Hopf-Pitchfork Bifurcation in a Symmetrically Conservative Two-Mass System with Delay

1 Introduction

Vibrations of many practical engineering systems have attracted considerable interest from many researchers due to their practical importance as well as the numerous issues that arise in the context of linear and non-linear dynamics theories. The examples include the vibrations of elastic beams supported by two springs, milling machines and rotors [1], [2], [3], [4], [5]. The two-mass system with three non-linear springs is investigated by Cveticanin [6] as a famous model of nonlinear free vibration, see Figure 1.

Figure 1:

The two-mass system with three non-linear springs.

It is described by (1)

where the double dot denotes the second derivative with respect to time t, k and k₁ which are the linear coefficients of spring stiffness, and both k₂ and k₃ are the non-linear coefficients of spring stiffness.

Considering the case where the small damping exists, the mathematical model of the system becomes

Recently, time delays were incorporated into symmetric models by many authors [7], [8], [9], [10]. In Figure 1, the forces of the spring act on the objects at a moment t, but they do not cause the objects to move immediately. As only the multiple reflections of stress waves can make the acceleration relatively uniform, the objects begin to move. We shall refer to delay as the time interval from the time an object experiences a force to the time it starts moving. Following the idea in the literature, we give a symmetrically conservative two-mass system together with the non-linear, time-delayed connections leading to the system that we shall study [11].

Rewriting (3) as

Hopf-pitchfork bifurcation was investigated for a long time as an important dynamical behaviour [12], [13], [14], [15], [16], [17], [18], [19], [20]. There are some works on Hopf-pitchfork bifurcation in systems with delay. In 2015, Wang and Wang [21] investigated the Hopf-pitchfork bifurcation in a two-neuron system with discrete and distributed delays. In 2012, Dong et al. [22] studied the Hopf-pitchfork bifurcation in an inertial two-neuron system with time delay and they [23] also analysed the Hopf-pitchfork bifurcation in a simplified Bidirectional associative memory neural network model with multiple delays. The Hopf-pitchfork bifurcation in van der Pol’s oscillator with non-linear delayed feedback was considered in [24].

The rest of the article is organised as follows: in Section 2, we give the existence condition of the Hopf-pitchfork bifurcation with interaction coefficient and delay as two parameters. In Section 3, we obtain and analyse the normal form and the unfolding for Hopf-pitchfork bifurcation in the symmetrically conservative two-mass system with time delay, as well as the Hopf-pitchfork diagrams. In Section 4, some numerical simulations are given to support the analytical results. In the final section, we give some conclusions and future works.

2 The Existence of Hopf-Pitchfork Bifurcation

In the following, if the characteristic in (4) has a simple root 0 and a simple pair of purely imaginary roots±iω₀ and all other roots of the characteristic equation have negative real parts, then the Hopf-pitchfork bifurcation will occur. The linearisation equation of system (4) at the origin is

the characteristic equation of the system (5) is

By Corollary 2.3 in [25], we know that as τ varies, the sum of the order of zeros of H(λ) on the open right half-plane can change only if a zero appears on or crosses the imaginary axis. Thus, we can obtain the following results.

Lemma 2.1: If k+2k₁=0, ε>0, k>0, then all the roots of (6) have negative real parts except a single zero root and a pair of purely imaginary roots, where

τ0=1ω0arccos(ω02k), ω0=−ε2+ε4+4k22.

Proof: From (6), if k+2k₁=0, then we obtain that

H(λ)=[λ2+ελ+ke−λτ][λ2+ελ]=0.

If τ=0, then

H(λ)=[λ2+ελ+k][λ2+ελ]=0.

Obviously, we can obtain that if τ = 0 and ε>0, k>0, then all the roots of (6) have negative real parts except a single zero eigenvalue. When τ≠0, let λ=iω (ω>0) into λ²+ελ+ke^−λ=0 and by equating both the real and imaginary parts to zero, we obtain

By adding the squares of two equations of (7), we obtain

which clearly shows, (8) has a positive solution ω₀ with

ω0=−ε2+ε4+4k22,

then we can solve that

τk=1ω0[arccosω02k+2kπ], k=0, 1, 2, ….

