Luo A.C.J. (Ed.) Dynamical Systems: Discontinuity, Stochasticity and Time-Delay
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1 General Solution of a Fractional-Order Vibration System 5
Let ˛ 2 R;m1<˛ m; m 2 N;a 2 R and f be a continuous function, then
Riemann–Liouville fractional-order derivative is defined by
RL
a
D
˛
f.t/ D
1
.m ˛/
d
m
dt
m
Z
t
a
f./
.t /
1C˛m
d; (1.5)
where .z/ is the Gamma function defined by
.z/ D
Z
1
0
e
t
t
z1
dt.<.z/>0/
satisfying .z C 1/ D z .z/. The derivative
RL
a
D
˛
f.t/ requires that f.t/ is con-
tinuous only, so it is widely used in analysis.
1.2.1 Linear Dependence and Linear Independence of Functions
Assume that x
1
.t/; x
2
.t/; : : : ; x
n
.t/ are n functions defined on Œa; b; they are called
linearly dependent in Œa; b, if there exist constants c
1
;c
2
;:::;c
n
that are not zero
simultaneously, such that
c
1
x
1
.t/ C c
2
x
2
.t/ CCc
n
x
n
.t/ 0.a t b/: (1.6)
Otherwise, x
1
.t/; x
2
.t/; : : : ; x
n
.t/ are called linearly independent in Œa; b.
To check the linear dependence in fractional calculus, it is convenient to use the
generalized Wronsky determinant W
˛
.x
1
;x
2
;:::;x
n
/ of x
1
.t/; x
2
.t/; : : : ; x
n
.t/,
defined by [9]
W
˛
.x
1
;x
2
;:::;x
n
/ D
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
x
1
.t/ x
2
.t/ x
n
.t/
a
D
˛
x
1
.t/
a
D
˛
x
2
.t/
a
D
˛
x
n
.t/
:
:
:
:
:
:
:
:
:
:
:
:
a
D
.n1/˛
x
1
.t/
a
D
.n1/˛
x
2
.t/
a
D
.n1/˛
x
n
.t/
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
:
(1.7)
The solutions x
1
.t/, x
2
.t/; : : : ; x
n
.t/ of (1.4) are linearly dependent in Œa; b if and
only if there is a t
0
2 Œa; b such that W
˛
.t
0
/ D 0.
1.2.2 Characteristic Polynomial
In classical calculus, the function e
t
plays an important role in solving ordinary
differential equations with constant coefficients, and it satisfies
d
dt
e
t
D e
t
:
6 Z.H. Wang and X. Wang
In fractional calculus, the ˛-exponential function [9]
e
.ta/
˛
D .t a/
˛1
1
X
kD0
k
.t a/
˛k
..k C 1/˛/
.t a/ (1.8)
satisfies the following fractional-order differential equation
a
D
˛
x.t/ D x.t/ .t > a/ (1.9)
in the sense of Riemann–Liouville fractional derivative. For the SFDE with constant
coefficients, one has
"
RL
a
D
n˛
C
n1
X
kD0
a
k
RL
a
D
k˛
#
e
.ta/
˛
D p./e
.ta/
˛
; (1.10)
where
p./ D
n
C
n1
X
kD1
a
k
k
(1.11)
is called the characteristic polynomial of (1.4).
Assume that (1.4) has a solution of the form c e
.ta/
˛
,then must be a root of
p./.Conversely,ifp./ D 0,then(1.4) has a solution c e
.ta/
˛
. Thus, (1.4)hasa
solution c e
.ta/
˛
if and only if is a root of p./.
The ˛-exponential function can also be defined as following [9]
Oe
.ta/
˛
D
1
X
kD0
k
.t a/
˛k
.k˛C 1/
.t a/;
which satisfies the fractional-order differential equation (1.9) in the sense of Caputo
fractional-order derivative. Caputo’s definition is given by
C
a
D
˛
f.t/ D
1
.m ˛/
Z
t
a
f
.m/
./
.t /
1C˛m
d: (1.12)
It requires that f.t/ has m-order continuous derivatives. For function f.t/, with
m-order continuous derivative and starting from standstill, however, the Caputo frac-
tional derivative gives the same value of Riemann–Liouville fractional derivative.
1 General Solution of a Fractional-Order Vibration System 7
1.2.3 General Solution of SFDE
In what follows, we shall discuss the general solution of (1.4) in two cases as
follows.
Case 1. The polynomial p./ has simple roots only.
