Commun Nonlinear Sci NumerXuechuan Wang ∗, Weicheng Pei, Satya N. Atluri Center for Advanced Research in the Engineering Sciences, Texas Tech University, Lubbock, TX 79415, USA a
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Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat
journal homepage: www.elsevier.com/locate/cnsns
Research paper
Bifurcation & chaos in nonlinear structural dynamics: Novel &
Although we have already obtained the iterative formula (13) through collocation, the matrix of generalized Lagrange
multipliers λ in it remains a puzzle. Directly solving the constraint Eq. (10) for λ is unlikely for most nonlinear cases.
The following introduces three approaches to bypass this dilemma, from which 3 OFAPI algorithms are developed. The first
algorithm OFAPI-1 is mathematically equivalent to Eq. (13) , but it involves inversion of Jacobian matrix. OFAPI-2 and 3
approximate λ with truncated polynomial and exponential series respectively, thus no matrix inversion is required therein.
Although the OFAPI algorithms-1, 2, and 3 are derived from the same optimal error-feedback iteration concept, they have
some differences in implementation and computational performances that the users of them should be aware of. As will
be shown in the flow chart overview of these three algorithms in Fig. 2 , in the implementations of OFAPI algorithms-1
and 3, the initial conditions have to be incorporated in the iteration formula and the excess collocation equations need
to be removed. In OFAPI algorithm-2, the initial conditions are naturally satisfied, thus the implementation of it is more
straightforward than OFAPI algorithms-1 and 3.
The convergence speed of the OFAPI algorithm-1 is supposed to be the fastest, since it is equivalent to Eq. (12) , where
the matrix of weighting functions λ is optimally derived. However, the OFAPI algorithm-1 involves inversion of the Jacobian
matrix ( E +
J ) , which is varying during the iteration process. For systems with high dimensions, computing the inversion of
X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 59
Fig. 2. Flow chart overview of the OFAPI algorithms.
the Jacobian matrix could be very time consuming. In addition, if the Jacobian matrix becomes ill-conditioned during the
iteration, the OFAPI algorithm-1 may easily diverge. On the contrary, the OFAPI algorithms-2 and 3 are free from inverting
matrices during the iteration process.
The formulations of these 3 algorithms are presented in this section. Further detailed comparisons between them are
made in Section 3 and 4 through numerical simulations.
60 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69
2.3.1. OFAPI algorithm-1
Differentiating Eq. (4) leads to
d
dt
[u 1 (t) u 2 (t)
]c
=
d
dt
[u 1 (t) u 2 (t)
]+ λ(t) R (t) +
∫ t
t i
∂λ
∂t R (τ ) dτ , (21)
where R ( τ ) is the residual function in Eq. (3) . According to Eq. (10) and using the theory of Magnus series, it is proved
[17] that
λ(t) = −I , and
∂λ
∂t = −J (t) λ. (22)
Substituting them into Eq. (21) , we have
d
dt
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]− J (t)
∫ t
t i
λ
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ . (23)
Noting that ∫ t
t i λ{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F (t)
]}dτ =
[u 1
u 2
]c −
[u 1
u 2
]accor ding to the optimal error-feedback itera-
tion formula (4) , Eq. (23) is rewritten as
d
dt
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]− J (t)
{[u 1
u 2
]c
−[
u 1
u 2
]}. (24)
After rearrangement, it is rewritten as
d
dt
[u 1 (t) u 2 (t)
]c
+ J (t)
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]+ J ( t )
[u 1 ( t ) u 2 ( t )
]. (25)
Eq. (25) can be regarded as equivalent to Eq. (4) . By collocating in the local time interval [ t i , t i +1 ] , an algebraic iterative
formula is obtained as. [u 1
′ ( t m
) u 2
′ ( t m
)
]c
+ J ( t m
)
[u 1 ( t m
) u 2 ( t m
)
]c
= −[−u 2 ( t m
) M
−1 C u 2 ( t m
) + M
−1 N ( t m
) − M
−1 F ( t m
)
]+ J ( t m
)
[u 1 ( t m
) u 2 ( t m
)
], (26)
where m = 1 , 2 , ...M, t m
∈ [ t i , t i +1 ] , and t i +1 − t i is a finite large time interval.