By the above analysis, we obtain

Theorem 2.2: If k + 2k₁ = 0 and ε > 0, k > 0 hold, then system (4) undergoes a Hopf-pitchfork bifurcation at equilibrium (0, 0, 0, 0) when τ = τ_k, and τ_k is defined in Lemma 2.1.

3 Normal Form for Hopf-Pitchfork Bifurcation

In this section, center manifold theory and normal form method [26], [27], [28], [29] are used to study the Hopf-pitchfork bifurcation. After scaling t → t/τ, system (4) can be written as

Let τ=τ₀+μ₁ and k=−2k₁+μ₂, where μ₁ and μ₂ are bifurcation parameters and expand the function g, then system (9) becomes

Choosing the phase space C = C([−1, 0]; R⁴) with supreme norm and X_t ∈ C is defined by X_t(θ) = X(t + θ), −τ ≤ θ ≤ 0 and ||X_t|| = sup|X_t(θ)|. Then system (10) becomes

where

L(μ)Xt:=(τ0+μ1)(x2(t)(2k1−μ2)x1(t−1)−k1(x1(t−1)−x3(t−1))−εx2(t)x4(t)(2k1−μ2)x3(t−1)−k1(x3(t−1)−x1(t−1))−εx4(t)),

and

F(Xt, μ)=(0−k3(τ0+μ1)x13(t−1)−k2(τ0+μ1)(x1(t−1)−x3(t−1))30−k3(τ0+μ1)x33(t−1)−k2(τ0+μ1)(x3(t−1)−x1(t−1))3),

where L(μ)φ=∫−10dη(θ, μ)φ(ξ)dξ, for φϵC([−1, 0], R⁴),

η(θ, μ)={0,θ=0−(τ0+μ1)A1,θ∈(−1, 0)−(τ0+μ1)(A1+A2),θ=−1

with

A1=(01000−ε00000−ε), A2=(0000k10k10k10k10).

Considering the following linear system

X˙(t)=L()Xt,

Define the bilinear form between C and C′ = C([0, τ]), C^n*) by

(ψ(s), φ(θ))=ψ()φ()−∫−10∫0θψ(ξ−θ)dη(θ, 0)φ(ξ)dξ,

where

φ(θ)=(φ1(θ), φ2(θ), φ3(θ))∈C, ψ(s)=(ψ1(s)ψ2(s)ψ3(s))∈C∗.

Because L(0) has a simple 0 and a pair of purely imaginary eigenvalues ± iω₀ (ω > 0) and all other eigenvalues have negative real parts. Let Λ = {0, iω₀, −iω₀} and P be the generalised eigenspace associated with Λ and P^* is the space adjoint with P. Then the C can be decomposed as C = P⊕Q where Q = {φ ∈ C:(ψ, φ) = 0 for all ψ ∈ P^*}. Choosing the bases Φ and Ψ for P and P^* such that (Ψ(s), Φ(θ)) = I, Φ˙=ΦJ, and −Ψ=JΨ, where J = diag(0, iω₀, −iω₀).

By calculating we choose

Φ(θ)=(1eiτ0ω0θe−iτ0ω0θ0iω0eiτ0ω0θ−iω0e−iτ0ω0θ−1eiτ0ω0θe−iτ0ω0θ0iω0eiτ0ω0θ−iω0e−iτ0ω0θ), −1≤θ≤0

and

Ψ(s)= (D1εD1−D1ε−D1D2(iω0+ε)e−iτ0ω0sD2e−iτ0ω0sD2(iω0+ε)e−iτ0ω0sD2e−iτ0ω0sD¯2(−iω0+ε)eiτ0ω0sD¯2eiτ0ω0sD¯2(−iω0+ε)eiτ0ω0sD¯2eiτ0ω0s),

where

D1=12ε,D2={4ω0i+2ε+4τ0k1e−iτ0ω0}−1.