Assume that
1
;
2
;:::;
n
are n different roots of p./,then(1.4)hasthecor-
responding particular solutions
x
1
.t/ D e
1
.ta/
˛
;x
2
.t/ D e
2
.ta/
˛
;:::; x
n
.t/ D e
n
.ta/
˛
: (1.13)
The functions in (1.13) are linearly independent and are the fundamental solutions
of (1.4). In this case, the general solution of (1.4)is[9]
x.t/ D c
1
e
1
.ta/
˛
C c
2
e
2
.ta/
˛
CCc
n
e
n
.ta/
˛
; (1.14)
where c
1
;c
2
;:::;c
n
are arbitrary constants.
Case 2. The polynomial p./ has repeated roots.
Suppose that is a root of p./, with multiplicity l; .1 < l n/, then according
to [9],
e
.ta/
˛
;
@
@
e
.ta/
˛
;
@
2
@
2
e
.ta/
˛
;:::;
@
l1
@
l1
e
.ta/
˛
(1.15)
are l linearly independent particular solutions of (1.4). Let
1
;
2
;:::;
k
be the
different roots with multiplicities l
1
;l
2
;:::; l
k
, respectively, and l
1
C l
2
CC
l
k
D n, then the general solution of (1.4) are the linear combination of the following
fundamental solutions
e
1
.ta/
˛
;
@
@
e
1
.ta/
˛
;
@
2
@
2
e
1
.ta/
˛
; :::;
@
l
1
1
@
l
1
1
e
1
.ta/
˛
e
2
.ta/
˛
;
@
@
e
2
.ta/
˛
;
@
2
@
2
e
2
.ta/
˛
; :::;
@
l
2
1
@
l
2
1
e
2
.ta/
˛
:
:
:
e
k
.ta/
˛
;
@
@
e
k
.ta/
˛
;
@
2
@
2
e
k
.ta/
˛
; :::;
@
l
k
1
@
l
k
1
e
k
.ta/
˛
where
@
j
@
j
e
k
.ta/
˛
D
h
@
j
@
j
e
.ta/
˛
i
ˇ
ˇ
ˇ
D
k
.
1.3 Analysis of the Characteristic Roots
The order ˛ is assumed to be rational:
˛ D
k
n
D kˇ
ˇ D
1
n
:
8 Z.H. Wang and X. Wang
Then the generalized Bagley–Torvik equation (1.3) can be recast into a SFDE
0
D
2nˇ
x.t/ C
0
D
kˇ
x.t/ C x.t/ D 0 (1.16)
and the characteristic equation reads
p./ WD
2n
C
k
C 1 D 0: (1.17)
Obviously, with s D 1=, one has p./ D 0 if and only if q.s/ D 0,where
q.s/ D 1 C s
2nk
C s
2n
. Thus, it is sufficient to consider the case 1 k<n
only. Based on the theory introduced above, the general solution of a sequential
fractional differential equation can be given directly if the characteristic roots are
in hand.
The case with repeated roots needs to be studied carefully, which can be carried
out in four steps as follows.
Step 1. p./ has no repeated roots with multiplicity greater than 2. In fact, if
p./ has repeated root , satisfying
8
<
:
p./ D 0 ,
2n
C
k
C 1 D 0
p
0
./ D 0 , 2n
2n1
C k
k1
D 0
p
00
./ D 0 , 2n.2n 1/
2n2
C k.k 1/
k2
D 0
; (1.18)
then, the latter two equations yield
2n
2n.2n 1/
D
k
k.k 1/
;
which is possible only if k D 2n.
Step 2. p./ has a repeated real root with multiplicity 2 if k is odd, and it has no
repeated real roots if k is even. In fact, the first two conditions in (1.18) implies that
2n
D
k
2n k
>0 and
k
D
2n
.2n k/
<0: (1.19)
If the repeated root is real, then k must be odd. In this case,
DO WD
2n
2n k
2n k
k
k
2n
: (1.20)
Step 3. p./ has no repeated roots of pure imaginary numbers if k is odd, and it
has at most two pairs of conjugate pure imaginary roots with multiplicity 2 if k is
even. In fact, if k is odd, then p.i !/ D 0 implies that
˙!
2n
˙ !
k
i C 1 D 0 ) ! D 0:
1 General Solution of a Fractional-Order Vibration System 9
Obviously, D 0 is not a root of p./ D 0.Whenk is even, let k D 2m, 1 m<
n=2, and denote
2
D s,thenp./ D q.s/ WD s
n
Cs
m
C1.Ifq.s/ has a repeated
negative root, then p./ has two pairs of repeated conjugate pure imaginary roots.
As known in Step 2, q.s/ has a repeated negative root only if n is even and m is odd.
It follows that p./ has a repeated complex roots with non-zero real part only, if
both n and m are even.