After rearranging the sequence of the collocation equations and using the relationship in Eq. (18), Eq. (26) is rewritten
as
( E +
J )
[U 1
U 2
]c
= −[−U 2
˜ M
−1 ˜ C U 2 +
˜ M
−1 ˜ N − ˜ M
−1 ˜ F
]+
J
[U 1
U 2
], (27)
where
U p =
[u p, 1 ( t 1 ) u p, 1 ( t 2 ) ... u p, 1 ( t M
) u p, 2 ( t 1 ) u p, 2 ( t 2 ) ... u p, 2 ( t M
) ... ]T
, p = 1 , 2 . (28)
The configuration of the matrices ˜ E , J , ˜ M , ˜ C , ˜ N , ˜ F are provided in Appendix C .
It should be noted that the initial conditions are not incorporated in Eq. (27) . For that, without loss of generality, we usu-
ally select the first collocation point at the initial boundary, of which the values u p, e ( t 1 ) are given. By doing that, Eq. (27) be-
comes overdetermined, thus it is necessary to drop excess collocation equations at time t 1 . Finally, Eq. (27) is modified as
[U 1
U 2
]c
r
=
[U 1
U 2
]r
− ( E
r +
J r ) −1
{˜ E
[U 1
U 2
]+
[−U 2
˜ M
−1 ˜ C U 2 +
˜ M
−1 ˜ N − ˜ M
−1 ˜ F
]}r
, (29)
The symbol [ •] r denotes the remained matrix after the (lM + 1) th rows and columns in [ •] are dropped, l = 0 , 1 , 2 , ... . If
[ •] r is a vector, we just need to remove the (lM + 1) th rows in [ •].
A flow diagram is presented in Fig. 2 to illustrate the OFAPI algorithm-1 along with the other two algorithms.
2.3.2. OFAPI algorithm-2
According to Eq. (10) , the Lagrange multipliers can be approximated by Taylor series as
λ(τ ) = T 0 + T 1 (τ − t) + T 2 (τ − t) 2 + ..., (30)
where T 0 = −I , T 1 = −J (t) , T 2 = −J (t) 2 / 2 , and so on. To generally obtain the Taylor series approximation of λ( τ ), one may
use the Differential Transform Method (DTM) [32] . Substituting it into Eq. (13) , the correctional formula is obtained as
X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 61
[u 1 ( t m
) u 2 ( t m
)
]c
=
[u 1 ( t m
) u 2 ( t m
)
]+
∫ t m
t i
[−I − J (t)(τ − t) − J (t)
2
2
(τ − t) 2 + · · ·
]{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ
(31)
In implementations, λ( τ ) is commonly approximated by truncated Taylor series [24] . The simplest could be the zeroth
order approximation
λ(τ ) = −I , (32)
or the first order approximation
λ(τ ) = −I − J (t)(τ − t) . (33)
Higher order approximations are possible, but they are rarely used in practice considering the computational complexity.
With the zeroth order approximation of λ( τ ), Eq. (31) becomes [u 1 ( t m
) u 2 ( t m
)
]c
=
[u 1 ( t m
) u 2 ( t m
)
]−
∫ t m
t i
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ . (34)
With the first order approximation of λ( τ ), Eq. (31) becomes [u 1 ( t m
) u 2 ( t m
)
]c
=
[u 1 ( t m
) u 2 ( t m
)
]+
∫ t m
t i
[ −I − J ( t m
)(τ − t m
)]
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ . (35)
where m = 1 , 2 , ...M, t m
∈ [ t i , t i +1 ] , and t i +1 − t i is a finite large time interval. By separating the terms involving t m
and τ ,
Eq. (35) leads to [u 1 ( t m
) u 2 ( t m
)
]c
=
[u 1 ( t m
) u 2 ( t m
)
]−
∫ t m
t i
R (τ ) dτ + J ( t m
) t m
∫ t m
t i
R (τ ) dτ − J ( t m
)
∫ t m
t i
τR (τ ) dτ , (36)
After some rearrangements, Eq. (36) can be rewritten as [U 1
U 2
]c
=
[U 1
U 2
]− ˜ H
R +
J T
H
R − ˜ J H
T
R , (37)
where ˜ H is the transformation matrix corresponding to integral, and
˜ T is the matrix related with time t . The configurations
of matrices ˜ H , ˜ T , J and
˜ R are provided in Appendix C . Fig. 2 shows the flow chart of OFAPI algorithm-2.
