Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2013, Article ID 790946, 14 pages
http://dx.doi.org/10.1155/2013/790946
Research Article
Deterministic and Stochastic Bifurcations of the Catalytic CO
Oxidation on Ir(111) Surfaces with Multiple Delays
Zaitang Huang and Weihua Lei
School of Mathematical Sciences, Guangxi Teachers Education University, Nanning 530023, China
Correspondence should be addressed to Zaitang Huang; zaitanghuang@163.com
Received 27 August 2012; Revised 19 November 2012; Accepted 18 December 2012
Academic Editor: Shukai Duan
Copyright © 2013 Z. Huang and W. Lei. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The main purpose is to investigate both deterministic and stochastic bifurcations of the catalytic CO oxidation. Firstly, super- and
subcritical bifurcations are determined by the signs of the Poincaré-Lyapunov coefficients of the center manifold scalar bifurcation
equations. Secondly, we explore the stochastic bifurcation of the catalytic CO oxidation on Ir(111) surfaces with multiple delays
according to the qualitative changes in the invariant measure, the Lyapunov exponent, and the stationary probability density of
system response. Some new criteria ensuring stability and stochastic bifurcation are obtained.
1. Introduction
For most of the practical cases, the dynamical systems will
be disturbed by some stochastic perturbation. There are
real benefits to be gained in using stochastic rather than
deterministic models. It could be intuitively thought that
the external noise would smear out the finer details of the
nonlinear system, such as bifurcations between well-defined
states. However, it could also happen that noise forces the
system far away from its deterministic attractors to explore
regions of phase space that are otherwise not visited or
interact with the nonlinearities giving rise to new phenomena
[1–10].
Now, a great care should be taken when reducing the
internal/external sources of noise and uncertainties such as
thermal noise, observation errors, and model misspecification. The CO oxidation on platinum group crystals offers an
experimental setup, where the nonlinear mechanisms have
been captured with high accuracy, and sources of internal
noise are controlled. The CO oxidation on crystals shows,
in the absence of noise, a wide variety of phenomena,
for example, oscillations, bistability, bifurcation, excitability,
spatiotemporal patterns, and turbulence. These reactions
have been receiving an increasing attention as a laboratory
analogy of catalysis used at the industrial level to manufacture
chemical products and process harmful species. Understanding the effects of external noise on this oxidation reaction can
be considered as an effort to bridge the quality gap. Besides
material and pressure gap, which have been discussed in the
literature before, the quality gap separates small experiments
in perfectly controlled settings and the industrial processes
that are always under the influence of uncertainties.
In this paper, we investigate the deterministic and
stochastic bifurcations of the catalytic CO oxidation on Ir(111)
surfaces with multiple delays. Mechanisms and parameters
are known with a high degree of accuracy, and the relevant parameters can be controlled. These ideal properties
make it possible to go back and forth between theory and
experiments. Our paper focuses on homogeneous states and
builds on the work of Hayase et al. [11–18] who studied
experimentally and numerically the effect of noise on CO
oxidation on Ir(111). They verified that the mathematical
model was correct and observed probability distributions that
are consistent with the presence of multiplicative noise.
The structure of the paper is as follows. In Section 2,
we analyzed the single and double Hopf bifurcation of
deterministic system. In Section 3, we analyzed the stochastic
stability and bifurcation of stochastic system.
2
Abstract and Applied Analysis
2. Deterministic System
or equivalently
In this section, we will investigate the delayed CO oxidation
on Ir(111) surfaces as follows:
𝑢̇ (𝑡) = 𝛼𝜅 (1 − 𝑢 − 𝑣) − 𝛽𝑢 − 𝛾𝑢 (𝑡 − 𝜏1 ) 𝑣 (𝑡 − 𝜏2 ) ,
𝑣̇ (𝑡) = 𝛿 (1 − 𝜅) (1 − 𝑢 − 𝑣)3 − 𝛾𝑢𝑣,
sin 𝜔𝜏1 =
Possible values of 𝜔𝜏1 are of the form 𝜔𝜏1 ∈ (2𝑘𝜋, 2𝑘𝜋 + 𝜋/2),
𝑘 = 0, 1, 2, . . .. Thus, we obtain the bifurcation parameter 𝛾 as
𝛾 = ( − 2𝛼𝜅𝜔2
𝑥1̇ (𝑡) = −𝛼𝜅 (𝑥1 + 𝑥2 ) − 𝛽𝑥1 − 𝛾𝑥1 (𝑡 − 𝜏1 ) [𝑥2 (𝑡 − 𝜏2 ) + 1] ,
−√4𝛼2 𝜅2 𝜔4 − 4 (𝛼2 𝜅2 − 𝜔2 ) (𝛽2 +2𝛼𝛽𝜅 + 𝛼2 𝜅2 + 𝜔4 ))
3
𝑥2̇ (𝑡) = −𝛿 (1 − 𝜅) (𝑥1 + 𝑥2 ) − 𝛾𝑥1 (𝑥2 + 1) .
−1
(2)
Linearized System. As the first step, we analyze the stability
of the trivial solution of the linearized system of (1), which is
given by
𝑥1̇ (𝑡) = − (𝛼𝜅 + 𝛽) 𝑥1 − 𝛼𝜅𝑥2 − 𝛾𝑥1 (𝑡 − 𝜏1 ) ,
𝑥2̇ (𝑡) = −𝛾𝑥1 ,
× (2 (𝛼2 𝜅2 − 𝜔2 )) (𝛼𝜅 < 𝜔) .
(10)
Now, we compute Hopf ’s transversality condition by
differentiating implicitly (6) with respect to 𝛾, obtaining
(3)
−1
{
in C := C([−𝜏, 0], R𝑛 ), and its adjoint form
𝑥1̇ (̂𝑡) = (𝛼𝜅 + 𝛽) 𝑥1 (̂𝑡) + 𝛼𝜅𝑥2 (̂𝑡) + 𝛾𝑥1 (̂𝑡 + 𝜏1 ) ,
𝑥2̇ (̂𝑡) = 𝛾𝑥1 (̂𝑡) ,
𝑑Δ (𝜆, 𝛾)
𝑑Δ (𝜆, 𝛾)
𝑑𝜆
= − {(
}
)(
) }
𝑑𝛾 𝜆=𝑖𝜔
𝑑𝛾
𝑑𝜆
𝜆=𝑖𝜔
= −{
(4)
𝐺 (𝑅) = −
(𝜓𝑗 (𝜗) , 𝜙𝑘 (𝜃)) = (𝜓𝑗 (0) , 𝜙𝑘 (0))
− 𝛾∫
−𝜏1
𝜓𝑗 (𝜁 + 𝜏1 ) 𝜙𝑘 (𝜁) 𝑑𝜁
𝑎𝑑 − 𝑏𝑐
,
𝑎2 + 𝑏2
𝑏 = 𝜔 cos 𝜔𝜏1 ,
𝑑 = 2𝜔 − 𝛾 sin 𝜔𝜏1 − 𝛾𝜏1 𝜔 cos 𝜔𝜏1 .
(11)
(6)
By the Hopf bifurcation, we let 𝜆 1,2 = ±𝑖𝜔(𝛾), with,
𝜔(𝛾) ≠ 0 and R𝑒{𝑑Δ(𝜆, 𝛾)/𝑑𝛾} ≠ 0, be the eigenvalue of (6).
All the remaining eigenvalues of Δ(𝜆, 𝛾) = 0 have negative
real parts. Then, by substituting 𝜆 1 = 𝑖𝜔(𝛾) into Δ(𝜆, 𝛾) =
0 and setting the real and imaginary parts of the resulting
algebraic equations to zero, we have
2
−𝜔 − 𝛾𝛼𝜅 + 𝜔𝛾 sin 𝜔𝜏1 + 𝜔 [(𝛼𝜅 + 𝛽) + 𝜔𝛾 cos 𝜔𝜏1 ] 𝑖 = 0.
(7)
Separating the real and imaginary parts of (7) gives the
following equations:
𝜔 [(𝛼𝜅 + 𝛽) + 𝜔𝛾 cos 𝜔𝜏1 ] = 0,
𝑀 (𝛾) = −
𝑐 = 𝛼𝜅 + 𝛾 cos 𝜔𝜏1 − 𝛾𝜏1 𝜔 sin 𝜔𝜏1 ,
(5)
has eigenvalues 𝜆 satisfying the transcendental characteristic
equation
−𝜔2 − 𝛾𝛼𝜅 + 𝜔𝛾 sin 𝜔𝜏1 = 0,
𝑎𝑐 + 𝑏𝑑
,
𝑎2 + 𝑏2
𝑎 = 𝜔 sin 𝜔𝜏1 − 𝛼𝜅,
𝑗, 𝑘 = 1, 2, . . . ,
Δ (𝜆, 𝑁) := 𝜆2 + 𝜆 [(𝛼𝜅 + 𝛽) + 𝛾𝑒−𝜆𝜏1 ] − 𝛾𝛼𝜅 = 0.
𝜆𝑒−𝜆𝜏1 − 𝛼𝜅
}
2𝜆 + (𝛼𝜅 + 𝛽) + 𝛾𝑒−𝜆𝜏1 − 𝛾𝜏1 𝑒−𝜆𝜏1 𝜆=𝑖𝜔
= 𝐺 (𝛾) + 𝑀 (𝛾) 𝑖,
̂ := C([−𝜏,
̂
in C
0], R𝑛 ) with respect to the associated bilinear
relation
0
(9)
𝛼𝜅 + 𝛽
cos 𝜔𝜏1 =
> 0.
𝛾𝜔
(1)
where 𝛾 = 𝛾𝑐 + 𝛾̃ is the bifurcation parameter. It is obvious
that there is a unique equilibrium point 𝐸(0, 1) in system (1).
Let 𝑥1 = 𝑢 − 0, 𝑥2 = 𝑣 − 1, and 𝑋 = [𝑥1 , 𝑥2 ]⊤ , then by
substituting the corresponding variables in (1), we have
𝜔2 + 𝛾𝛼𝜅
> 0,
𝛾𝜔
(8)
Lemma 1. The transversality conditions R𝑒(𝑑𝜆/𝑑𝛾)𝜆=𝜔𝑖 =
−𝑎𝑐 − 𝑏𝑑 ≠ 0 hold. If the choice of values for the model
parameters is such that the inequality
(𝜔 sin 𝜔𝜏1 − 𝛼𝜅) (𝛼𝜅 + 𝛾 cos 𝜔𝜏1 − 𝛾𝜏1 𝜔 sin 𝜔𝜏1 )
+ 𝜔 cos 𝜔𝜏1 (2𝜔 − 𝛾 sin 𝜔𝜏1 − 𝛾𝜏1 𝜔 cos 𝜔𝜏1 ) < 0
(12)
holds.
This means that the pair of eigenvalues 𝜆 1,2 = 𝑣(𝛾) + 𝑖𝜔(𝛾)
of Δ(𝜆, 𝛾) = 0 with 𝑣(𝛾𝑐 ) = 0, 𝜔(𝛾𝑐 ) ≠ 0 will cross the
imaginary axis from left to right in the complex plane with
a nonzero speed as the bifurcation parameter 𝛾 varies near its
critical value 𝛾𝑐 . Also, it ensures that the steady-state stability
of the system is lost at 𝛾 = 𝛾𝑐 , and the instability in the sense of
supercritical and subcritical bifurcations will persist for larger
values of 𝛾. The exact nature of such bifurcations satisfying
Hopf ’s conditions ((6)–(12)) will depend on the nonlinearity
in the delay system.