Step 4. p./ has two pairs of repeated conjugate complex roots with non-zero
real part if n D 2
i
a, k D 2
j
b,wherei>jand b is odd. Actually, let n D 2a,
m D 2b,(1 b<n=4), then p./ D q.s/ D r.t/ WD t
a
C t
b
C 1 with t D
4
.
If r.t/ has a repeated negative root, then q.s/ has two pairs of repeated conjugate
pure imaginary roots ˙i with >0. As known in Step 2, r.t/ has a repeated
negative root only if a is even and b is odd. In this case, p./ has repeated roots as
following
p
2
p
2
.1 ˙ i/;
p
2
p
2
.1 ˙ i/:
Thus, the claim is confirmed if the same procedure is carried out repeatedly.
As for ¤O, the number of real roots of p./ keeps unchanged in .0; O/
and . O; C1/, respectively, since the roots depend continuously on >0.When
D 0, p./ D
2n
C1 has no real roots; thus, p./ has n pairs of conjugate simple
complex roots for all 2 .0; O/.Ifp./ has a repeated real root at DO,then
p./ has exactly two simple negative roots and .n 1/ pairs of conjugate simple
complex roots for > O.Otherwise,p./ has n pairs of conjugate simple complex
roots for > O.
1.4 The General Solution
Based on the above analysis, we can express the general solution of the SFDE di-
rectly by using the roots of p./,intermsof˛-exponential function e
t
˛
or Oe
t
˛
,and
then prove that some .2n 2/ free constants depend on the rest two free constants.
For simplicity, the general solution, in the terms of e
t
˛
, of the following equation
Rx.t/ C
0
D
˛
x.t/ C x.t/ D 0
>0; ˛D
2
3
(1.21)
is only addressed. With ˇ D1=3,(1.21) can be changed to
0
D
6ˇ
x.t/ C
0
D
2ˇ
x.t/ C x.t/ D 0. For each ¤O and >0, the characteristic function
p./ D
6
C
2
C 1 has no repeated roots but three pairs of conjugate simple
roots
1
;
N
1
;
2
;
N
2
;
3
;
N
3
, and the general solution of the SFDE can be written as
x.t/ D c
1
x
1
.t/ CNc
1
Nx
1
.t/ C c
2
x
2
.t/ CNc
2
Nx
2
.t/ C c
3
x
3
.t/ CNc
3
Nx
3
.t/; (1.22)
10 Z.H. Wang and X. Wang
where x
i
.t/ D e
i
t
1=3
.i D 1; 2; 3/.Fort !C0, one has
x.t/ D .c
1
CNc
1
C c
2
CNc
2
C c
3
CNc
3
/
t
2=3
.1=3/
C .c
1
1
CNc
1
N
1
C c
2
2
CNc
2
N
2
C c
3
3
CNc
3
N
3
/
t
1=3
.2=3/
C
c
1
2
1
CNc
1
N
2
1
C c
2
2
2
CNc
2
N
2
2
C c
3
2
3
CNc
3
N
2
3
1
.1/
C
c
1
3
1
CNc
1
N
3
1
C c
2
3
2
CNc
2
N
3
2
C c
3
3
3
CNc
3
N
3
3
t
1=3
.4=3/
C
c
1
4
1
CNc
1
N
4
1
C c
2
4
2
CNc
2
N
4
2
C c
3
4
3
CNc
3
N
4
3
t
2=3
.5=3/
C O.t
2
/:
(1.23)
In order that jx.0/j < 1, jPx.0/j < 1, it is required to have
8
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
:
c
1
CNc
1
C c
2
CNc
2
C c
3
CNc
3
D 0
c
1
1
CNc
1
N
1
C c
2
2
CNc
2
N
2
C c
3
3
CNc
3
N
3
D 0
c
1
3
1
CNc
1
N
3
1
C c
2
3
2
CNc
2
N
3
2
C c
3
3
3
CNc
3
N
3
3
D 0
c
1
4
1
CNc
1
N
4
1
C c
2
4
2
CNc
2
N
4
2
C c
3
4
3
CNc
3
N
4
3
D 0
: (1.24)
In this case, it holds jRx.0/j< 1as well. Straightforward computation gives the coef-
ficient determinant of the above linear system with respect to <.c
2
/; =.c
2
/; <.c
3
/
and =.c
3
/ as following
D D 2.
2
3
/.
N
2
N
3
/.
N
2
3
/.
2
N
3
/
.
