c} (u, v), d_{0010} v_{1} + M_{3 d} (u, v))}^{T}, \\ P (u, v, ε) = {(M_{1 a} (u, v), - b_{0100} u_{2} + M_{1 b} (u, v))}^{T}, \\ Q (u, v, ε) = {(M_{4 c} (u, v), - d_{0001} v_{2} + M_{4 d} (u, v))}^{T} . \end{matrix}

(14)

For simplicity, we use the following transformation:

\begin{matrix} b_{0100} ⟶ ε b_{0100}, d_{0001} ⟶ ε d_{0001}, \\ i_{m n p q} ⟶ ε i_{m n p q}, \end{matrix}

(15)

when

i = a, b, {(m + n)}^{p + q} = 1,

(16)

i = c, d, {(p + q)}^{m + n} = 1, m + n \neq 0 .

(17)

Then system ((6a) and (6b)) can be rewritten as follows:

\frac{d u}{d τ} = f (u) + ε P (u, v, ε),

(18a)

\frac{d v}{d τ} = g (v) + ε Q (u, v, ε) .

(18b)

The perturbed system ((18a) and (18b)) satisfies the following two qualities.

(A) The planar autonomous system

\frac{d u}{d τ} = f (u)

(19)

is a Hamiltonian system with the Hamiltonian H(u) = u₁² + u₂², and there exists an open interval J⊂R, such that system ((18a) and (18b)) has a family of periodic orbits ${L_{h} : h \in J}$ , where

L_{h} = {(u_{1}, u_{2}) : u_{1}^{2} + u_{2}^{2} = 2 h} .

(20)

(B) When β₁≠0, v = 0 is a 1-order weak focus of planar autonomous system

\begin{matrix} \frac{d v}{d τ} = g (v), \\ D g (0) = (\begin{pmatrix} 0 & - c_{0001} \\ d_{0010} & 0 \end{pmatrix}) . \end{matrix}

(21)

Since $c_{0001} = d_{0010} = \tilde{n} / \tilde{m}$ , systems (18a) and (18b) and (7a) and (7b) are of the same kind. We could use the method in Section 3.1 to study the bifurcation of periodic solution of system ((18a) and (18b)), that is, the existence of periodic solutions of system ((6a) and (6b)).

In system ((18a) and (18b)), by direct computation, we have

\begin{matrix} T (h) = 2 π, \\ G (θ, h) = \sqrt{2 h} {(\cos θ, \sin θ)}^{T} . \end{matrix}

(22)

By Lemma 1, we have that the sufficient condition for the existence of periodic solutions of the perturbed system ((18a) and (18b)) in a neighborhood of

Γ_{h_{0}} \equiv {(u_{1}, u_{2}, v_{1}, v_{2}) : u_{1}^{2} + u_{2}^{2} = 2 h_{0}, v_{1} = v_{2} = 0}

(23)

is that

\begin{matrix} M (h_{0}) = \int_{0}^{2 π} f (G (θ, h_{0})) \land P (G (θ, h_{0}), 0,0) d θ = - π {\sqrt{2 h_{0}}}^{2} b_{0100} - {\sqrt{2 h {}_{0}}}^{4} (\frac{3 π}{4} b_{0300} + \frac{π}{4} b_{2100}) - {\sqrt{2 h_{0}}}^{4} (\frac{π}{4} a_{1200} + \frac{3 π}{4} a_{3000}) = 0, \\ M^{'} (h_{0}) = - 2 b_{0100} - 2 h_{0} (3 π b_{0300} + π b_{2100}) - 2 h_{0} (π a_{1200} + 3 π a_{3000}) \neq 0, \\ ω = c_{0001} \neq k . \end{matrix}

(24)

Solving the equation M(h₀) = 0, we have

h_{0} = \frac{2 b_{0100}}{3 a_{3000} + a_{1200} + b_{2100} + 3 b_{0300}},

(25)

that is, in system ((18a) and (18b)), when

\begin{matrix} c_{0001} = d_{0010} \neq k, \\ h_{0} = \frac{2 b_{0100}}{3 a_{3000} + a_{1200} + b_{2100} + 3 b_{0300}}, h_{0} > 0, \end{matrix}

(26)

system ((7a) and (7b)) has a unique periodic orbit with period near T(h₀) in a neighborhood of

L_{h_{0}} = {(u_{1}, u_{2}) : u_{1}^{2} + u_{2}^{2} = 2 h_{0}} .

(27)

This periodic orbit is bifurcated from the periodic orbit of the unperturbed system.