Abstract and Applied Analysis
3
̃ we
Rewriting (10) as a quadratic in 𝜔2 and letting 𝜔2 = 𝜔,
have
̃ 2 + 𝛼1 𝜔
̃ + 𝛼2 = 0,
𝜔
𝛼1 = 2𝛼𝛾𝜅 − 𝛾2 ,
2
𝛼2 = 𝛼2 𝛾2 𝜅2 + (𝛼𝜅 + 𝛽) ,
̃1 =
𝜔
𝜏1,𝑘,min =
(14)
1
̃2 = (−𝛼1 − √𝛼12 − 4𝛼2 ) ,
𝜔
2
𝛼1 < 0 | 2𝛼𝜅 < 𝛾,
2
𝛼12 − 4𝛼2 > 0 | 𝛾4 − 2𝛼𝜅𝛾3 − 4(𝛼𝜅 + 𝛽) > 0,
𝛼1 < 0 | 2𝛼𝜅 < 𝛾,
(15)
2
𝛼12 − 4𝛼2 = 0 | 𝛾4 = 2𝛼𝜅𝛾3 + 4(𝛼𝜅 + 𝛽) .
̃2
̃1 and 𝜔
In addition, one has to show that the substitution of 𝜔
into (10) will ensure Hopf ’s transversality conditions, that is,
̃1 = 𝜔2 and R𝑒(𝑑𝜆/𝑑𝛾)𝜆=𝑖𝜔 ,
R𝑒(𝑑𝜆/𝑑𝛾)𝜆=𝑖𝜔 . Therefore, with 𝜔
we obtain
2𝜔 + 𝛼1 = 0 | 𝜔 =
2
,
(16)
2
𝛼12 − 4𝛼2 = 0 | 𝛾4 = 2𝛼𝜅𝛾3 + 4(𝛼𝜅 + 𝛽) .
By setting 𝜔min = 𝜔 and 𝛾min = 𝛾, we have
𝜔min =
2 − 2𝛼𝛾
√𝛾min
min 𝜅
2
,
(17)
2
4
3
= 2𝛼𝜅𝛾min
+ 4(𝛼𝜅 + 𝛽) .
𝛾min
These are the minimum values separating the regimes of
stable and unstable solutions of the linearized delay equation
at the Hopf bifurcation point 𝛾 = 𝛾𝑐 , 𝜆 = ±𝜔𝑖(𝛾min ) =
2 − 2𝛼𝛾
±√𝛾min
min 𝜅/2. Below and equal to these minimum
values, all the infinite numbers of eigenvalues of the transcendental characteristic equation (6) will lie in the left-hand side
of the complex plane.
Next, deriving an expression for 𝜏1 begins by further
rewiring (8) as follows:
tan 𝜔𝜏1 =
𝜔min
(arctan (
(20)
With these minimum values for the gain 𝜔min , time
delay 𝜏1,𝑘,min , and the response frequency 𝜔min =
2 − 2𝛼𝛾
𝜔min = √𝛾min
min 𝜅/2, fine-tuning the model parameters in the vicinity of their critical settings can control
such oscillations. To find out whether the bifurcations are
supercritical and/or subcritical, we carry out a local center
manifold analysis of the nonlinear delay equation (1) as
presented in the preceding section.
2.1. The Single Hopf Bifurcation. To obtain the explicit analytical expressions for the stability condition of the Hopf
bifurcation solutions (limit cycles), system (1) should be
reduced to its center manifold [19–21]. While studying the
critical infinite dimensional problem on a two-dimensional
center manifold, we begin by defining the elements of the
initial continuous function 𝜙(𝜃) ∈ C and its projections
𝜙𝑃 (𝜃), 𝜙𝑄(𝜃) onto the center and stable subspaces 𝑃, 𝑄 ∈
C. For the eigenvalues 𝜆 = ±𝑖𝜔, we have 𝜙𝑃 (𝜃) = Φ(𝜃)𝑏
with 𝜙𝑘𝑃 (𝜃) = 𝑒𝜆 𝑘 𝜃 , −𝜏 ≤ 𝜃 ≤ 0 of the adjoint-generalized
̂
̂
eigenspace 𝑃 ∈ C and 𝜓𝑃 (𝑠) = Ψ̂𝑏, 𝜓𝑗𝑃 (𝑠)𝑒−𝜆 𝑗𝑠 , 0 ≤ 𝑠 ≤ 𝜏,
̂ The
𝑗, 𝑘 = 1, 2 of the adjoint-generalized eigenspace 𝑃̂ ∈ C.
values of the basis Φ(𝜃) ∈ C for 𝑃 are thus given by
Φ (𝜃) = [
cos 𝜔𝜃 − sin 𝜔𝜃 𝑏
][ ],
𝜔 sin 𝜔𝜃 𝜔 cos 𝜔𝜃 𝑏
(21)
̂ for 𝑃̂ is of the form
and, similarly, the basis Ψ(𝑠) ∈ C
Ψ (𝜃) = [𝜓1 , 𝜓2 ] = [
cos 𝜔𝜃 𝜔 sin 𝜔𝜃
],
− sin 𝜔𝜃 𝜔 cos 𝜔𝜃
0 ≤ 𝑠 ≤ 𝑅.
(22)
2
𝜔 + 𝛾𝛼𝜅
,
𝛼𝜅 + 𝛽
(18)
These bases form the inner product matrix
thus, we obtain
𝜏1,𝑘
2
𝜔min
+ 𝛾𝛼𝜅
) + 2𝑘𝜋)
𝛼𝜅 + 𝛽
2 − 2𝛼𝛾
√𝛾min
min 𝜅/2, one can describe all the critical settings
of the model parameters in the delay equation (1) at which
the unstable-free oscillation persists. Should unstable
oscillations appear at some desired frequency such that
if the following holds
√𝛾2 − 2𝛼𝛾𝜅
1
𝑘 = 0, 1, 2, . . . , 𝑛, . . . .
1
(−𝛼1 + √𝛼12 − 4𝛼2 ) ,
2
2
2 − 2𝛼𝛾
√𝛾min
min 𝜅/2, we get
(13)
whose real and positive solutions are of the form
=
and by using the minimum value for 𝜔min
𝜔2 + 𝛾𝛼𝜅
1
= (arctan (
) + 2𝑘𝜋)
𝜔
𝛼𝜅 + 𝛽
𝑘 = 0, 1, 2, . . . , 𝑛, . . . ,
(Ψ (𝑠) , Φ (𝜃)) = [
(19)
𝜓1 (𝑠)
] [𝜙1 𝜙2 ]
𝜓2 (𝑠)
(𝜓 (𝑠) 𝜙1 (𝜃)) (𝜓1 (𝑠) 𝜙2 (𝜃))
],
= [ 1
(𝜓2 (𝑠) 𝜙1 (𝜃)) (𝜓2 (𝑠) 𝜙2 (𝜃))
(23)
4
Abstract and Applied Analysis
̂
Consequently, the constant matrices 𝐵 ∈ C and 𝐵̂ ∈ C
are equivalent, and the elements of 𝐵 at the Hopf bifurcation
are 𝐵 = [[0, 𝜔]⊤ , [−𝜔, 0]⊤ ].
With the algebraic simplifications
in which
(𝜓1 (𝑠) 𝜙1 (𝜃)) = (cos 𝜔𝑠 cos 𝜔𝜃 + 𝜔2 sin 𝜔𝑠 sin 𝜔𝜃) ,
(𝜓1 (𝑠) 𝜙2 (𝜃)) = − (cos 𝜔𝑠 sin 𝜔𝜃 − 𝜔2 sin 𝜔𝑠 cos 𝜔𝜃) ,
(𝜓2 (𝑠) 𝜙1 (𝜃)) = − (sin 𝜔𝑠 cos 𝜔𝜃 − 𝜔2 cos 𝜔𝑠 sin 𝜔𝜃) ,
(24)
𝑒𝐵𝜃 = [
2
(𝜓2 (𝑠) 𝜙2 (𝜃)) = (sin 𝜔𝑠 sin 𝜔𝜃 + 𝜔 cos 𝜔𝑠 cos 𝜔𝜃) .
The substitution of the elements of (Ψ, Φ)𝑛𝑠𝑔 ≠ 0
(𝜓 , 𝜙 ) (𝜓1 , 𝜙2 )
],
(Ψ, Φ)𝑛𝑠𝑔 = [ 1 1
(𝜓2 , 𝜙1 ) (𝜓2 , 𝜙2 )
(29)
one can easily see that
(25)
Φ (𝜃) = Φ (0) 𝑒𝐵𝜃 = [
where
1
1
(𝜓1 , 𝜙1 ) = 1 − 𝛾 {( − 𝜔) sin 𝜔𝜏1 + 𝜏1 (1 + 𝜔2 ) cos 𝜔𝜏1 } ,
2
𝜔
2
(𝜓1 , 𝜙2 ) = 𝛾 (1 + 𝜔 ) sin 𝜔𝜏1 ,
(𝜓2 (𝑠) 𝜙1 (𝜃)) = −𝛾 (1 + 𝜔2 ) sin 𝜔𝜏1 ,
(𝜓2 (𝑠) 𝜙2 (𝜃)) = 𝜔2 + 𝛾 {(
cos 𝜔𝜃 − sin 𝜔𝜃
],
sin 𝜔𝜃 cos 𝜔𝜃
1
− 𝜔) sin 𝜔𝜏1
𝜔
cos 𝜔𝜃 − sin 𝜔𝜃
],
𝜔 sin 𝜔𝜃 𝜔 cos 𝜔𝜃
−𝑅 ≤ 𝜃 ≤ 0,
(30)
and it follows that 𝐽(𝑡, 𝜇)Φ(𝜃) = Φ(0)𝑒𝐵(𝜃+𝑡) is indeed the
solution operator of the linearized delay equations in C.
̂
Similarly, we have Ψ(𝑠) = Ψ(0)𝑒−𝐵𝑠 , 0 ≤ 𝑠 ≤ 𝑅, and
𝐵(𝜃+𝑡)
̂ ̂𝑡, 𝑅)Ψ(𝑠) = Φ(0)𝑒
𝐽(
is the solution operator for the
̂
adjoint delay equations in C.
By the change of variable 𝑥𝑡𝑃 (𝜃) = Φ(𝜃)𝑧(𝑡) + 𝑥𝑡𝑄(𝜃) with
̃ 𝜙𝑃 (𝜃)), we obtain the following as a
𝑧(𝑡) ∈ R2 , 𝑧(𝑡) = (Ψ(𝑠),
first-order approximation in 𝜀, for 𝜃 = −𝜏1 , −𝜏2 :
−𝜏1 (1 + 𝜔2 ) cos 𝜔𝜏1 } .