N
2
N
2
C
N
3
N
3
C
2
3
C
N
2
N
3
C
2
N
3
C
N
2
3
/: (1.25)
The assumptions
i
¤
j
and
i
¤
N
j
imply that D ¤ 0. Thus, <.c
2
/; =.c
2
/;
<.c
3
/ and =.c
3
/ can be determined uniquely from (1.24), in terms of <.c
1
/ and
=.c
1
/. As a result, the general solution of (1.21) depends on <.c
1
/ and =.c
1
/ only
and can be determined fully by the initial displacement and initial velocity.
The above procedure can be carried out to obtain the general solution of (1.3)in
a similar way for DO, as well for any given rational number ˛ 2 .0; 2/.The
computational complexity increases sharply as n increases.
Acknowledgement This work was supported by NSF of China under Grant 10825207 and in part
by FANEDD of China under Grant 200430.
1 General Solution of a Fractional-Order Vibration System 11
References
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2. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential
equations. Elsevier, Amsterdam
3. Das S (2008) Functional fractional calculus for system identification and controls. Springer,
Berlin
4. Bagley RL, Torvik PJ (1984) On the appearance of the fractional derivative in the behavior of
real materials. ASME J Appl Mech 51:294–298
5. Bagley RL, Torvik PJ (1983) Fractional calculus – a different approach to the analysis of vis-
coe1astically damped structures. AIAA J 21(5):741–748
6. Wang ZH, Hu HY (2010) Stability of a linear oscillator with damping force of fractional-order
derivative. Sci China: Phys, Mech Astron 53(2):1–8, doi:10.1007/s11433-009-0291-y
7. Suarez LE, Shokooh A (1997) An eigenvector expansion method for the solution of motion
containing fractional derivatives. ASME J Appl Mech 64:629–635
8. Lorenzo CF, Hartley TT (2008) Initialization of fractional-order operators and fractional differ-
ential equations. ASME J Comput Nonlinear Dyn 3:021101
9. Bonilla B, Rivero M, Trujillo JJ (2007) Linear differential equations of fractional orders. In:
Sabatier J, Agrawal OP, Tenreiro Machado JA (eds) Advances in fractional calculus. Springer,
Dordrecht, pp 77–91
Chapter 2
An Analytic Proof for the Sensitivity of Chaos
to Initial Condition and Perturbations
J.H. Peng and J.S. Tang
Abstract An analytic method to prove the sensitivity of chaotic motion to initial
states and perturbations is proposed in this paper. With the fundamental perturbation
method, a second order nonlinear differential equation is expanded into a series of
perturbation equations, and by means of variation of constants, the general solutions
of the perturbation equations are obtained. Based on these general solutions, we
prove that the chaotic motion is sensitively dependent on the initial conditions and
perturbations.
2.1 Introduction
Since Lorentz discovered the chaotic motion of air flow in his famous article
“Deterministic nonperiodic flow” [1] and Li and Yorke first introduced the term
chaos [2], considerable efforts, both theoretical and experimental, have been de-
voted to the study of chaotic motion of nonlinear dynamical system. Many important
properties and applications of chaos have been obtained [3–12].
The property that chaotic motion is sensitively dependent on initial conditions is
the most important nature of chaos [13]. With numeric computation technique the
property of various chaotic systems has been checked [1, 2] and by means with the
Lyapunov characteristic exponents, it can be verified quantitatively [14,15]. But all
these methods are numerical ones and we have not analytically proved the property
of chaotic motion being sensitively dependent on the initial conditions still. In this
paper, we propose an analytical proof for the property. Firstly, we expand the second
nonlinear differentiate equation into a series perturbation equations, then, with the
trial and error procedure and variation of constants method, we derive the general
J.H. Peng (
)
Department of Mechanical Engineering, Shaoyang Polytechnic, Hunan,
People’s Republic of China
and
Department of Physics, Shaoyang University, Shaoyang, Hunan, People’s Republic of China
e-mail: pengjhua@sina.com
A.C.J. Luo (ed.), Dynamical Systems: Discontinuity, Stochasticity and Time-Delay,
DOI 10.1007/978-1-4419-5754-2
2,
c
Springer Science+Business Media, LLC 2010
13
14 J.H. Peng and J.S. Tang
solutions of each perturbation equation, and, finally based on the above solutions,
we prove the sensitivity of chaos to initial conditions and perturbations analytically.