2.3.3. OFAPI algorithm-3
Considering Eq. (23) , if the Lagrange multiplier λ is approximated by Taylor series, we have
d
dt
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − F
]− J (t)
∫ t
t i
[ T 0 + T 1 (τ − t) + ... ]
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ
(38)
If λ is simply approximated by T 0 , Eq. (38) becomes
d
dt
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]− J (t)
∫ t
t i
T 0
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ. (39)
Since T 0 = −I , Eq. (39) is further rewritten as
d
dt
[u 1 (t) u 2 (t)
]c
= −[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]+ J (t)
∫ t
t i
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ (40)
By using u p ( p = 1 , 2 ) as trial functions and making collocations, the OFAPI algorithm-3 is obtained from Eq. (40) as [u 1
′ ( t m
) u 2
′ ( t m
)
]c
= −[−u 2 ( t m
) M
−1 C u 2 ( t m
) + M
−1 N ( t m
) − M
−1 F ( t m
)
]+ J ( t m
)
∫ t m
t i
{[u
′ 1
u
′ 2
]+
[−u 2
M
−1 C u 2 + M
−1 N − M
−1 F
]}dτ
(41)
where u p ′ ( t m
) and the integrals can be obtained as is stated in Eqs. (18 ) and ( 19 ). After rearrangements, Eq. (41) can be
written as
˜ E
[U 1
U 2
]c
= −[−U 2
˜ M
−1 ˜ C U 2 +
˜ M
−1 ˜ N − ˜ M
−1 ˜ F
]+
J H
R . (42)
The configurations of matrices ˜ E , ˜ M , ˜ C , ˜ N , J , ˜ H , and
˜ R are provided in Appendix C .
62 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69
Fig. 3. Computational error of Hamiltonian using the HHT- α and the 3 OFAPI algorithms, and ode45 in MATLAB.
Table 1
Comparison of HHT, OFAPI and ode45 on solving Duffing equation.
Computational time (s) Iteration steps Step size
HHT, α= 0 24 299,827 0.01
HHT, α= −0 . 1 26 298,448 0.01
OFAPI 2 6076 2
ode45(MATLAB) 11 1,033,129 0.001
Note that the initial conditions are not guaranteed by Eq. (42) , just like Eq. (27) in OFAPI algorithm-1, thus it should be
further modified as
˜ E
r
[U 1
U 2
]c
r
=
{−[−U 2
˜ M
−1 ˜ C U 2 +
˜ M
−1 ˜ N − ˜ M
−1 ˜ F
]+
J H
R
}r
− ˜ E
d
[U 1
U 2
]c
d
, (43)
The symbol [ •] r denotes the remained matrix after the (lM + 1) th rows and columns in [ •] are dropped, l = 0 , 1 , 2 , ... .
If [ •] r is a vector, we just need to remove the (lM + 1) th rows in [ •]. The symbol [ •] d denotes the part of [ •] that were
dropped to obtain [ •] r . The flow chart of this algorithm is provided in Fig. 2 .
2.4. Performance of OFAPI on energy preservation
To investigate the energy conservation properties of the HHT- α and the 3 OFAPI algorithms, a simple undamped duffing
equation is used for demonstration.
x ′′ − x + x 3 = 0 .
The Hamiltonian energy of this system is
H =
x ′ 2
2
− x 2
2
+
x 4
4
.