(26)
̂ of the adjoint equation
Then, the basis Ψ(𝑠) for 𝑃̂ ∈ C
̂ computing Ψ
̃
̃ =
(2) is normalized to Ψ = [𝜓̃1 , 𝜓̃2 ]⊤ ∈ C,
−1
(Ψ, Φ)𝑛𝑠𝑔 Ψ(𝑠) to yield
̃ (𝑠) = [𝜓̃11 (𝑠) 𝜓̃12 (𝑠)] ,
Ψ
𝜓̃21 (𝑠) 𝜓̃22 (𝑠)
̃11 (𝑠) =
Ψ
̃12 (𝑠) =
Ψ
̃21 (𝑠) =
Ψ
̃22 (𝑠) =
Ψ
1
{(𝜓2 , 𝜙2 ) sin 𝜔𝑠 − (𝜓1 , 𝜙2 ) cos 𝜔𝑠} ,
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
{(𝜓2 , 𝜙1 ) cos 𝜔𝑠 + (𝜓1 , 𝜙1 ) cos 𝜔𝑠} ,
𝜔
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
{(𝜓2 , 𝜙1 ) sin 𝜔𝑠 − (𝜓1 , 𝜙1 ) cos 𝜔𝑠} ,
where the substitution of the new elements (𝜓𝑗 (𝑠), 𝜙𝑘 (𝜃)),
𝑗, 𝑘 = 1, 2 into (3) will lead to the identity matrix
1
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
[
0
(31)
and for 𝜃 = 0,
det ((Ψ, Φ)−1
𝑛𝑠𝑔 ) = {(𝜓1 , 𝜙1 ) (𝜓2 , 𝜙2 ) − (𝜓1 , 𝜙2 ) (𝜓2 , 𝜙1 )} ,
(27)
(Ψ, Φ)id =
𝑧1 (𝑡) cos 𝜔𝜏 + 𝑧2 (𝑡) sin 𝜔𝜏2
𝑥 (𝑡 − 𝜏2 )
]=[
],
[ 1
−𝑧1 (𝑡) 𝜔 sin 𝜔𝜏2 + 𝑧2 (𝑡) 𝜔 cos 𝜔𝜏2
𝑥2 (𝑡 − 𝜏2 )
[
𝜔
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
𝑥1 (𝑡 − 𝜏1 )
𝑧1 (𝑡) cos 𝜔𝜏 + 𝑧2 (𝑡) sin 𝜔𝜏1
]=[
],
−𝑧1 (𝑡) 𝜔 sin 𝜔𝜏1 + 𝑧2 (𝑡) 𝜔 cos 𝜔𝜏1
𝑥2 (𝑡 − 𝜏1 )
{(𝜓2 , 𝜙2 ) cos 𝜔𝑠 + (𝜓1 , 𝜙2 ) sin 𝜔𝑠} ,
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
𝜔
0 ≤ 𝑠 ≤ 𝑅,
[
0
]
det ((Ψ, Φ)−1
𝑛𝑠𝑔 )
𝑧 (𝑡)
𝑥1 (𝑡)
]=[ 1
].
𝑥2 (𝑡)
𝜔𝑧2 (𝑡)
(32)
Therefore, we obtain (1) of the center manifold as follows:
𝑧̇1 (𝑡) = −𝜔𝑧2 + Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾) ,
𝑧̇2 (𝑡) = 𝜔𝑧1 + Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾) ,
(33)
where the perturbation functions Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾) and
Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾) denote the following:
Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾) = 𝛾̃ (𝑐10 𝑧1 + 𝑐100 𝑧2 ) + 𝑐11 𝑧13 + 𝑐12 𝑧12 𝑧2
+ 𝑐13 𝑧1 𝑧22 + 𝑐14 𝑧23 + 𝑐15 𝑧12 + 𝑐16 𝑧1 𝑧2 + 𝑐17 𝑧22 ,
Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾) = 𝛾̃ (𝑐20 𝑧1 + 𝑐200 𝑧2 ) + 𝑐21 𝑧13 + 𝑐22 𝑧12 𝑧2
1 0
= [
].
0 1
(28)
+ 𝑐23 𝑧1 𝑧22 + 𝑐24 𝑧23 + 𝑐25 𝑧12 + 𝑐26 𝑧1 𝑧2 + 𝑐27 𝑧22 .
(34)
Abstract and Applied Analysis
5
Here, the lengthy expressions of 𝑐𝑖𝑗 are omitted here, and
which in polar coordinate, 𝑧1 = 𝑎 sin 𝜃, 𝑧2 = −𝑎 cos 𝜃, 𝜃 =
𝜔𝑡 + 𝜑 is equivalent to
𝑎̇ (𝑡) = 𝜈 (𝛾) 𝑎 + Π
(1)
3
5
(𝜏1 , 𝜏2 , 𝛾) 𝑎 +
𝜑̇ (𝑡) = 𝜔 + Π(2) (𝜏1 , 𝜏2 , 𝛾) 𝑎2 +
⃝ (𝑎 , 𝜏1 , 𝜏2 , 𝛾) ,
⃝ (𝑎4 , 𝜏1 , 𝜏2 , 𝛾) .
(35)
The so-called Poincaré-Lyapunov coefficient Π(1) (𝜏1 , 𝜏2 , 𝛾) is
independently determined by the following formula:
Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾)
=
+
(𝑧1 , 𝑧2 , 𝛾) +
Δ𝑓𝑧(2)
2 𝑧2 𝑧2
1
(𝑧1 , 𝑧2 , 𝛾)
{Δ𝑓𝑧(1)
1 𝑧2
16𝜔
(36)
− Δ𝑓𝑧(2)
(𝑧1 , 𝑧2 , 𝛾)
1 𝑧2
− Δ𝑓𝑧(1)
(𝑧1 , 𝑧2 , 𝛾) Δ𝑓𝑧(2)
(𝑧1 , 𝑧2 , 𝛾)
1 𝑧1
1 𝑧1
(𝑧1 , 𝑧2 , 𝛾) Δ𝑓𝑧(2)
(𝑧1 , 𝑧2 , 𝛾)} .
+ Δ𝑓𝑧(1)
2 𝑧2
2 𝑧2
(37)
(39)
Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾)
=
1
{3 (𝑐11 + 𝑐24 ) + 𝑐13 + 𝑐23 }
8
1
+
{𝑐 (𝑐 + 𝑐 ) − 𝑐26 (𝑐25 + 𝑐27 )
8𝜔 16 15 17
(40)
This coefficient together with the amplitude bifurcation equation (35) produces the nonlinear bifurcations for multiple
delays as shown in the following: (a) Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾) <
0 → supercritical bifurcation; (b) Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾) > 0 →
subcritical bifurcation.
(a) If Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾) < 0, then system (1) occurs supercritical bifurcation.
𝜕Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 𝑐20 𝛾̃ + 3𝑐21 𝑧12 + 2 (𝑐25 + 𝑐22 𝑧2 ) 𝑧1
𝜕𝑧1
+ (2𝑐27 + 3𝑐24 𝑧2 ) 𝑧2 ,
𝜕Δ𝑓(2) (𝑧1 , 𝑧1 , 𝛾)
= 2 (3𝑐21 𝑧1 + 𝑐22 𝑧2 + 𝑐25 ) ,
𝜕𝑧1 𝜕𝑧1
Theorem 2. Let 𝑎𝑐 + 𝑏𝑑 > 0.
+ (2𝑐17 + 3𝑐14 𝑧2 ) 𝑧2 ,
𝜕Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 𝑐200 𝛾̃ + 𝑐23 𝑧12 + (𝑐26 + 2𝑐23 𝑧2 ) 𝑧1
𝜕𝑧2
𝜕Δ2 𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 2 (𝑐22 𝑧1 + 𝑐23 𝑧2 ) + 𝑐26 ,
𝜕𝑧1 𝜕𝑧2
− 𝑐15 𝑐25 + 𝑐17 𝑐27 } .
𝜕Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾)
= 𝑐10 𝛾̃ + 3𝑐11 𝑧12 + 2 (𝑐15 + 𝑐12 𝑧2 ) 𝑧1
𝜕𝑧1
+ (𝑐26 + 𝑐23 𝑧2 ) 𝑧2 ,
𝜕Δ3 𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 2𝑐23 ,
𝜕𝑧1 𝜕𝑧2 𝜕𝑧2
Substituting these quantities into the formulate in (36) will
yield
As is well known [19–24], the amplitude equation (35)
together with (36) can determine the stability behaviour of
the center manifold ODEs (33), where all the partial deriva=
tives of Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾), Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾), for example, Δ𝑓𝑧(2)
1 𝑧2
(2)
{𝜕(𝜕Δ𝑓𝑧1 𝑧2 /𝜕𝑧1 )/𝜕𝑧2 }(0,0,𝛾𝑐 ) , are evaluated at the bifurcation
point (𝑧1 , 𝑧2 ) → (0, 0). Thus, the partial derivatives are given
as follows:
𝜕Δ𝑓(1) (𝑧1 , 𝑧2 , 𝛾)
= 𝑐100 𝛾̃ + 𝑐13 𝑧12 + (𝑐16 + 2𝑐13 𝑧2 ) 𝑧1
𝜕𝑧2
𝜕Δ𝑓(1) (𝑧1 , 𝑧1 , 𝛾)
= 2 (3𝑐11 𝑧1 + 𝑐12 𝑧2 + 𝑐15 ) ,
𝜕𝑧1 𝜕𝑧1
𝜕Δ𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 2 (𝑐23 𝑧1 + 3𝑐24 𝑧2 + 𝑐27 ) .
𝜕𝑧2 𝜕𝑧2
(𝑧1 , 𝑧2 , 𝛾) + Δ𝑓𝑧(2)
(𝑧1 , 𝑧2 , 𝛾))
× (Δ𝑓𝑧(2)
1 𝑧1
1 𝑧2
+ (𝑐16 + 𝑐13 𝑧2 ) 𝑧2 ,
𝜕Δ3 𝑓(1) (𝑧1 , 𝑧2 , 𝛾)
= 2𝑐13 ,
𝜕𝑧1 𝜕𝑧2 𝜕𝑧2
𝜕Δ2 𝑓(1) (𝑧1 , 𝑧2 , 𝛾)
= 2 (𝑐12 𝑧1 + 𝑐13 𝑧2 ) + 𝑐16 ,
𝜕𝑧1 𝜕𝑧2
𝜕Δ3 𝑓(2) (𝑧1 , 𝑧2 , 𝛾)
= 6𝑐24 ,
𝜕𝑧1 𝜕𝑧1 𝜕𝑧1
(𝑧1 , 𝑧2 , 𝛾)}
× (Δ𝑓𝑧(1)
(𝑧1 , 𝑧2 , 𝛾) + Δ𝑓𝑧(1)
(𝑧1 , 𝑧2 , 𝛾))
1 𝑧1 𝑧1
2 𝑧2
𝜕Δ3 𝑓(1) (𝑧1 , 𝑧2 , 𝛾)
= 6𝑐11 ,
𝜕𝑧1 𝜕𝑧1 𝜕𝑧1
𝜕Δ𝑓(1) (𝑧2 , 𝑧2 , 𝛾)
= 2 (𝑐13 𝑧1 + 3𝑐14 𝑧2 + 𝑐17 ) ,
𝜕𝑧2 𝜕𝑧2
1
(𝑧1 , 𝑧2 , 𝛾) + Δ𝑓𝑧(1)
(𝑧1 , 𝑧2 , 𝛾)
{Δ𝑓𝑧(1)
1 𝑧1 𝑧1
1 𝑧2 𝑧2
16
+Δ𝑓𝑧(2)
1 𝑧1 𝑧2
from which we readily obtain the required values for the
formula in (36) as
(38)
(b) If Π(1) (𝑎, 𝜏1 , 𝜏2 , 𝛾) > 0, then system (1) occurs subcritical bifurcation.
2.2. The Double Hopf Bifurcation. For the eigenvalue 𝜆 1,2 =
±𝑖𝜔1 , 𝜆 1,2 = ±𝑖𝜔2 of the equation Δ(𝜆, 𝛾) = 0 in (6), we
have the base function Φ(𝜃) = [𝜙1 (𝜃), 𝜙2 (𝜃), 𝜙3 (𝜃), 𝜙4 (𝜃)],
6
Abstract and Applied Analysis
−𝜏 ≤ 𝜃 ≤ 0, Ψ(𝑠) = [𝜓1 (𝑠), 𝜓2 (𝑠), 𝜓3 (𝑠), 𝜓4 (𝑠)]⊤ , 0 ≤ 𝑠 ≤ 𝜏
in C([−𝜏, 0], R𝑛 ) and C([−𝜏, 0], R𝑛 ), accordingly,
Φ (𝜃) = [
cos 𝜔1 𝜃 − sin 𝜔𝜃1 cos 𝜔2 𝜃 − sin 𝜔2 𝜃
],
sin 𝜔1 𝜃 cos 𝜔1 𝜃 sin 𝜔2 𝜃 cos 𝜔2 𝜃
Again, the substitution of the elements 𝜓̃𝑗 (𝑠), 𝜙𝑘 (𝜃), 𝑗, 𝑘 =
̃ Φ(𝜃)) will yield the identity matrix, namely,
1, 2, 3, 4 of (Ψ(𝑠),
(41)
⊤
cos 𝜔1 𝑠 − sin 𝜔𝑠 cos 𝜔2 𝑠 − sin 𝜔2 𝑠
Ψ (𝑠) = [
] ,
sin 𝜔1 𝑠 cos 𝜔1 𝑠 sin 𝜔2 𝑠 cos 𝜔2 𝑠
(Ψ, Φ)𝑛𝑠𝑔
and they form the inner product matrix (Ψ(𝑠), Φ(𝜃)) =
(𝜓𝑗 (𝑠)𝜙𝑘 (𝜃)), 𝑗, 𝑘 = 1, 2, 3, 4, namely,
Λ
1[
0
[
= [
Λ 0
[0
0
Λ
0
0
0
0
Λ
0
0
0]
] = 𝐼,
0]
Λ]
(46)
2
2
2
2
+ 𝜓12
) (𝜓33
+ 𝜓34
).