To consider system (11), we need to enlarge the space C to the following BC:

BC={φ is continuous functions on[−1, 0) and  limθ→0−φ(θ) exists}

Its elements can be written as ϕ = φ + Y₀c, with φ ∈ C, c ∈ R⁴ and

Y0(θ)={0,θ∈[−1, 0)I,θ=0.

In BC, (11) becomes an abstract ODE:

where u∈C and A is defined by

A=C1→BC, Au=u˙+Y0[L0u−u˙(0)],

and

F˜(u, μ)=[L(μ)−L0]u+F(u, μ).

Then the enlarged phase space BC can be decomposed as BC = P⊕Kerπ. Let Xt=Φz(t)+y˜(θ) where z(t) = (z₁, z₂, z₃)^T, namely

{x1(θ)=z1+eiτ0ω0θz2+e−iτ0ω0θz3+y1(θ),x2(θ)=iω0eiτ0ω0θz2−iω0e−iτ0ω0θz3+y2(θ),x3(θ)=−z1+eiτ0ω0θz2+e−iτ0ω0θz3+y3(θ),x4(θ)=iω0eiτ0ω0θz2−iω0e−iτ0ω0θz3+y4(θ).

Let

Ψ(0)=(ψ11ψ12ψ13ψ14ψ21ψ22ψ23ψ24ψ31ψ32ψ33ψ34)=(D1εD1−D1ε−D1D2(iω0+ε)D2D2(iω0+ε)D2D2¯(−iω0+ε)D2¯D2¯(−iω0+ε)D2¯)

Equation (12) is, therefore, decomposed into the system

where y˜(θ)∈Q1:=Q∩C1⊂Kerπ,AQ1 is the restriction of A as an operator from Q₁ to the Banach space Kerπ. Neglecting the higher-order terms with respect to parameters μ₁ and μ₂, (13) can be written as

(z˙1z˙2z˙3)=(ψ11ψ12ψ13ψ14ψ21ψ22ψ23ψ24ψ31ψ32ψ33ψ34)(F21+F31+O(∥x∥4)F22+F32+O(∥x∥4)F23+F33+O(∥x∥4)F24+F34+O(∥x∥4))

where

F12=μ1(iω0z2−iω0z3+y2(0)),F22=(2k1μ1−μ2τ0−μ1μ2)(z1+e−iτ0ω0z2+eiτ0ω0z3+y1(−1))− k1μ1(2z1+y1(−1)−y3(−1))−εμ1(iω0z2−iω0z3+y2(0)),F32=μ1(iω0z2−iω0z3+y4(0)),F42=(2k1μ1−μ2τ0−μ1μ2)(−z1+e−iτ0ω0z2+eiτ0ω0z3+y3(−1))− k1μ1(−2z1+y3(−1)−y1(−1))−εμ1(iω0z2−iω0z3+y4(0)),F23=−k3(τ0+μ1)(z1+e−iτ0ω0z2+eiτ0ω0z3+y1(−1))3− k2(τ0+μ1)(2z1+y1(−1)−y3(−1))3,F43=−k3(τ0+μ1)(−z1+e−iτ0ω0z2+eiτ0ω0z3+y3(−1))3− k2(τ0+μ1)(−2z1+y3(−1)−y1(−1))3.

According to [24], (Im(M21))c is spanned by

{z12e1, z2z3e1, z1μie1, μ1μ2e1, z1z2e2, z2μie2, z1z3e3, z3μie3}, i=1, 2

with e₁=(1, 0, 0)^T, e₂=(0, 1, 0)^T, e₃=(0, 0, 1)^T.