Starting from the initial state x (0) = 1 . 5 , x ′ (0) = 0 , the system is integrated using the HHT- α, the 3 OFAPI, and the
ode45 methods. The step size of HHT- α method is �t = 0 . 01 and the simulation is carried in the time interval t ∈ [0, 10 0 0].
For the 3 OFAPI algorithms, the step size is selected as �t = 2 , with 32 collocation points in each step. The absolute and
relative accuracies of ode45 are both set as 1 E − 15 . The computational errors of the Hamiltonian are recorded and plotted
in Fig. 3 . It can be seen that both the OFAPI algorithms and the ode45 are much superior to the HHT- α algorithm on
energy conservation. Notably, the present 3 OFAPI algorithms behave even better than ode45, with a negligible error of
Hamiltonian of 1 E − 13 . Among all these methods, the HHT- α method with α = −0 . 1 is the worst on energy conservation,
as can be seen in Fig. 3 (a). After dropping out the numerical damping by setting α = 0 , the performance of HHT- α method
is much improved, with the error of Hamiltonian being 1 E − 5 .
Aside from the much higher computational accuracy (energy preservation), the OFAPI is also much more efficient as
shown in Table 1 .
X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 63
3. Nonlinear vibrations of a buckled beam
The governing equation of a buckled one-dimensional Bernoulli beam is written in non-dimensional form as
w + w
i v + P w
′′ + c ˙ w − 1
2
w
′′ ∫ 1
0
w
′ 2 dx = F (x ) cos t, (44)
BC ’ s : w = w
′ = 0 at x = 0 and x = 1 ,
where w is the transverse displacement of the beam, and P is the axial load on the beam. F ( x ) is the transverse distributed
load on the beam and is the frequency of the applied load F ( x ). The overdot denotes the derivative with respect to time
t , while the prime denotes the derivative with respect to the spatial coordinate x . To semi-discretize Eq. (44) , the mode
decomposition method introduced by Emma et al. [29] is applied herein.
To solve the preceding partial differential equation, we first assume the spatial modes:
w (x, t) = w s (x ) + v (x, t) = w s (x ) +
N ∑
n =1
φn (x ) q n (t) , (45)
where w s is the static postbuckling displacement, v ( x, t ) is the superposed dynamic response around the buckled configura-
tion. φn ( x ) are the mode shapes of vibration around the buckled configuration, and q n ( t ) are the amplitudes of φn ( x ).
The buckled configuration w s ( x ) can be obtained by first solving the static buckling problem where the time derivatives
and the dynamic load are removed in Eq. (44) .
w
i v + P w
′′ − 1
2
w
′′ ∫ 1
0
w
′ 2 dx = 0 . (46)
Mathematically, there could be various buckled mode shapes, depending on the corresponding load. However, in struc-
tural mechanics, the first buckled mode shape is mostly of importance, from which w s ( x ) is obtained as [29]
w s (x ) =
1
2
b(1 − cos 2 πx ) . (47)
The non-dimensional transverse displacement b at the mid-span of the beam is related to the load P via [29]
b 2 = 4(P − P c ) / π2 , (48)
where P c is the critical load corresponding to the first Euler buckled mode, namely P c = 4 π2 .
Substituting the assumed solution of w ( x, t ) into the governing equation and dropping all the nonlinear, damping, and
forcing terms, we have the following linear partial differential equation that can be tackled using the linear vibration mode
theory.
v + v i v + 4 π2 v ′′ − 2 b 2 π3 cos 2 πx
∫ 1
0
v ′ sin 2 πx dx = 0 . (49)
By assuming v (x, t) = φ(x ) e iωt and substituting it into Eq. (49) , the mode shape is obtained as
φ(x ) = φh + φp = ( c 1 sin s 1 x + c 2 cos s 1 x + c 3 sinh s 2 x + c 4 cosh s 2 x ) + c 5 cos 2 πx, (50)
where s 1 , 2 = (±2 π2 +
√
4 π2 + ω
2 ) 1 / 2 , and c 5 should satisfy the following equation:
(2 b 2 π4 − ω
2 ) c 5 = 2 b 2 π3
∫ 1
0
φ′ h sin 2 πx dx. (51)
Using the boundary conditions and Eq. (51) , the mode shapes φn ( x ) can be obtained.