Λ = (𝜓11
(Ψ (𝑠) , Φ (𝜃)) =
(𝜓1 (𝑠) 𝜙1 (𝜃))
[(𝜓2 (𝑠) 𝜙1 (𝜃))
(𝜓3 (𝑠) 𝜙1 (𝜃))
[(𝜓4 (𝑠) 𝜙1 (𝜃))
(𝜓1 (𝑠) 𝜙2 (𝜃))
(𝜓2 (𝑠) 𝜙2 (𝜃))
(𝜓3 (𝑠) 𝜙2 (𝜃))
(𝜓4 (𝑠) 𝜙2 (𝜃))
(𝜓1 (𝑠) 𝜙3 (𝜃))
(𝜓2 (𝑠) 𝜙3 (𝜃))
(𝜓3 (𝑠) 𝜙3 (𝜃))
(𝜓4 (𝑠) 𝜙3 (𝜃))
(𝜓1 (𝑠) 𝜙4 (𝜃))
(𝜓2 (𝑠) 𝜙4 (𝜃))]
.
(𝜓3 (𝑠) 𝜙4 (𝜃))
(𝜓4 (𝑠) 𝜙4 (𝜃))]
Then, the constant matrix 𝐵 is given by
0 −𝜔1 0 0
[𝜔1 0 0 0 ]
]
𝐵=[
[ 0 0 0 −𝜔2 ] .
[ 0 0 𝜔2 0 ]
(42)
The substitution of the elements (Ψ(𝑠), Φ(𝜃)) = (𝜓𝑗 (𝑠)𝜙𝑘 (𝜃))
into the bilinear pairing (5) and integrating, we obtain the
nonsingular matrix
0
𝜓11 −𝜓12 0
[𝜓21 𝜓22 0
0 ]
],
[
(Ψ, Φ)𝑛𝑠𝑔 = [
0
0 𝜓23 −𝜓34 ]
0 𝜓43 𝜓44 ]
[ 0
where the constant values of this matrix are given by
(43)
The change of variable 𝑥𝑡𝑃 (𝜙(𝜃), 𝛾) = Φ(𝜃)𝑧(𝑡) + 𝑥𝑡𝑄(𝜙(𝜃), 𝛾),
̃
𝜙𝑃 (𝜃)) will yield
𝑧(𝑡) ∈ R4 , 𝑧(𝑡) = (Ψ(𝜃),
Φ (𝜃) 𝑧 (𝑡) = [
𝜓11 = 𝜓22 = 1 − 𝛾𝜏1 cos 𝜔1 𝜏1 ,
𝜓12 = −𝛾𝜏1 sin 𝜔1 𝜏1 ,
𝜓12 = −𝜓21 ,
𝜓34 = −𝛾𝜏1 sin 𝜔2 𝜏1 ,
1
2
2
{ [ (𝜓22
cos 𝜔1 𝑠 + 𝜓21
sin 𝜔1 𝑠) ,
2 + 𝜓2
𝜓11
12
thereby, for −𝜏 ≤ 𝜃 < 0 and 𝜏1 ≤ 𝜏 ≤ 𝜏2 , we have
cos 𝜔1 𝜏1 sin 𝜔1 𝜏1 cos 𝜔2 𝜏1 sin 𝜔2 𝜏1
Φ (−𝜏1 ) 𝑧 (𝑡) = [
]
− sin 𝜔1 𝜏1 cos 𝜔1 𝜏1 sin 𝜔2 𝜏1 cos 𝜔2 𝜏1
𝑧1 (𝑡)
[𝑧2 (𝑡)]
]
×[
[𝑧3 (𝑡)] ,
[𝑧4 (𝑡)]
⊤
2
2
(𝜓22
sin 𝜔1 𝑠 − 𝜓21
cos 𝜔1 𝑠) , 0, 0] } ,
𝜓̃2 =
2
𝜓11
1
2
2
{ [ (𝜓12
cos 𝜔1 𝑠 − 𝜓11
sin 𝜔1 𝑠) ,
2
+ 𝜓12
Φ (−𝜏2 ) 𝑧 (𝑡)
⊤
2
2
(𝜓12
sin 𝜔1 𝑠 + 𝜓11
cos 𝜔1 𝑠) , 0, 0] } ,
𝜓̃3 =
1
2
2
{ [0, 0, (𝜓44
cos 𝜔2 𝑠 + 𝜓43
sin 𝜔2 𝑠) ,
2 + 𝜓2
𝜓33
34
=[
cos 𝜔1 𝜏2 sin 𝜔1 𝜏2 cos 𝜔2 𝜏2 sin 𝜔2 𝜏2
]
− sin 𝜔1 𝜏2 cos 𝜔1 𝜏2 − sin 𝜔2 𝜏2 cos 𝜔2 𝜏2
𝑧1 (𝑡)
[𝑧2 (𝑡)]
]
×[
[𝑧3 (𝑡)] ,
[𝑧4 (𝑡)]
⊤
2
2
sin 𝜔2 𝑠 − 𝜓43
cos 𝜔2 𝑠)] } ,
(𝜓44
𝜓̃4 =
(48)
𝜓43 = −𝜓34 .
̂ is normalized to the new basis Ψ
̃ =
Then, Ψ ∈ C
[𝜓̃1 (𝑠), 𝜓̃3 (𝑠), 𝜓̃3 (𝑠), 𝜓̃4 (𝑠)], 0 ≤ 𝑠 ≤ 𝜏, where the elements
̂ are given by
𝜓̃𝑗 (𝑠), 𝑗 = 1, 2, 3, 4 of Ψ ∈ C
𝜓̃1 =
cos 𝜔1 𝜃 − sin 𝜔1 𝜃 cos 𝜔2 𝜃 − sin 𝜔2 𝜃
]
sin 𝜔1 𝜃 cos 𝜔1 𝜃 sin 𝜔2 𝜃 cos 𝜔2 𝜃
𝑧1 (𝑡)
[𝑧2 (𝑡)]
]
×[
[𝑧3 (𝑡)] ,
[𝑧4 (𝑡)]
(44)
𝜓33 = 𝜓44 = 1 − 𝛾𝜏1 cos 𝜔2 𝜏1 ,
(47)
1
2
2
{ [0, 0, (𝜓34
cos 𝜔2 𝑠 − 𝜓33
sin 𝜔2 𝑠) ,
2 + 𝜓2
𝜓11
12
⊤
2
2
(𝜓34
sin 𝜔2 𝑠 + 𝜓33
cos 𝜔1 𝑠)] } .
(45)
𝑧1 (𝑡)
𝑧2 (𝑡)]
1 0 1 0 [
],
Φ (0) 𝑧 (𝑡) = [
][
[
0 1 0 1 𝑧3 (𝑡)]
[𝑧4 (𝑡)]
(49)
Abstract and Applied Analysis
7
Δ𝑔4 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾)
and so
(4) 3
(4) 2
(4) 2
(4) 2
(4)
𝑧1 + 𝑐112
𝑧1 𝑧2 + 𝑐113
𝑧1 𝑧3 + 𝑐114
𝑧1 𝑧2 + 𝑐115
𝑧1 𝑧22
= 𝑐111
𝛾𝑥1 (𝑡 − 𝜏1 ) = 𝛾 [𝑧1 cos 𝜔1 𝜏1 + 𝑧2 sin 𝜔1 𝜏1
+𝑧3 cos 𝜔2 𝜏1 + 𝑧4 sin 𝜔2 𝜏1 ] ,
(4) 2
(4)
(4)
(4)
(4) 2
+ 𝑑111
𝑧1 + 𝑑112
𝑧1 𝑧2 + 𝑑113
𝑧1 𝑧3 + 𝑑114
𝑧1 𝑧4 + 𝑑222
𝑧2
𝛾𝑥1 (𝑡 − 𝜏1 ) 𝑥2 (𝑡 − 𝜏2 ) = 𝛾 [𝑧1 cos 𝜔1 𝜏1 + 𝑧2 sin 𝜔1 𝜏1
(4) 2
(4) 2
+ 𝑑333
𝑧3 + 𝑑444
𝑧4 + ⋅ ⋅ ⋅ ,
+𝑧3 cos 𝜔2 𝜏1 + 𝑧4 sin 𝜔2 𝜏1 ]
(52)
× [𝑧1 cos 𝜔1 𝜏2 + 𝑧2 sin 𝜔1 𝜏2
+𝑧3 cos 𝜔2 𝜏2 + 𝑧4 sin 𝜔2 𝜏2 ] .