(Im(M31))c is spanned by

{z13e1, z1z2z3e1, z12z2e2, z22z3e2, z12z3e3, z2z32e3}

Then we get

g21(x, 0, μ)=Proj(Im(M21))cf21(x, 0, μ)=ProjS1f21(x, 0, μ) +O(|μ|2)g31(x, 0, μ)=Proj(Im(M31))cf˜31(x, 0, μ)=ProjS1f˜31(x, 0, 0) +O(|μ|2|x|+|μ||x|2)

where S₁ and S₂ are spanned, respectively, by

z1μie1, z2μie2, z3μie3, i=1, 2

and

z13e1, z1z2z3e1, z12z2e2, z22z3e2, z12z3e3, z2z32e3

On the center manifold, (11) can be transformed as the following normal form

z˙=Bz+12!g21(z, 0, μ)+13!g31(z, 0, 0)+h.o.t,

with

g31(z, 0, 0)=proj(Im(M31))cf31(z, 0, 0).

Following Theorem 2.1 in [30], we know that the dynamical behaviour of (11) near X_t=0 is governed by the general normal form of the third order

where

b11=0,b12=−2D1τ0,c11=D1(−2k3−16k2)τ0,c12=−12D1k3τ0,b21=D2(4k1e−iτ0ω0−2ω02),b22=−2D2τ0e−iτ0ω0,c21=−6D2k3τ0e−iτ0ω0,c22=−6D2k3τ0e−iτ0ω0.

Through the change of variables z₁=ω₁, z₂=ω₂+iω₃, z₃=ω₂−iω₃, and then a change to cylindrical coordinates according to ω₁=ζ, ω₂=rcosθ, ω₃=rsinθ, r>0, the system (14) becomes

Removing the azimuthal term and introducing the transformation ζ^=ζ|c11| and r^=r|Re(c22)|, (15) becomes, after dropping the hats,

where c₁ = Re(b₂₁)μ₁ + Re(b₂₂μ₂), c₂ = b₁₂μ₂.

If c₁₁ < 0 and Re(c₂₂) < 0, then (16) becomes

where σ=Re(c21)c11, δ=c12Re(c22).

In (17), M₀=(r, Z)=(0, 0) is always an equilibrium and the other equilibria are

M1=(c1, 0) for c1>0,M2±=(0, ±c2) for c2>0,M3±=(σc2−c1σδ−1, ±δc1−c2σδ−1) for σc2−c1σδ−1>0, δc1−c2σδ−1>0.

From [31], we know that if c₁₁ < 0 and Re(c₂₂) < 0, then (17) has five distinct types of unfolding with respect to different signs of σ, δ, σ, −δ, and σδ−1, which are shown in section 8.6.2 of [31], corresponding to Table 1.

By analysing, we know that only three cases I, II and III arise. In addition, we briefly list some results below:

1. System (17) undergoes a pitchfork bifurcation at the trivial equilibrium M₀ on the curves

   L1={(c1, c2):c1=0, c2≠0} and L2={(c1, c2):c2=0, c1≠0}.

2. System (17) undergoes a pitchfork bifurcation at the half-trivial equilibrium M2± and M₁, respectively, on the curves

L3={(c1, c2):c1=σc2, c2>0} and L4={(c1, c2):c2=δc1, c1>0}.

Similar to [21], we have the following theorem:

Theorem 3.1: The detailed dynamics of system (4) in D₁−D₈ near the original parameters (k^*, τ₀) are as follows:

1. In D₁, the trivial equilibrium (corresponding to M₀) becomes a source from a saddle, a pair of unstable semi-trivial equilibria (corresponding to M2±) appear, and the periodic orbit (corresponding to M₁) remains stable.

2. In D₂, the semi-trivial equilibria (corresponding to M2±) becomes stable from its unstable state, a pair of unstable periodic orbits (corresponding to M3±) appear, and the periodic orbit (corresponding to M₁) remains stable.

3. In D₃, the unstable periodic orbits (corresponding to M3±) disappear, the periodic orbit (corresponding to M₁) becomes unstable, and the semi-trivial equilibria (corresponding to M2±) remain stable.

4. In D₄, the periodic orbit (corresponding to M₁) disappears, the trivial equilibrium (corresponding to M₀) becomes a saddle from a source, and the semi-trivial equilibria (corresponding to M2±) remain stable.