The resulting linear vibration mode shapes φn ( x ) are used to construct the solution v ( x, t ). Using the multi-mode Galerkin
discretization, where the weighting functions are the same as the trial functions φn ( x ), the partial differential Eq. (44) is then
reduced to a system of coupled Duffing equations.
q n + ω
2 n q n = −c q n + b
N ∑
i, j
A ni j q i q j +
N ∑
i, j,k
B ni jk q i q j q k + f n cos t, n = 1 , 2 , ..., N. (52)
In this paper, four modes are retained in the reduced model. The buckling level is selected as b = 4 . Correspondingly
the natural frequencies of the four linear vibration modes are obtained as ω 1 = 30 . 7067 , ω 2 = 44 . 3627 , ω 3 = 108 . 3322 , and
ω 4 = 182 . 1178 . The parameters A nij , and B nijk are provided in Appendices A and B .
64 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69
Fig. 4. Bifurcation diagrams obtained by (a) force-sweeping, (b) frequency-sweeping process. Same results obtained for HHT- α (with very small step size)
and the 3 OFAPI algorithms.
4. Numerical results and discussion
Considering the discretized nonlinear system (52) , several types of nonlinear resonances of the buckled beam may occur
due to the external harmonic excitation. The primary resonance is mostly observed when the excitation frequency is
close to one of the mode frequencies ω n , which often leads to a periodic motion of large amplitude in that mode. For the
existence of quadratic nonlinearities and cubic nonlinearities, the subharmonic resonances and superharmonic resonances
may also occur for that has an integer relationship to ω n .
In the following, the resonance of the buckled beam under harmonic excitations is investigated. The frequency is
selected as being close to the natural frequency of the first vibration mode ω 1 . The external force is supposed to be uniform
over the length of the beam, thus F ( x ) in Eq. (44) is constant. Through Galerkin discretization, the forces imposed on the
four linear vibration modes are obtained as f 1 = −0 . 850654 F , f 2 = 0 , f 3 = 0 . 309884 F , and f 4 = 0 .
In the numerical simulations, the force-sweep as well as the frequency-sweep processes are used to obtain an overview
of the nonlinear dynamical behaviors of the buckled beam when subjected to a primary resonance excitation of its first
vibration mode. Considering that the magnitudes of the coefficients ω
2 n , bA nij , and B nijk are roughly between 10 3 and 10 4
in the restoring forces, the amplitude of the external force F is set to vary between F = 40 and F = 600 in the force-sweep
process . In the frequency-sweep process, F is fixed at F = 400 , while the frequency is varied between = 30 and = 28 .
The numerical methods including HHT- α (with the values of α selected as 0 and −0.1), OFAPI, and ode45 function built
in MATLAB are used to investigate the buckled beam system. In this paper, the basis functions used in OFAPI method are
selected as the Chebyshev polynomials of the first kind and the collocation points are selected as the Chebyshev-Gauss-
Lobatto nodes.
4.1. Nonlinear dynamical behaviors including bifurcation and chaos
With the discretized equations derived above, the nonlinear vibrations of a buckled beam are first investigated under
uniform harmonic excitations, in which the frequency of the external load F ( x )cos t is = 30 . A force sweeping approach
is first adopted herein to capture the bifurcation phenomenon. For simplicity, a periodic motion is referred to as period- n
motion if its period is nT , where T = 2 π/ . It is shown in Fig. 4 (a) that a period-one motion is obtained for F = 40 . As
the excitation force F increases, the first period-doubling bifurcation occurs at about F = 420 . By further increasing the
force amplitude, the second period-doubling occurs at about F = 454 . The next bifurcation occurs at F = 461 , leading to the
period-eight motion. Similar to the force sweeping approach, a period doubling bifurcation route to chaos is also revealed
by using frequency sweeping approach in Fig. 4 (b) with the amplitude fixed at F = 400 .