(50)
The ODEs of the center manifold 𝑀𝛾 ∈ C([−𝜏, 0], R4 ) are
given by as follows:
3
𝑧̇3 (𝑡) = −𝜔2 𝑧2 + Δ𝑔 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾) ,
(51)
4
𝑧̇4 (𝑡) = 𝜔2 𝑧2 + Δ𝑔 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾) ,
where the perturbations in these equations denote
Δ𝑔1 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾)
(1) 3
(1) 2
(1) 2
(1) 2
(1)
𝑧1 + 𝑐112
𝑧1 𝑧2 + 𝑐113
𝑧1 𝑧3 + 𝑐114
𝑧1 𝑧2 + 𝑐115
𝑧1 𝑧22
= 𝑐111
(1)
(1)
(1) 3
(1) 3
(1) 3
+ 𝑐116
𝑧1 𝑧32 + 𝑐117
𝑧1 𝑧42 + 𝑐222
𝑧2 + 𝑐333
𝑧3 + 𝑐444
𝑧4
(1) 2
(1)
(1)
(1)
(1) 2
+ 𝑑111
𝑧1 + 𝑑112
𝑧1 𝑧2 + 𝑑113
𝑧1 𝑧3 + 𝑑114
𝑧1 𝑧4 + 𝑑222
𝑧2
(1) 2
(1) 2
+ 𝑑333
𝑧3 + 𝑑444
𝑧4 + ⋅ ⋅ ⋅ ,
Δ𝑔2 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾)
(2) 3
(2) 2
(2) 2
(2) 2
(2)
𝑧1 + 𝑐112
𝑧1 𝑧2 + 𝑐113
𝑧1 𝑧3 + 𝑐114
𝑧1 𝑧2 + 𝑐115
𝑧1 𝑧22
= 𝑐111
(2)
(2)
(2) 3
(1) 3
(1) 3
+ 𝑐116
𝑧1 𝑧32 + 𝑐117
𝑧1 𝑧42 + 𝑐222
𝑧2 + 𝑐333
𝑧3 + 𝑐444
𝑧4
(2) 2
(2)
(2)
(2)
(2) 2
+ 𝑑111
𝑧1 + 𝑑112
𝑧2 𝑧2 + 𝑑113
𝑧1 𝑧3 + 𝑑114
𝑧1 𝑧4 + 𝑑222
𝑧2
(2) 2
(2) 2
+ 𝑑333
𝑧3 + 𝑑444
𝑧4 + ⋅ ⋅ ⋅ ,
Δ𝑔3 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾)
(3) 3
(3) 2
(3) 2
(3) 2
(1)
𝑧1 + 𝑐112
𝑧1 𝑧2 + 𝑐113
𝑧1 𝑧3 + 𝑐114
𝑧1 𝑧2 + 𝑐115
𝑧1 𝑧22
= 𝑐111
(3)
(3)
(3) 3
(3) 3
(1) 3
+ 𝑐116
𝑧1 𝑧32 + 𝑐117
𝑧1 𝑧42 + 𝑐222
𝑧2 + 𝑐333
𝑧3 + 𝑐444
𝑧4
(3) 2
(3)
(3)
(3)
(3) 2
+ 𝑑111
𝑧1 + 𝑑112
𝑧1 𝑧2 + 𝑑113
𝑧1 𝑧3 + 𝑑114
𝑧1 𝑧4 + 𝑑222
𝑧2
(3) 2
(3) 2
+ 𝑑333
𝑧3 + 𝑑444
𝑧4 + ⋅ ⋅ ⋅ ,
and the above coefficients in these equations denote the
constant values in (52). Here, the lengthy expressions of
𝑘
𝑘
𝑐𝑖𝑗𝑙
and 𝑑𝑖𝑗𝑙
are omitted here. These equations are further
written into the desired standard form of amplitude and
phase relations using the polar coordinates 𝑧𝑗 = 𝑎𝑗 sin 𝜃𝑗 ,
𝑧𝑗 = 𝑎𝑗 cos 𝜃𝑗 , 𝜃𝑗 = 𝜔𝑗 𝑡 + 𝜑𝑗 , 𝑗 = 1, 2, 3, 4. Again, the
application of the integral averaging method to the resulting
amplitude and phase equations will yield
𝑑𝑎1
(11)
(11)
𝑎1 + Π(11) (𝑎, 𝛾) 𝑎13 + 𝛽112
𝑎1 𝑎22 + ⋅ ⋅ ⋅ ,
= 𝛽111
𝑑𝑡
𝑧̇1 (𝑡) = −𝜔1 𝑧2 + Δ𝑔1 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾) ,
𝑧̇2 (𝑡) = 𝜔1 𝑧1 + Δ𝑔2 (𝑧1 , 𝑧2 , 𝑧3 , 𝑧4 , 𝛾) ,
(4)
(4)
(4) 3
(4) 3
(4) 3
+ 𝑐116
𝑧1 𝑧32 + 𝑐117
𝑧1 𝑧42 + 𝑐222
𝑧2 + 𝑐333
𝑧3 + 𝑐444
𝑧4
𝑑𝜑1
(12)
(12)
𝑎1 + Π(12) (𝑎, 𝛾) 𝑎13 + 𝛽112
𝑎1 𝑎22 + ⋅ ⋅ ⋅ ,
= 𝛽111
𝑑𝑡
𝑑𝑎2
(21)
(21)
𝑎2 + Π(21) (𝑎, 𝛾) 𝑎23 + 𝛽112
𝑎1 𝑎22 + ⋅ ⋅ ⋅ ,
= 𝛽111
𝑑𝑡
(53)
𝑑𝜑2
(22)
(22)
𝑎1 + Π(22) (𝑎, 𝛾) 𝑎23 + 𝛽112
𝑎1 𝑎22 + ⋅ ⋅ ⋅ ,
= 𝛽111
𝑑𝑡
where the coefficients in these equations are the resulting constant values after averaging. Similarly, the coefficients in these
averaged equations, in particular the cubic ones Π(11) (𝑎, 𝛾),
Π(21) (𝑎, 𝛾), will determine whether the double Hopf interactions are of the form (i) subsupercritical bifurcation type, (ii)
supersubcritical bifurcation type, (iii) subsubcritical bifurcation type, and (iv) supersupercritical bifurcation type. Like
the Poincaré-Lyapunov coefficient Π(𝑎, 𝛾) in (36) for the
single Hopf, there are also explicit formulas for computing
the double Hopf coefficients Π(11) (𝑎, 𝛾), Π(21) (𝑎, 𝛾), which are
extremely too long to be displayed here. For the account of
these formulas for the double Hopf coefficients, again, the
publications [22, 23] are excellent references. Classical studies
of dynamical systems resulting to equations of the form (53)
have demonstrated unprecedented dynamics ranging from
global dynamic bifurcations to ultimately chaotic dynamics
(see [22, 24]). There exist reliable and robust mathematical
tools to examine the long-term behaviour of such dynamics.
Campbell et al. [24], in particular, have shown numerically
that equations of the form (53) exhibit four double Hopf ’s
interactions, namely, two super-super-, one super-sub-, and
one sub-sub-critical bifurcations. Other interesting dynamics
such as large limit cycles, tori, and period doubling were also
characterized by the authors.
3. Stochastic System
In the section, we present some preliminary results to be used
in a subsequent section to establish the stochastic stability and
8
Abstract and Applied Analysis
stochastic bifurcation. Before proving the main theorem, we
give some lemmas and definitions.
Definition 3 (see [1] (D-bifurcation)). Dynamical bifurcation
is concerned with a family of random dynamical systems
which is differential and has the invariant measure 𝜇𝛼 . If
there exists a constant 𝛼𝐷 satisfying in any neighborhood of
𝛼𝐷, there exists another constant 𝛼 and the corresponding
invariant measure 𝜈𝛼 ≠ 𝜇𝛼 satisfying 𝜈𝛼 → 𝜇𝛼 as 𝛼 → 𝛼𝐷.
Then, the constant 𝛼𝐷 is a point of dynamical bifurcation.
Definition 4 (see [1] (P-bifurcation)). Phenomenological bifurcation is concerned with the change in the shape of
density (stationary probability density) of a family of random
dynamical systems as the change of the parameter. If there
exists a constant 𝛼0 satisfying in any neighborhood of 𝛼𝐷,
there exist other two constants 𝛼1 , 𝛼2 , and their corresponding stationary density functions 𝑝𝛼1 , 𝑝𝛼2 satisfying 𝑝𝛼1 and
𝑝𝛼2 are not equivalent. Then, the constant 𝛼0 is a point of
phenomenological bifurcation.
Definition 5 (see [1] (stochastic pitchfork bifurcation)). In the
viewpoint of phenomenological bifurcation, the stationary
solution of the Fokker-Planck-Kolmogorov (FPK) equation
which is corresponded with the stochastic differential equation changes from one peak into two peaks.
In the viewpoint of dynamical bifurcation, there exists a
constant 𝜇0 that satisfies the following conditions:
(i) when 𝜇 < 𝜇0 , the stochastic differential equation has
only one invariant measure 𝜈0 ; moreover, 𝜈0 is stable
(i.e., the maximal Lyapunov exponent is negative).
(ii) when 𝜇 = 𝜇0 , the invariant measure 𝜈0 loses its stability and becomes unstable (i.e., the maximal Lyapunov
exponent is positive); moreover, the rotation number
is zero.
(iii) when 𝜇 > 𝜇0 , the stochastic differential equation has
three invariant measures 𝜈0 , 𝜈1 , and 𝜈2 ; both 𝜈1 and
𝜈2 are stable (i.e., their maximal Lyapunov exponents
are both negative).
(iv) the global attractors of the stochastic differential
equation change from a single-point set into a collection of one-dimensional (the closure of the unstable
manifold of the unstable invariant measure).
If the stochastic bifurcation of a stochastic differential
equation has the above characters, then the stochastic differential equation admits stochastic pitchfork bifurcation at
𝜇 = 𝜇0 .
Definition 6 (see [1] (the stochastic Hopf bifurcation)). In the
viewpoint of phenomenological bifurcation, the stationary
solution of the FPK equation which is corresponded to the
stochastic differential equation changes from one peak into a
crater.
In the viewpoint of dynamical bifurcation, the following
can be noticed:
(i) if one of the invariant measures of the stochastic
differential equation loses its stability and becomes
unstable (i.e., two Lyapunov exponents are positive),
then the rotation number is not zero. Meanwhile,
there at least appears one new stable invariant measure,
(ii) the global attractors of the stochastic differential
equation change from a single-point set into a random
topological disk (the closure of the unstable manifold
of the unstable invariant measure).
If the stochastic bifurcation of a stochastic differential
equation has the above characters, then the stochastic differential equation admits the stochastic Hopf bifurcation.
Definition 7 (see [1] (stochastically stable)). The trivial solution 𝑥(𝑡, 𝑡0 , 𝑥0 ) of stochastic differential equation is said to be
stochastically stable or stable in probability if for every pair of
𝜀 ∈ (0, 1) and 𝛼 > 0, there exists 𝛿 = 𝛿(𝜀, 𝛿, 𝑡0 ) > 0 such that
󵄨
󵄨
𝑃 {󵄨󵄨󵄨𝑥 (𝑡, 𝑡0 , 𝑥0 )󵄨󵄨󵄨 < 𝛼 ∀𝑡 ≥ 𝑡0 } ≥ 1 − 𝜀,
(54)
whenever |𝑥0 | < 𝛿. Otherwise, it is said to be stochastically
unstable.
In this section, taking some stochastic factors into
account, we introduce randomness into the model (1) by
replacing the parameters 𝛾 by 𝛾 → 𝛾 + 𝛼𝜉(𝑡). This is only a
first step in introducing stochasticity into the model. Ideally,
we would also like to introduce stochastic environmental
variation into the other parameters such as the transmission
coefficient 𝛼 and 𝛿, but this would make the analysis much
too difficult. Hence, we get the following system of stochastic
differential equations:
𝑢̇ (𝑡) = 𝛼𝜅 (1 − 𝑢 − 𝑣) − 𝛽𝑢 − 𝛾𝑢 (𝑡 − 𝜏1 ) 𝑣 (𝑡 − 𝜏2 )
+ (𝜎1 + 𝜎2 𝑢 + 𝜎3 𝑣) 𝜉 (𝑡) ,
𝑣̇ (𝑡) = 𝛿 (1 − 𝜅) (1 − 𝑢 − 𝑣)3 − 𝛾𝑢𝑣
(55)
+ (𝜎4 + 𝜎5 𝑢 + 𝜎6 𝑣) 𝜉 (𝑡) .
Here, 𝜉(𝑡) is the multiplicative random excitation and the
external random excitation directly, and 𝜉(𝑡) is independent,
in possession of zero mean value and standard variance
Gaussian white noises, that is, E[𝜉(𝑡)] = 0, E[𝜉(𝑡)𝜉(𝑡 + 𝜏)] =
𝛿(𝜏). Note that the intensity of the noise 𝜎𝑖 (𝑖 = 1, 2, . . . , 6),
𝛾 = 𝛾𝑐 + 𝛾̃ is the bifurcation parameter. From (3)–(33), we
obtain (55) of the stochastic center manifold:
𝑧̇1 (𝑡) = − 𝜔𝑧2 + 𝑐10 𝑧1 + 𝑐100 𝑧2 + 𝑐11 𝑧13 + 𝑐12 𝑧12 𝑧2
+ 𝑐13 𝑧1 𝑦22 + 𝑐14 𝑧23 + 𝑐15 𝑧12 + 𝑐16 𝑧2 𝑧1
+ 𝑐17 𝑧22 + (𝑘11 𝑧1 + 𝑘12 𝑧2 + 𝑘13 ) 𝜉 (𝑡) ,
𝑧̇2 (𝑡) = 𝜔𝑧1 + 𝑐20 𝑧1 + 𝑐200 𝑧2 + 𝑐21 𝑧13 + 𝑐22 𝑧12 𝑧2
+ 𝑐23 𝑧1 𝑧22 + 𝑐24 𝑧23 + 𝑐25 𝑧12 + 𝑐26 𝑧2 𝑧1
+ 𝑐27 𝑧22 + (𝑘21 𝑧1 + 𝑘22 𝑧2 + 𝑘23 ) 𝜉 (𝑡) .