In system ((18a) and (18b)),

\begin{matrix} d_{1} = 2 π (C h_{0} - d_{0001}), \\ d_{2} = π (A h_{0} - 2 b_{0100}), \end{matrix}

(28)

where

\begin{matrix} C = c_{0210} + c_{2010} + d_{0201} + d_{2001}, \\ A = 2 b_{2100} + 3 b_{0300} + 3 a_{3000} + 2 a_{1200} . \end{matrix}

(29)

By Lemma 2, when ω = c₀₀₀₁ is an irrational number, the stability of the periodic orbit L_ε in the neighborhood and the nontrivial invariant torus S_2ε is showed in Table 1.

Table 1:

The stability in system ((18a) and (18b)).

Number	Condition	Conclusion

1	2d₂ε<d₁ε<0	L_ε is asymptotically stable, and the nontrivial invariant torus S_2ε is unstable.

2	d₂ε<0<d₁ε	L_ε is unstable, and the nontrivial invariant torus S_2ε is stable.

3	d₁d₂≠0, d₁ε<2d₂ε或0<2d₂ε<d₁ε	L_ε is unstable, and the nontrivial invariant torus S_2ε is unstable.

4. Numerical Simulation

In this section, we present one group of phase diagrams of the unperturbed system and four groups of phase diagrams of the perturbed system of different parameters to verify the results and compare relative parameters to find out which relative parameter influences the existence or stability of the periodic solution.

For simplicity, we note

\begin{matrix} UPE = (μ, σ_{1}, α, α_{1}, α_{2}), \\ UPS = (σ_{2}, β, α_{3}, α_{4}, α_{5}, α_{6}, β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, β_{6}) . \end{matrix}

(30)

(1) When ε = 0, there is a family of closed orbits of system ((18a) and (18b)) on the plane v₁ = 0, v₂ = 0. In other words, there is a family of closed orbits of system (3) on the plane x₃ = 0, x₄ = 0. Refer to Figure 1.

Figure 1:

Periodic solutions of the unperturbed system.

(2) When ε≠0, the group of parameters of UPE is equal to UPE1, where

UPE 1 = (2,1, 4, - 3,2) .

(31)

By computation, there does not exist h0, such that M(h₀) = 0. Hence, there is no periodic orbit of system ((6a) and (6b)) under this group of parameters. Refer to Figure 2.

Figure 2:

Orbit of the unperturbed system under parameter conditions UPE1.

(3) When ε≠0, the group of parameters of UPE is equal to UPE2 and UPS is equal to UPS1, where

\begin{matrix} UPE 2 = (2,1, 2,2, - \frac{1}{2}), \\ UPS 1 = (2,1, - 2, - 2, - 3, - \frac{5}{2}, 3,2, - 1, - 2,4, - 1) . \end{matrix}

(32)

By computation, there exists h1, such that M(h₁) = 0, M′(h₁)≠0, and d1, d2 satisfy the condition of number 1 in Table 1, so there exists a periodic solution with period near 2π in a neighborhood of L_{h
₁}. The solution is asymptotically stable, and the nontrivial invariant torus is unstable. Refer to Figure 3.

Figure 3:

Periodic orbit of the unperturbed system under parameter conditions UPE2 and UPS1.

(4) When ε≠0, the group of parameters of UPE is equal to UPE2, and UPS is equal to UPS2, where

UPS 2 = (2,1, - 2, - 2, - 3, - \frac{5}{2}, 3,2, 1, - 2, - 4, - 1) .

(33)

By computation, there exists h1, such that M(h₁) = 0, M′(h₁)≠0, and d1, d2 satisfy the condition of number 2 in Table 1, so there exists a periodic solution with period near 2π in a neighborhood of L_{h
₁}. The solution is unstable, and the nontrivial invariant torus is stable. Refer to Figure 4.

Figure 4:

Periodic orbit of the unperturbed system under parameter conditions UPE2 and UPS2.

(5) When ε≠0, the group of parameters of UPE is equal to UPE2, and UPS is equal to UPS3, where

UPS 3 = (2,1, 2, - 2,3, \frac{5}{2}, - 3,2, 1,2, 4,1) .

(34)

By computation, there exists h1, such that M(h₁) = 0, M′(h₁)≠0, and d1, d2 satisfy the condition of number 3 in Table 1, so there exists a periodic solution with period near 2π in a neighborhood of L_{h
₁}. The solution is unstable, and the nontrivial invariant torus is unstable. Refer to Figure 5.

Figure 5:

Periodic orbit of the unperturbed system under parameter conditions UPE2 and UPS3.

In this section, we divide the parameters into two groups, noted UPE and UPS. When the group UPE is equal to UPE1, there is no periodic solution of system ((18a) and (18b)). When the group UPE is equal to UPE2, each system, with the different parameters of UPS, has a unique periodic solution which bifurcated from the same periodic orbit h1. When the group UPE is equal to UPE2, and the group UPS is equal to UPS1, UPS2, and UPS3, respectively, the stability of the periodic solution and the nontrivial invariant torus are different from each other. From above, we get that the group of UPE decides the existence and its relative location, and the group of UPS decides the stability of the periodic solution and the nontrivial invariant torus.