2.2 Perturbation Equations
The governing equation of the second order nonlinear differential dynamical system
being studied can be expressed as
(
Rx D f.x/C "g.t; Px/
x.0/ D a; Px.0/ D b
;f;g2 C
r
; (2.1)
where ".>0/ is a small arbitrary parameter, f.x/is a nonlinear function of x, g.t; Px/
is the perturbation acting on the system. The fundamental perturbation method is
employed to solve (2.1). Setting
x.t/ D x
0
.t/ C "x
1
.t/ C "
2
x
2
.t/ C: (2.2)
Substituting (2.2)into(2.1) and expanding f.x/as well as g.t; Px/ into the Taylor’s
series of x and Px yields
Rx
0
.t/ C " Rx
1
.t/ C "
2
Rx
2
.t/ CDf.x
0
/ C
@f . x
0
/
@x
."x
1
C "
2
x
2
C/
C
1
2Š
@
2
f.x
0
/
@x
2
."x
1
C "
2
x
2
C/
2
C
C"
g.t; Px
0
/ C
@g.t; Px
0
/
@ Px
." Px
1
C "
2
Px
2
C/
C
1
2Š
@
2
g.t; Px
0
/
@ Px
2
." Px
1
C "
2
Px
2
C/
2
C
;
(2.3)
(
x
0
D x
0
.0/ C "x
1
.0/ C "
2
x
2
.0/ C
Px
0
DPx
0
.0/ C " Px
1
.0/ C "
2
Px
2
.0/ C
; (2.4)
equating the sum of ith-order terms of " in (2.3) to zero, we obtain the zeroth-order
perturbation equation and ith-order perturbation equations as follows
"
0
WRx
0
D f.x
0
/; (2.5)
"
1
W
8
<
:
Rx
1
D
@f . x
0
/
@x
x
1
C h
1
.t; x
0
; Px
0
/
h
1
.t; x
0
; Px
0
/ D g.t; Px
0
/
; (2.6)
2 Sensitivity of Chaos to Initial Condition and Perturbation 15
"
2
W
8
ˆ
<
ˆ
:
Rx
2
D
@f . x
0
/
@x
x
2
C h
2
.t; x
0
;x
1
IPx
0
; Px
1
/
h
2
.t; x
0
; Px
0
;x
1
/ D
@g.t; Px
0
/
@ Px
C
1
2
@
2
f.x
0
/
@x
2
x
2
1
; (2.7)
"
i
W
8
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
Rx
i
D
@f . x
0
/
@x
x
i
C h
i
.t; x
0
;x
1
;:::;x
i1
IPx
0
; Px
1
I:::IPx
i1
/
h
i
.t; x
0
;x
1
;:::;x
i1
IPx
0
; Px
1
I:::IPx
i1
/
D
@g.t; Px
0
/
@ Px
Px
i1
C
@
2
g.t; Px
0
/
@ Px
2
. Px
1
Px
i2
CPx
2
Px
i3
C/ C
C
1
.i1/Š
@
i1
g.t; Px
0
/
@ Px
i1
Px
i1
1
C
d
2
f.x
0
/
dx
2
.x
1
x
i1
C x
2
x
i2
C/ C
C
1
.i 2/Š
d
i1
f.x
0
/
dx
i1
x
i2
1
x
2
C
1
iŠ
d
i
f.x
0
/
dx
i
x
i
1
:(2.8)
2.3 General Solutions of Perturbation Equations
Equation (2.5) is the unperturbed equation of original system (2.1). If we can get
the solutions, x
i
.t/ .i D 0; 1; 2; : : :/, of various equations given earlier, then
the solution of (2.1) is obtained by inserting x
i
.t/ into (2.2). Noting that
h
i
.t; x
0
; Px
0
Ix
1
; Px
1
I:::Ix
i1
; Px
i1
/.i D 1;2;:::/ is the non-homogeneous term
of the ith-order perturbation equation .i D 1;2;:::/, hence the homogeneous parts
of the ith-order perturbation equations .i D 1;2;:::/ are linear differential equa-
tions. The general solution of the i th-order perturbation equation .i D 1;2;:::/
can be derived from the fundamental solutions of the homogeneous parts of the
ith-order equation with the method of variation of constants that is well known to
us. For the sake of determining the general solution of the ith-order perturbation
equation .i D 1;2;:::/, assume the formal solution of zeroth-order equation (2.5)
is x
0
.t/. According to the characteristic of the homogeneous parts of i th-order
perturbation equation
Rx
i
D
@f . x
0
/
@x
x
i
;iD 1;2;::: (2.9)
we suppose the trial solutions of them being
x
i1
.t/ DPx
0
.t/; x
i2
.t/ DPx
0
.t/
Z
ΠPx
0
.t/
2
dt; i D 1;2;::: (2.10)
We verify these solutions satisfying (2.9) first. On the one hand, it is easy to get the
second derivative as follows
Rx
i1
.t/ D«x
0
.t/: (a)
Inserting it into (2.9), we have
«x
0
.t/ D
@f . x
0
/
@x
Px
0
.t/: (b)