Here, the lengthy expressions of 𝑐𝑖𝑗 and 𝑘𝑖𝑗 are omitted.
(56)
Abstract and Applied Analysis
9
Setting the coordinate transformation 𝑧1 = 𝑟 cos 𝜃, 𝑧2 =
𝑟 sin 𝜃 and by substituting the variable in (34), we obtain
where 𝑊𝑟 (𝑡) and 𝑊𝜃 are the independent and standard
Wiener processes. Set the parameter as follows:
𝜇1 =
𝑟 ̇ (𝑡) = 𝑟𝑐10 cos2 𝜃 + 𝑟 (𝑐100 + 𝑐20 ) sin 𝜃 cos 𝜃 + 𝑟𝑐200 sin2 𝜃
3
4
+ 𝑟 [𝑐11 cos 𝜃 + (𝑐12 + 𝑐21 ) cos 𝜃 sin 𝜃
2
3
+ (𝑐13 + 𝑐22 ) cos 𝜃sin 𝜃 + (𝑐14 + 𝑐23 ) cos 𝜃sin 𝜃
+𝑐24 sin4 𝜃] + 𝑟2 [𝑐15 cos3 𝜃
1 2
2
),
(𝑘 + 𝑘23
2 13
2
2
2
2
+ 3𝑘22
+ 𝑘12
+ 𝑘21
+ 2𝑘12 𝑘21 + 2𝑘11 𝑘22 ,
𝜇4 = 3𝑘11
1
(𝑘 + 𝑘22 ) (𝑘21 − 𝑘12 ) ,
4 11
+(𝑐17 +𝑐26 ) cos 𝜃sin2 𝜃 +𝑐27 sin3 𝜃]
2
2
2
2
+ 𝑘22
+ 3𝑘12
+ 3𝑘21
− 2𝑘12 𝑘21 − 2𝑘11 𝑘22 ,
𝜇6 = 𝑘11
𝜇7 = 3𝑐11 + 3𝑐21 + 𝑐13 + 𝑐22 ,
𝜃̇ (𝑡) = − 𝜔 + 𝑐20 cos2 𝜃 + (𝑐200 − 𝑐10 ) sin 𝜃 cos 𝜃 − 𝑐100 sin2 𝜃
+ 𝑟2 [𝑐21 cos4 𝜃 + (𝑐21 − 𝑐12 ) cos3 𝜃 sin 𝜃
+ (𝑐23 + 𝑐12 ) cos2 𝜃sin2 𝜃
𝑑𝑟 = [(𝜇1 +
+ (𝑐27 − 𝑐16 ) cos 𝜃sin2 𝜃 − 𝑐17 sin3 𝜃]
+ [𝑘21 cos2 𝜃 + (𝑘22 − 𝑘11 ) cos 𝜃 sin 𝜃 − 𝑘12 sin2 𝜃] 𝜉 (𝑡)
(57)
It is difficult to calculate the exact solution for system
(57) today. According to the Khasminskii limit theorem,
when the intensities of the white noises 𝜎𝑖 (𝑖 = 1, . . . , 6)
are small enough, the response process {𝑟(𝑡), 𝜃(𝑡)} weakly
converged to the two-dimensional Markov diffusion process
[1–5]. Through the stochastic averaging method, we obtained
the It̂o stochastic differential equation (55) as follows:
𝜇2
𝜇
𝜇
) 𝑟 + 3 + 7 𝑟3 ] 𝑑𝑡
8
𝑟
8
𝜇4 2 1/2
1/2
𝑟 ) 𝑑𝑊𝑟 + (𝑟𝜇5 ) 𝑑𝑊𝜃 ,
8
𝜇8 2
1/2
𝑑𝜃 = (𝜇10 + 𝑟 ) 𝑑𝑡 + (𝑟𝜇5 ) 𝑑𝑊𝑟
8
+ (𝜇3 +
𝜇3 𝜇6 1/2
+ ) 𝑑𝑊𝜃 ,
𝑟2
8
3.1. Stochastic D-Bifurcation. In the section, we will see
how the introduction of randomness changes the stochastic
behavior significantly from both the dynamical and phenomenological points of view.
Proof. When 𝜇3 = 0, 𝜇7 = 0, Then, system (58) becomes
+ 𝑟 [𝑏25 cos 𝜃 + (𝑐26 − 𝑐15 ) cos2 𝜃 sin 𝜃
1
[𝑘 cos 𝜃 − 𝑘13 sin 𝜃] 𝜉 (𝑡) .
𝑟 23
𝜇8 = 3𝑐21 − 3𝑐14 + 𝑐23 − 𝑐12 .
(59)
Theorem 8 (D-bifurcation). Let 𝜇3 = 0, 𝜇7 = 0. Then, system
(55) undergoes stochastic D-bifurcation.
+ (𝑐24 + 𝑐23 ) cos 𝜃sin3 𝜃 + 𝑐14 sin4 𝜃]
+(
𝜇3 =
𝜇5 =
+𝑘22 sin2 𝜃] 𝜉 (𝑡) + [𝑘13 cos 𝜃 + 𝑘23 sin 𝜃] 𝜉 (𝑡) ,
𝑑𝑟 = [(𝜇1 +
1
(𝑐 + 𝑐 ) ,
2 20 100
+ (𝑐16 + 𝑐25 ) cos2 𝜃 sin 𝜃
+ 𝑟 [𝑘11 cos2 𝜃 + (𝑘12 + 𝑘21 ) cos 𝜃 sin 𝜃
+
𝜇10 = −𝜔 +
2
2
2
2
+ 5𝑘22
+ 3𝑘21
+ 3𝑘12
+ 6𝑘12 𝑘21 − 2𝑘11 𝑘22 ,
𝜇2 = 5𝑘11
3
2
1
(𝑐 + 𝑐 ) ,
2 10 200
(58)
1/2
𝜇2
𝜇
) 𝑟] 𝑑𝑡 + ( 4 𝑟2 ) 𝑑𝑊𝑟 .
8
8
(60)
When 𝜇4 = 0, (60) is a deterministic system, and there is
no bifurcation phenomenon. Here, we discuss the situation
𝜇4 ≠ 0, let
𝑚 (𝑟) = (𝜇1 +
𝜇2 𝜇4
− ) 𝑟,
8 16
𝜎 (𝑟) = (
𝜇4 1/2
) 𝑟.
8
(61)
The continuous random dynamical system generated by (60)
is
𝑡
𝑡
0
0
𝜑 (𝑡) 𝑥 = 𝑥 + ∫ 𝑚 (𝜑 (𝑠) 𝑥) 𝑑𝑠 + ∫ 𝜎 (𝜑 (𝑠) 𝑥) ∘ 𝑑𝑊𝑟 , (62)
where ∘𝑑𝑊𝑟 is the differential in the sense of Statonovich, and
it is the unique strong solution of (60) with the initial value
𝑥. And 𝑚(0) = 0, 𝜎(0) = 0, so 0 is a fixed point of 𝜑. Since
𝑚(𝑟) is bounded and for any 𝑟 ≠ 0, it satisfies the ellipticity
condition 𝜎(𝑟) ≠ 0. This ensures that there is at most one
stationary probability density. According to the Itô equation
of amplitude 𝑟(𝑡), we obtain its Fokker-Planck-Kolmogorov
equation corresponding to (60) as follows:
𝜇
𝜇
𝜕𝑝
𝜕
𝜕2
= − {[(𝜇1 + 2 ) 𝑟] 𝑝} + 2 {[ 4 𝑟2 ] 𝑝} .
𝜕𝑡
𝜕𝑟
8
𝜕𝑟
8
(63)
Let 𝜕𝑝/𝜕𝑡 = 0, then we obtain the solution of system (63)
𝑡
2𝑚 (𝑢)
󵄨
󵄨
𝑑𝑢) .
𝑝 (𝑡) = 𝑐 󵄨󵄨󵄨󵄨𝜎−1 (𝑡)󵄨󵄨󵄨󵄨 exp (∫ 2
0 𝜎 (𝑢)
(64)
10
Abstract and Applied Analysis
The above equation (63) has two kinds of equilibrium
states: fixed point and nonstationary motion. The invariant
measure of the former is 𝛿0 , and its probability density is 𝛿𝑥 .
The invariant measure of the latter is 𝜈, and its probability
density is (64). In the following, we calculate the Lyapunov
exponent of the two invariant measures.
Using the solution of linear Itô stochastic differential
equation, we obtain the solution of system (60):
𝑡
Proof. By simplifying (64), we can obtain
𝑝st (𝑟) = 𝑐𝑟2(8𝜇1 +𝜇2 −𝜇4 )/𝜇4 ,
󸀠
𝑡
(65)
+ ∫ 𝜎󸀠 (0) 𝑑𝑊𝑟 ) .
0
The Lyapunov exponent with regard to 𝜇 of the dynamical
system 𝜑 is defined as:
1
𝜆 𝜑 (𝜇) = lim
𝑡 → +∞ 𝑡
ln ‖𝑟 (𝑡)‖ ,
𝑝st (𝑟) = 𝑜 (𝑟𝑣 )
𝜆 𝜑 (𝛿0 )
𝑡
𝑡
1
(ln ‖𝑟 (0)‖ + 𝑚󸀠 (0)∫ 𝑑𝑠 + 𝜎󸀠 (0) ∫ 𝑑𝑊𝑟 (𝑠))
𝑡 → +∞ 𝑡
0
0
= lim
𝑊 (𝑡)
= 𝑚 (0) + 𝜎 (0) lim 𝑟
𝑡 → +∞
𝑡
Remark 10. Unfortunately, these two definitions (D-bifurcation and P-bifurcation) do not necessarily yield the same
result.
Theorem 11 (stochastic pitchfork bifurcation). Let 𝛾𝛼𝜅 = 0,
𝜇3 = 0, 𝜇7 < 0. Then, system (55) undergoes stochastic
pitchfork bifurcation.
Proof. When 𝜇3 = 0, 𝜇7 ≠ 0, then (58) can be rewritten as
𝑑𝑟 = [(𝜇1 +
󸀠
= 𝑚 (0)
1/2
𝜇2
𝜇
𝜇
) 𝑟 + 7 𝑟3 ] 𝑑𝑡 + ( 4 𝑟2 ) 𝑑𝑊𝑟 .
8
8
8
(71)
Let 𝛼 = 𝜇1 + 𝜇2 /8, 𝜙 = (√−𝜇7 /8)𝑟, 𝜇7 < 0, then the system
(71) becomes
𝜇
𝜇
= 𝜇1 + 2 − 4 .