5. In D₅, (4) has only one trival equilibrium M₀, which is a sink.

6. In D₆, the trivial equilibrium (corresponding to M₀) becomes a saddle from a sink, and a stable periodic orbit (corresponding to M₁) appears.

7. In D₇, (4) has a pair of stable periodic orbits (corresponding toM3±), a pair of unstable semi-trivial equilibria (corresponding toM2±), an unstable periodic orbit (corresponding to M₁), and an unstable trivial equilibrium (corresponding to M₀).

8. In D₈, (4) has a pair of stable periodic orbits (corresponding toM3±), an unstable periodic orbit (corresponding to M₁), and an unstable trivial equilibrium (corresponding to M₀).

Theorem 3.2: If the assumptions of Lemma 2.1 are satisfied, σ ≥ δ, σδ < 1, and k₃ > 0 hold, then system (2) undergoes a Hopf-pitchfork bifurcation of case II at equilibrium (0, 0, 0, 0), which is shown in Figure 2, where σ, δ are expressed as (16).

Figure 2:

Bifurcation diagrams of system (17) with parameter (c₁, c₂) around (0, 0).

Noticing that if case II arises, then the detailed dynamics of system (2) in D₁, D₂, D₃, D₅ and D₆ are the same as that in case I except in D₇. In D₇ system (2) has a pair of stable periodic orbits (corresponding toM3±), a pair of unstable semi-trivial equilibria (corresponding toM2±), an unstable periodic orbit (corresponding to M₁) and an unstable trivial equilibrium (corresponding to M₀).

In addition, τ=τ₀+μ₁ and k=−2k₁+μ₂, under original coordinates for (k₁, τ₀), the bifurcation critical lines are as follows:

L1:τ=Re(b22)Re(b21)(k+2k1)+τ0,

corresponding to

μ1=Re(b22)Re(b21)μ2;L2:k=−2k1

corresponding to

μ2=0;L3:τ=[Re(c21)b12c11Re(b21)−Re(b22)Re(b21)](k+2k1)+τ0,

corresponding to

μ1=[σb12Re(b21)−Re(b22)Re(b21)]μ2;L4:τ=Re(c22)b12−c12Re(b22)c12Re(b21)(k+2k1)+τ0

corresponding to

μ1=b12−δRe(b22)δRe(b21)μ2.

By the above analysis, we can obtain the bifurcation set. H₀ represents a branch curve (Figs. 3 and 4 ).

Figure 3:

Phase portraits in D1−D_8.

Figure 4:

The bifurcation set. H0=[Re(c21)b12c11Re(b21)−Re(b22)Re(b21)](k+2k1)+τ0.

4 Numerical Simulations

In this section, we give some examples to verify the theoretical results:

1. We chose k=2.01, k₁=−1, k₂=1, k₃=−1, (μ₁, μ₂)= (−0.05, 0.01). By direct calculation, we obtained ω₀=1.2535, τ₀=0.5372, D₁=0.5, D₂=0.0079–0.1570i, b₁₁=0, b₁₂=−0.5372, c₁₁=−3.7604, c₁₂=3.2232, b₂₁=0.3420+1.0039i; b₂₂=0.0985+0.1371i, c₂₁=c₂₂=−0.2956–0.4114i, Re(c₂₂)<0, c₁₁<0, c₁=−0.0161<0, c₂=−0.0054<0, σ=0.0786>0, δ=−10.9039.

   Here, the bifurcation critical lines are, respectively,

   L1:τ=0.2880(k+2)+0.5372, i.e. μ1=0.2880μ2;L2:k=2, i.e. μ2=0; L3:τ=−0.4115(k+2)+0.5372, i.e. μ1=−0.4115μ2, μ2>0;L4:τ=−0.1440(k+2)+0.5372, i.e. μ1=−0.1440μ2, μ2>0.

   By Table 1 and Theorem 3.1, D₅ of case III occurs. Figures 5 and 6 both show a stable trivial equilibrium.