The results in Fig. 4 can be obtained by using both the HHT- α and the 3 OFAPI algorithms. However, the time step
size has to be selected very small to accurately obtain the steady periodic motions in using the HHT- α algorithm, of which
the computational time (as shown in detail in Table 3 to follow) will be much more prolonged. On the contrary, the step
size of the present OFAPI algorithms can be selected to be relatively very large (several hundreds of times larger than the
time step in the HHT- α, as shown in Table 3 to follow), and all the 3 OFAPI can easily achieve high accuracy in predicting
these limit cycle oscillations. It will be shown in the next subsection that all the 3 OFAPI algorithms have a far better
performance than the HHT- α algorithm in terms of accuracy, computational time, and speed of convergence, in predicting
the nonlinear dynamical responses of the buckled beam involving bifurcation and chaos.
X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 65
Fig. 5. The same chaotic motion revealed by HHT- α (with very small step size) and the 3 OFAPI algorithms. (a) Phase portrait, (b) time responses,
(c) Poincare map, (d) FFT curves.
As the period-doubling bifurcation proceeds, more sinks appear in the chaotic regime of Poincare map of the dynamical
system, and eventually the motion becomes completely chaotic. In Fig. 5 , the chaotic motion is presented through phase
portrait, response curve, Poincare map, and FFT curve. Both the HHT- α method with damping coefficients α = −0 . 1 , and the
OFAPI algorithms are used to capture the chaotic motion. The step size of HHT method is �t = T / 10 0 0 , while that of OFAPI
method is T /5, where T is the period of the first vibration mode, T = 2 π/ ω 1 .
4.2. Comparison between the HHT- α and the 3 OFAPI algorithms
The discretized model is solved using both the HHT- α and the 3 OFAPI algorithms. It is found that all the algorithms can
predict the limit cycle oscillations and chaos. However, the computational performances of these algorithms are very differ-
ent. In the analysis below, various step sizes are used to test the stability of the algorithms. It is found through simulations
that all the 3 OFAPI algorithms converge for both periodic and chaotic motions with step sizes as large as T /5. The largest
step size of HHT- α method depends on the nonlinear algebraic equations (NAEs) solver. By using Newton-Raphson method,
it is found that the step size should be no greater than T /20, otherwise the solution of nonlinear equilibrium equations in
the HHT- α may not be found [19] .
Additional simulations were conducted to reveal the steady periodic motions of the buckled beam under different ex-
ternal excitations, using HHT- α and the 3 OFAPI algorithms separately. After the motion settles down, the extremes of the
displacement q 1 ( t ) obtained by the HHT, and the 3 OFAPI algorithms are recorded in Table 2 . The exact values are assumed
to be provided by ode45, and they are fully consistent with the results obtained by the 3 OFAPI algorithms, with the time
66 X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69
Table 2
Extremes of the steady periodic motions obtained by HHT- α and OFAPI methods.
X. Wang et al. / Commun Nonlinear Sci Numer Simulat 65 (2018) 54–69 69
Appendix C
Table C1
The constant and varying matrices in the OFAPI algorithms.
Constant matrices Varying matrices
˜ E = I 2 L ×2 L � ( LB ) B −1 , ˜ J = J ( t ) , ˜ M = M � I M×M ,
˜ N = N (t ) ,
˜ C = C � I M×M , ˜ R =
E [ U 1
U 2 ] + [
−U 2
˜ M
−1 ˜ C U 2 +
˜ M
−1 ˜ N − ˜ F ]
t = [ t 1 , t 2 , ..., t M ] T
,
ˆ t = diag(t ) , ˜ t = I 2 L ×2 L � t , ˜ H = I 2 L ×2 L � ( L −1 B ) B −1 . ˜ F = F (t )
M is the number of collocation points in each time interval; L
is the length of variable vector x in Eq. (1) . � denotes the Kro-
necker product.
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