8 16
(67)
For the invariant measure which regard (65) as its density, we
obtain the Lyapunov exponent:
= ∫ [𝑚󸀠 (𝑟) +
32√2𝜇43/2 (𝜇1
(72)
which is solved by
𝜇2
𝜇 1/2
) 𝑡 + ( 4 ) 𝑊𝑡 (𝜔)))
8
8
𝑡
× ( (1 + 2𝜙2 ∫ exp (2 ((𝜇1 +
2
16
= − 32√2𝜇43/2 𝑚(𝑟)2 exp [ 𝑚 (𝑟)]
𝜇4
𝜇2
𝜇 1/2
) 𝜙 − 𝜙3 ] 𝑑𝑡 + ( 4 ) 𝜙 ∘ 𝑑𝑊𝑡 ,
8
8
= (𝜙 exp ((𝜇1 +
𝜎 (𝑟) 𝜎󸀠 (𝑟)
] 𝑝 (𝑟) 𝑑𝑟
2
𝑚 (𝑟)
] 𝑝 (𝑟) 𝑑𝑟
= − 2∫ [
𝑅 𝜎 (𝑟)
𝑑𝜙 = [(𝜇1 +
𝜙 󳨀→ 𝜓𝛼 (𝑡, 𝜔) 𝜙
1 𝑡 󸀠
∫ [𝑚 (𝑟) + 𝜎 (𝑟) 𝜎󸀠 (𝑟)] 𝑑𝑠
𝑡 → +∞ 𝑡 0
𝜆 𝜑 (𝜈) = lim
× exp [
(70)
where 𝑣 = 2(8𝜇1 +𝜇2 −𝜇4 )/𝜇4 . Obviously, when 𝑣 < −1, that is,
𝜇1 +𝜇2 /8−𝜇4 /16 < 0, 𝑝st (𝑟) is a 𝛿 function. When −1 < 𝑣 < 0,
that is, 𝜇1 + 𝜇2 /8 − 𝜇4 /16 > 0, 𝑟 = 0 is a maximum point of
𝑝st (𝑟) in the state space; thus, the system admits D-bifurcation
when 𝑣 = −1, that is, 𝜇1 + 𝜇2 /8 − 𝜇4 /16 = 0, is the critical
condition of D-bifurcation at the equilibrium point. When
𝑣 > 0, there is no point that makes 𝑝st (𝑟) has the maximum
value; thus, the system does not admits P-bifurcation.
󸀠
= −
𝑟 󳨀→ 0,
(66)
substituting (65) into (66), note that 𝜎(0) = 0, 𝜎󸀠 (0) = 0, we
obtain the Lyapunov exponent of the fixed point:
𝑅
(69)
where 𝑐 is a normalization constant; thus, we have
󸀠
𝜎 (0) 𝜎 (0)
] 𝑑𝑠
𝑟 (𝑡) = 𝑟 (0) exp (∫ [𝑚 (0) +
2
0
󸀠
Theorem 9. Let 𝜇3 = 0, 𝜇7 = 0. Then, system (55) does not
undergo stochastic P-bifurcation.
0
(68)
𝜇
𝜇 2
+ 2 − 4)
8 16
𝜇
𝜇
16
(𝜇 + 2 − 4 )] .
𝜇4 1 8 16
Let 𝛼 = 𝜇1 +𝜇2 /8−𝜇4 /16. We can obtain that the invariant
measure of the fixed point is stable when 𝛼 < 0, but the
invariant measure of the nonstationary motion is stable when
𝛼 > 0, so 𝛼 = 𝛼𝐷 = 0 is a point of 𝐷-bifurcation.
𝜇2
)𝑡
8
𝜇 1/2
+( 4 ) 𝑊𝑠 (𝜔))) 𝑑𝑠)
8
1/2
−1
) .
(73)
We now determine the domain 𝐷𝛼 (𝑡, 𝜔), where
𝐷𝛼 (𝑡, 𝜔) := {𝜙 ∈ R : (𝑡, 𝜔, 𝜙) ∈ 𝐷} (𝐷 = R × Ω × 𝑋)
is (in general possibly empty) the set of initial values 𝜙 ∈ R
for which the trajectories still exist at time 𝑡 and the range
𝑅𝛼 (𝑡, 𝜔) of 𝜓𝛼 (𝑡, 𝜔) : 𝐷𝛼 (𝑡, 𝜔) → 𝑅𝛼 (𝑡, 𝜔).
We have
𝐷𝛼 (𝑡, 𝜔) = {
R,
𝑡 ≥ 0,
(−𝑑𝛼 (𝑡, 𝜔) , 𝑑𝛼 (𝑡, 𝜔)) , 𝑡 < 0,
(74)
Abstract and Applied Analysis
11
(i) For 𝛼 ≤ 0, the only invariant measure is 𝜇𝜔𝛼 = 𝛿0 .
(ii) For 𝛼 > 0, we have the three invariant forward
𝛼
= 𝛿±𝑘𝛼 (𝜔) , where
Markov measures, 𝜇𝜔𝛼 = 𝛿0 and 𝜐±,𝜔
where
𝑑𝛼 (𝑡, 𝜔)
󵄨󵄨 𝑡
𝜇
󵄨
= 1 × ( (2 󵄨󵄨󵄨󵄨∫ exp (2 (𝜇1 + 2 ) 𝑡
8
󵄨󵄨 0
𝑘𝛼 (𝜔) := (2 ∫
−∞
󵄨󵄨 1/2
𝜇4 1/2
󵄨
+ 2( ) 𝑊𝑠 (𝜔)) 𝑑𝑠󵄨󵄨󵄨󵄨) )
8
󵄨󵄨
exp (2 (𝜇1 +
−1
𝜇2
)𝑡
8
𝜇 1/2
+2( 4 ) 𝑊𝑡 (𝜔)) 𝑑𝑠)
8
(80)
−1/2
.
We have E𝑘𝛼2 (𝜔) = 𝛼. Solving the forward Fokker-PlanckKolmogorov equation
> 0,
𝑅𝛼 (𝑡, 𝜔) = 𝐷𝛼 (−𝑡, 𝜗 (𝑡) 𝜔)
={
0
𝐿∗ 𝑝𝛼 = − (((𝜇1 +
(−𝑟𝛼 (𝑡, 𝜔) , 𝑟𝛼 (𝑡, 𝜔)) , 𝑡 > 0,
R,
𝑡 ≤ 0,
󸀠
𝜇2
𝜇
) 𝜙 + 4 𝜙 − 𝜙3 ) 𝑃𝛼 (𝜙))
8
16
󸀠󸀠
𝜇
+ 4 (𝜙2 𝑃𝜈1 (𝜙)) = 0,
16
(75)
(81)
yield
where
(i) 𝑝𝛼 = 𝛿0 for all 𝛼,
(ii) for 𝑝𝛼 > 0,
𝑟𝛼 (𝑡, 𝜔) = 𝑑𝛼 (−𝑡, 𝜗 (𝑡) 𝜔)
= (exp ((𝜇1 +
𝜇2
𝜇 1/2
) 𝑡 + ( 4 ) 𝑊𝑡 (𝜔)))
8
8
2
𝑞𝛼+
󵄨󵄨 𝑡
𝜇
󵄨
× ( (2 󵄨󵄨󵄨󵄨∫ exp (2 (𝜇1 + 2 ) 𝑡
8
󵄨󵄨 0
−1
󵄨󵄨 1/2
𝜇 1/2
󵄨
+ 2( 4 ) 𝑊𝑠 (𝜔)) 𝑑𝑠󵄨󵄨󵄨󵄨) )
8
󵄨󵄨
> 0.
(76)
{ 𝑁 𝜙(2(𝜇1 +𝜇2 /8)/𝜇4 )−1 exp ( 16𝜙 ) , 𝜙 > 0,
(𝜙) = { 𝛼
(82)
𝜇4
0,
𝜙 ≤ 0,
{
and 𝑞𝛼+ (𝜙) = 𝑞𝛼+ (−𝜙), where 𝑁𝛼− = Γ(𝜈1 /𝜇4 )(𝜇4 /8)(𝜇1 +𝜇2 /8)/𝜇4 .
𝛼
= 𝛿±𝑘𝛼 (𝜔) are
Naturally, the invariant measures 𝜐±,𝜔
those corresponding to the stationary measures 𝑞𝛼+ . Hence,
all invariant measures are Markov measures.
We determine all invariant measures (necessarily Dirac
measure) of local RDS 𝜒 generated by the SDE
𝜇2
𝜇 1/2
(83)
) 𝜙 − 𝜙3 ] 𝑑𝑡 + ( 4 ) 𝜙 ∘ 𝑑𝑊,
8
8
on the state space R, 𝜇1 + 𝜇2 /8 ∈ R. We now calculate the
Lyapunov exponent for each of these measures.
The linearized RDS 𝜒𝑡 = 𝐷Υ(𝑡, 𝜔, 𝜙)𝜒 satisfies the
linearized SDE
𝜇
𝜇 1/2
2
𝑑𝜒𝑡 = [(𝜇1 + 2 ) − 3(Υ (𝑡, 𝜔, 𝜙)) 𝜒𝑡 ] 𝑑𝑡 + ( 4 ) 𝜒𝑡 ∘ 𝑑𝑊.
8
8
(84)
𝑑𝜙 = [(𝜇1 +
We can now determine that
𝐸𝛼 (𝜔) := ⋂ 𝐷𝛼 (𝑡, 𝜔)
𝑡∈R
(77)
and obtain
𝜇2
−
−
{
{ (−𝑑𝛼 (𝑡, 𝜔) , 𝑑𝛼 (𝑡, 𝜔)) , 𝛼 = 𝜇1 + 8 > 0,
𝐸𝛼 (𝜔) = {
𝜇
{
𝛼 = 𝜇1 + 2 ≤ 0,
{0} ,
8
{
(78)
where
Hence,
𝐷Υ (𝑡, 𝜔, 𝜙) 𝜒 = 𝜒 exp ((𝜇1 +
𝜇2
𝜇 1/2
) 𝑡 + ( 4 ) 𝑊𝑡 (𝜔)
8
8
𝑡
0 < 𝑑𝛼± (𝑡, 𝜔)
(85)
2
−3 ∫ (Υ (𝑠, 𝜔, 𝜙)) 𝑑𝑠) .
0
󵄨󵄨 ±∞
𝜇
󵄨
= 1 × ( (2 󵄨󵄨󵄨󵄨∫ exp (2 (𝜇1 + 2 ) 𝑡
8
󵄨󵄨 0
+(
󵄨󵄨 1/2 −1
𝜇4 1/2
󵄨
) 𝑊𝑠 (𝜔)) 𝑑𝑠󵄨󵄨󵄨󵄨) )
8
󵄨󵄨
< ∞.
(79)
The ergodic invariant measures of system (71) are as follows.
Thus, if 𝜈𝜔 = 𝛿𝜙0 (𝜔) is a Υ-invariant measure, its Lyapunov
exponent is
1
󵄩
󵄩
𝜆 (𝜇) = lim log 󵄩󵄩󵄩𝐷Υ (𝑡, 𝜔, 𝜙) 𝜒󵄩󵄩󵄩
𝑡→∞ 𝑡
𝑡
𝜇
2
(86)
= (𝜇1 + 2 ) − 3 lim ∫ (Υ (𝑠, 𝜔, 𝜙)) 𝑑𝑠
𝑡→∞ 0
8
𝜇
= (𝜇1 + 2 ) − 3E𝜙02 ,
8
providing the IC 𝜙02 ∈ 𝐿1 (P) is satisfied.
12
Abstract and Applied Analysis
𝛼
(i) For 𝜇1 + 𝜇2 /8 ∈ R, the IC for 𝜈±,𝜔
= 𝛿0 is trivially
satisfied, and we obtain
𝜆 (𝜈1𝛼 )
𝜇
= 𝜇1 + 2 .
8
is stable for 𝜇1 + 𝜇2 /8 < 0 and unstable for
So,
𝜇1 + 𝜇2 /8 > 0.
𝛼
(ii) For 𝜇1 + 𝜇2 /8 > 0, 𝜈2,𝜔
= 𝛿𝑑𝜔𝛼 is F0−∞ measurable;
hence, the density 𝑝𝛼 of 𝜌𝛼 = E𝜈2𝛼 satisfies the FokkerPlanck-Kolmogorov equation
󸀠
𝜇2
𝜇
) 𝜙 + 4 𝜙 − 𝜙3 ) 𝑝𝛼 (𝜙))
8
16
󸀠󸀠
𝜇
+ 4 (𝜙2 𝑝𝛼 (𝜙)) = 0,
16
(88)
which has the unique probability density solution
𝑃𝛼 (𝜙) = 𝑁𝛼 𝜙(2(𝜇1 +𝜇2 /8)/𝜇4 )−1 exp (
𝜙2 8
),
𝜇4
𝜙 > 0.