5. Conclusion

In this paper, we apply the bifurcation theory to research the existence and stability of the periodic solution of the viscoelastic belt system. At first, we transformed the average equations. Then according to Melnikov function, we can get existence conditions and relative location of periodic solution of the perturbed system and judge the periodic solution and the stability of nontrivial invariant torus. Finally by analyzing and comparing the relationship between parameters and the periodic solutions, we can come to the conclusion that the group of UPE influences existence and relative location of periodic solution, and the group of UPS affects the stability of the periodic solution and the nontrivial invariant torus. The results play a significant role in studying the periodic solution of the viscoelastic belt system.

Footnotes

Appendices

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China through Grant nos. 11072007, 11372014, and 11290152 and Beijing Natural Science Foundation through Grant no. 1122001.

References

Chow

S.-N.

Deng

, and Fiedler

, “Homoclinic bifurcation at resonant eigenvalues,” Journal of Dynamics and Differential Equations, vol. 2, no. 2, pp. 177–244, 1990.

Perdios

Zagouras

C. G.

, and Ragos

, “Three-dimensional bifurcations of periodic solutions around the triangular equilibrium points of the restricted three-body problem,” Celestial Mechanics and Dynamical Astronomy, vol. 51, no. 4, pp. 349–362, 1991.

Mehri

and Mahdavi-Amiri

, “Periodic solutions of certain three dimensional autonomous systems,” Physica D, vol. 112, pp. 117–120, 1998.

Liu

and Han

, “Poincaré bifurcation of a three-dimensional system,” Chaos, Solitons & Fractals, vol. 23, no. 4, pp. 1385–1398, 2005.

Liu

and Han

, “Bifurcation of periodic solutions and invariant tori for a four-dimensional system,” Nonlinear Dynamics, vol. 57, no. 1–2, pp. 75–83, 2009.

Llibre

and Makhlouf

, “Bifurcation of limit cycles from a 4-dimensional center in 1: n resonance,” Applied Mathematics and Computation, vol. 215, no. 1, pp. 140–146, 2009.

Llibre

Mereu

A. C.

, and Teixeira

M. A.

, “Limit cycles of resonant four-dimensional polynomial systems,” Dynamical Systems, vol. 25, no. 2, pp. 145–158, 2010.

Lima

M. F. S.

and Llibre

, “Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 9985–9996, 2011.

Barreira

Llibre

, and Valls

, “Bifurcation of limit cycles from a 4-dimensional center in R^m in resonance 1: N,” Journal of Mathematical Analysis and Applications, vol. 389, no. 2, pp. 754–768, 2012.

10.

Liu

, and Xu

, “Noise-induced chaos in the elastic forced oscillators with real-power damping force,” Nonlinear Dyanmics, vol. 71, no. 3, pp. 457–467, 2013.

11.

Ulsoy

A. G.

Whitesell

J. E.

, and Hooven

M. D.

, “Design of belt-tensioner systems for dynamic stability,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, vol. 107, no. 3, pp. 282–290, 1985.

12.

Wickert

J. A.

Jr. and Mote

C. D.

, “Current research on the vibration and stability of axially moving materials,” Shock and Vibration Digest, vol. 20, pp. 3–13, 1988.

13.

Zhang

and Zu

J. W.

, “Non-linear vibrations of viscoelastic moving belts—part I: free vibration analysis,” Journal of Sound and Vibration, vol. 216, no. 1, pp. 75–90, 1998.

14.

Zhang

and Zu

J. W.

, “Non-linear vibrations of viscoelastic moving belts—part II: forced vibration analysis,” Journal of Sound and Vibration, vol. 216, no. 1, pp. 93–105, 1998.

15.

Zhang

Wen

H. B.

, and Yao

M. H.

, “Periodic and chaotic oscillation of a parametrically excited viscoelastic moving belt with 1:3 internal resonance,” Acta Mechanica Sinica, vol. 36, no. 4, pp. 443–454, 2004.

16.

Liu

Y. Q.

Zhang

Gao

M. J.

, and Yang

X. L.

, “Transverse nonlinear nonplanar dynamics of an axially moving viscoelastic belt with integral constitutive law,” Acta Mechanica Sinica, vol. 40, no. 3, pp. 421–432, 2008.

Bifurcation of Periodic Solutions and Numerical Simulation for the Viscoelastic Belt

Abstract

1. Introduction

2. The System and Nonsingular Linear Transformation

3. Bifurcation of Periodic Solution

3.1. Lemmas

3.2. Dynamic Analysis of the System

4. Numerical Simulation

5. Conclusion

Footnotes

Appendices

Conflict of Interests

Acknowledgments

References