2. We chose k = 1.99, k₁ = −1, k₂ = 1, k₃ = −1, (μ₁, μ₂) = (−0.05, −0.01). By direct calculation, we obtained ω₀ = 1.2457, τ₀ = 0.5430, D₁ = 0.5, D₂ = 0.0076–0.1573i, b₁₁ = 0, b₁₂ = −0.5430, c₁₁ = −3.8009, c₁₂ = 3.2579, b₂₁ = 0.3466+0.9979i, b₂₂ = 0.1005+0.1384i, c₂₁ = c₂₂ = −0.3015–0.4151i, Re(c₂₂) < 0, c₁₁ < 0, c₁ = −0.0183 < 0, c₂ = 0.0054>0, σ = 0.0793>0, δ = −10.8056 < 0, δc₁−c₂ = 0.1927, σc₂−c₁ = 0.0188>0. By Table 1 and Theorem 3.1, D₄ of case III occurs. Figures 7 and 8 both depict a pair of stable fixed points.

3. We chose k = 1.99, k₁ = −1, k₂ = 1, k₃ = −1, (μ₁, μ₂) = (0.05, −0.01). By direct calculation, we obtained ω₀ = 1.2457, τ₀ = 0.5430, D₁ = 0.5, D₂ = 0.0076–0.1573i, b₁₁ = 0, b₁₂ = −0.5430, c₁₁ = −3.8009, c₁₂ = 3.2579, b₂₁ = 0.3466+0.9979i, b₂₂ = 0.1005+0.1384i, c₂₁ = c₂₂ = −0.3015–0.4151i, Re(c₂₂) < 0, c₁₁ < 0, c₁ = 0.0163>0, c₂ = 0.0054>0, σ = 0.0793>0, δ = −10.8056 < 0, δc₁−c₂ = −0.1818 < 0, σc₂−c₁ = −0.0159 < 0. By Table 1 and Theorem 3.1, D₇ of case III occurs. Figures 9 and 10 both depict a pair of stable periodic orbits.

Figure 5:

Wave plot of x₁, the trivial equilibrium of system (4) is locally stable in D₅: (μ₁, μ₂) = (−0.05, 0.01) with the initial values (0.2, 0.2, 0.04, −0.4) for the red line and (−0.2, −0.2, −0.04, 0.4) for the blue line, respectively.

Figure 6:

Phase plane of (x₂, x₃, x₄).

Figure 7:

Wave plot of x₁, two stable nontrivial equilibria of system (4) coexist in D₄: (μ₁, μ₂) = (−0.05, −0.01) with the initial values (0.2, 0.2, 0.04, −0.4) for the red line and (−0.2, −0.2, −0.04, 0.4) for the blue line.

Figure 8:

Phase plane of (x₂, x₃, x₄).

Figure 9:

Wave plot of x₄, a pair of stable periodic orbits of system (4) in D₇: (μ₁, μ₂) = (0.05, −0.01) with the initial values (0.6, −0.65, 0.6, −0.65) for the red line and (−0.6, 0.65, −0.6, 0.65) for the blue line.

Figure 10:

Phase plane of (x₁, x₂, x₃).

Therefore, it is clear that numerical simulations agree with analytical predictions.

5 Conclusions

In this paper, the two-mass system with three non-linear springs and time delay is considered. Our contributions include the following:

1. By analysing the distribution of the eigenvalues of the characteristic equation of its linearised equation, we find the critical values for the occurrence of Hopf-pitchfork bifurcation.

2. By using the normal form method and the center manifold theorem, we obtain the codimension-2 unfolding with original parameters for Hopf-pitchfork bifurcation in which the unfolding form for case III is seldom given in delayed differential equations. Furthermore, by analysing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviours of the original system are found, such as the coexistence of two stable nontrivial equilibria and a pair of stable periodic orbits.

We only show the unfolding for Hopf-pitchfork bifurcation in which Re(c₂₂)<0 and the corresponding simulations. In the future work, we can investigate the case for Re(c₂₂)>0, as well as the other dynamical behaviours of the system.