(89)
E𝜈2𝛼 𝜙 =
2
E(𝑑−𝛼 )
∞
2 𝛼
= ∫ 𝜙 𝑝 (𝜙) 𝑑𝜙 < ∞,
0
(90)
the IC is satisfied. The calculation of the Lyapunov exponent
is accomplished by observing that
(96)
The two families of densities (𝑞𝛼+ )𝛼>0 clearly undergo a Pbifurcation at the parameter value 𝛼𝑃 = 𝜇4 /8, which is the
same value as the transcritical case, since the SDE linearized
at 𝜙 = 0 is the same in the next section. In both cases, we have
a D-bifurcation of the trivial reference measure 𝛿0 at 𝛼𝐷 = 0
and a P-bifurcation of 𝛼𝑃 = (𝜇1 + 𝜇2 /8)2 /2. Then, system (55)
has stochastic pitchfork bifurcation.
3.2. P-Bifurcation. In the following, we investigate the steadystate probability density 𝑝st (𝑟) of the linear Itô stochastic differential equation. Calculating extreme values of the steadystate probability density is one of the most popular efficient
methods in studying the bifurcation of a nonlinear dynamical
system. The steady-state probability density is an important
characteristic value of stochastic bifurcation.
𝑑𝑟 = [(𝜇1 +
1/2
𝜇2
𝜇
𝜇
) 𝑟 + 7 𝑟3 ] 𝑑𝑡 + ( 4 𝑟2 ) 𝑑𝑊𝑟 .
8
8
8
exp (2 (𝜇1 + 𝜇2 /8) 𝑡 + 2(𝜇4 /8)
1/2
𝑡
2 ∫−∞ exp (2 (𝜇1 + 𝜇2 /8) 𝑠 + 2(𝜇4 /8)
𝑊𝑡 (𝜔))
1/2
𝑊𝑠 (𝜔)) 𝑑𝑠
Ψ󸀠 (𝑡)
,
=
2Ψ
(91)
where
Ψ (𝑡) = ∫
𝑡
−∞
exp ((𝜇1 +
𝜇2
𝜇 1/2
) 𝑠 + ( 4 ) 𝑊𝑠 (𝜔)) 𝑑𝑠. (92)
8
8
2
E(𝑑−𝛼 ) =
𝜇
1
1
lim log Ψ (𝑡) = 𝜇1 + 2 ,
𝑡
→
∞
2
𝑡
8
(93)
finally
𝜆 (𝜈2𝛼 ) = −2 (𝜇1 +
𝜇2
) < 0.
8
2
2
𝜇2
,
8
(95)
(98)
with the initial value condition 𝜇7 = 0, 𝑝(𝑟, 𝑡|𝑟0 , 𝑡0 ) → 𝛿(𝑟 −
𝑟0 ), 𝑡 → 𝑡0 , where 𝑝(𝑟, 𝑡|𝑟0 , 𝑡0 ) is the transition probability
density of diffusion process 𝑟(𝑡). The invariant measure of
𝑟(𝑡) is the steady-state probability density 𝑝st (𝑟) which is the
solution of the degenerate system as follows:
0= −
𝜇
𝜇
𝜕
{[(𝜇1 + 2 ) 𝑟 − 4 𝑟 − 𝑟3 ] 𝑝 (𝑟)}
𝜕𝑟
8
2𝜇7
(99)
Through calculation, we can obtain
𝑝st (𝑟) =
(94)
𝛼
= 𝛿𝑑𝜔𝛼 is F0−∞ measurable.
(iii) For (𝜇1 + 𝜇2 /8) > 0, 𝜈2,𝜔
Since L(𝑑+𝛼 ) = L(𝑑−𝛼 )
E(−𝑑−𝛼 ) = E(𝑑−𝛼 ) = 𝜇1 +
𝜇 𝜕2
− 4 2 (𝑟2 𝑝 (𝑟)) ,
2𝜇7 𝜕𝑟
𝜇 𝜕2
− 4 2 (𝑟2 𝑝 (𝑟)) .
2𝜇7 𝜕𝑟
Hence, by the ergodic theorem
(97)
According to the Itô equation of amplitude 𝑟(𝑡), we obtain
its Fokker-Planck-Kolmogorov equation form (97) as follows:
𝜇
𝜕𝑝
𝜇
𝜕
= −
{[(𝜇1 + 2 ) 𝑟 − 4 𝑟 − 𝑟3 ] 𝑝 (𝑟)}
𝜕𝑡
𝜕𝑟
8
2𝜇7
2
𝑑−𝛼 (𝜗𝑡 𝜔)
=
𝜇2
) < 0.
8
Case I. When 𝜇3 = 0, 𝜇7 ≠ 0, then (38) can be rewritten as
Since
2
𝜆 (𝜈2𝛼 ) = −2 (𝜇1 +
(87)
𝜈1𝛼
𝐿∗𝜈1 = − (((𝜇1 +
thus
exp (𝑟2 𝜇7 /𝜇4 ) 𝑟−1−(2(𝜇1 +𝜇2 /8)𝜇7 /𝜇4 )
(𝜇1 +𝜇2 /8)𝜇7 /𝜇4
Γ [− (𝜇1 + 𝜇2 /8) 𝜇7 /𝜇4 ] (−𝜇7 /𝜇4 )
.
(100)
According to Namachivaya’s theory, the extreme value
of an steady-state probability density contains the most
important essence of the nonlinear stochastic system. In
other words, the steady-state probability density can uncover
the characteristic information of the steady state. When the
intension of the noise tends to zero, the extreme values of
Abstract and Applied Analysis
13
0.8
0.8
Pst (r)
Pst (r)
0.6
0.4
0.2
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
0.4
0.3
0.6
0.5
r
0.7
0.8
0.9
r
(a)
(b)
0.8
Pst (r)
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1
r
(c)
Figure 1: P-Bifurcation of system (55) with 𝜇1 = −0.2, 𝜇3 = 0, 𝜇4 = 5.1183, 𝜇7 = −20.4732; (a) P-bifurcation of 𝑝𝛼 (𝑟) at 𝛼𝑝 < 𝜎2 /2, 𝜇2 = 5.6,
where 𝛼 = 𝜇1 + 𝜇2 /8, 𝜎2 = −𝜇4 /𝜇7 ; (b) P-bifurcation of 𝑝𝛼 (𝑟) at 𝛼𝑝 = 𝜎2 /2, 𝜇2 = 9.5907, where 𝛼 = (𝜇1 + 𝜇2 /8), 𝜎2 = −𝜇4 /𝜇7 ; (c) P-bifurcation
of 𝑝𝛼 (𝑟) at 𝛼𝑝 > 𝜎2 /2, 𝜇2 = 17.6, where 𝛼 = 𝜈1 /8, 𝜎2 = −𝜈3 /𝜈2 .
𝑝st (𝑟) approximate to show the behavior of the deterministic
system. If the process 𝑟(𝑡) is ergodic, then 𝑝st (𝑟) can be
regarded as the time measurement for staying in the neighborhood of 𝑟(𝑡) according to the Oseledec ergodic theorem.
From the analysis above, if 𝑝st (𝑟) has a maximum value
at 𝑟∗ , the sample trajectory will stay for a longer time in
the neighborhood of 𝑟∗ , that is, 𝑟∗ is stable in the meaning
of probability (with a bigger probability). If 𝑝st (𝑟) has a
minimum value (zero), it is just the opposite.
We now calculate the most possible amplitude 𝑟∗ of
system (97), that is, 𝑝st (𝑟) has a maximum value at 𝑟∗ . So, we
have
𝑑𝑝st (𝑟) 󵄨󵄨󵄨󵄨
= 0,
󵄨
𝑑𝑟 󵄨󵄨󵄨𝑟=𝑟∗
󵄨
𝑑2 𝑝st (𝑟) 󵄨󵄨󵄨
󵄨󵄨
<0
𝑑𝑟2 󵄨󵄨󵄨𝑟=𝑟∗
(101)
and the solution 𝑟 = 𝑟̃ = √(𝜈1 𝜈2 + 4𝜈3 )/8𝜈2 . The probabilities
and the positions of the Hopf bifurcation occur with different
parameter.
Since
󵄨
𝑑2 𝑝st (𝑟) 󵄨󵄨󵄨
󵄨󵄨
𝑑𝑟2 󵄨󵄨󵄨=̃𝑟
𝜇
1
= (24+(3(𝜇1 +𝜇2 /8)𝜇2 /𝜇3 ) exp ( (4 + 7 ))
8
𝜇3
× 𝜇7 (−
𝜇7 −(𝜇1 +𝜇2 /8)𝜇7 /𝜈4
)
𝜇4
−2(𝜇1 +𝜇2 /8)𝜇7 /4𝜇4
× (√
(8 (𝜇1 + 𝜇2 /8) 𝜇7 + 4𝜇4 )
)
𝜇7
)
−1
× (Γ [−
(𝜇1 + 𝜇2 /8) 𝜇7
𝜇 (𝜇1 +𝜇2 /8)𝜇7 /𝜇4
](− 7 )
)
𝜇4
𝜇4
< 0,
(102)
thus, what we need is 𝑟∗ = 𝑟̃. The conclusion is to go
all the way with what has been obtained by the singular
boundary theory. The original nonlinear stochastic system
has a stochastic Hopf bifurcation at 𝑟 = 𝑟̃, where
𝑥12 + 𝑥22 =
8 (𝜇1 + 𝜇2 /8) 𝜇7 + 4𝜇4
.
8𝜇7
(103)
We now choose some values of the parameters in the
equations, draw the graphics of 𝑝st (𝑟) (Figure 1). It is worth
putting forward that calculating the Hopf bifurcation with
the parameters in the original system is necessary. If we now
have values of the original parameters in system (55), then
𝛼 = 0.778205, 𝛽 = 0.024530, 𝜅 = 0.08, 𝛾 = 5.640593, 𝛿 =
0.200101, 𝜎1 = 0.1, 𝜎2 = 0.1, 𝜎3 = 0.2, 𝜎4 = 0.3, 𝜎5 = 0.1, 𝜎6 =
14
Abstract and Applied Analysis
0.5. After further calculations, we obtain 𝜇1 = −0.2, 𝜇2 =
9.5907, 𝜇3 = 0, 𝜇4 = 5.1183, 𝜇7 = −20.4732, then
2
𝑝 (𝑟) = 0.28209𝑒−4𝑟 .
(104)
What is more is that 𝑟̃ = 0.28209, where 𝑝st (𝑟) has the
maximum value.
4. Conclusion
In this work, we investigate both the deterministic and
stochastic bifurcations of the catalytic CO oxidation on Ir(111)
surfaces with multiple delays. By replacing 𝛾̃ with 𝛾 − 𝛾𝑐 ,
it can be readily seen that the addition of the noise gives
rise to a shift in the critical bifurcation point 𝛾𝑐 at the
Hopf bifurcation. The results of our computational studies
using the Lyapunov exponents and the integral stochastic
averaging combined with the Hopf bifurcation produced
stability boundaries for structural systems that were much
sharper and well defined than those obtained in the earlier
stability studies of the same systems. The influence of noise
on the reaction-diffusion system model has been studied by
the authors [11–18], but explicit and derived conditions for
stochastic stability and bifurcation, as the ones displayed here,
were not transparent in their publications.
Acknowledgments
This paper was supported by the National Natural Science
Foundation of China (no. 11201089) and the Guangxi Natural
Science Foundation (no. 2011GXNSFB018065).
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