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Journal ArticleDOI

Chaos in fractional-order autonomous nonlinear systems

01 Mar 2003-Chaos Solitons & Fractals (Pergamon)-Vol. 16, Iss: 2, pp 339-351
TL;DR: In this paper, the authors numerically investigate chaotic behavior in autonomous nonlinear models of fractional order and show that chaotic attractors can be obtained with system orders as low as 2.1.
Abstract: We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0; 1� , based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic ‘‘jerk’’ model. In both models, numerical simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2 þ e ,1 > e > 0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated. 2002 Elsevier Science Ltd. All rights reserved.

Summary (2 min read)

1. Introduction

  • Chaotic systems have been a focal point of renewed interest for many researchers in the past few decades.
  • In their study the authors will focus on two types of autonomous chaotic systems that are of practical interest: an electronic chaotic oscillator model [10], and a mechanical chaotic ‘‘jerk’’ model [11,12].
  • The authors derive the approximate linear transfer functions for the fractional integrator of order that varies from 0.1 to 0.9, and study the resulting behavior of the entire system for each case under the effect of different types of nonlinearities.
  • The authors will follow the algorithm in [13] to calculate linear transfer function approximations of (1).

2.1. Fractional chaotic oscillator

  • The authors consider the one-parameter, third-order chaotic oscillator of canonical structure reported in [10] as being the simplest possible structure for a chaotic oscillator.
  • The output in- tegrator of this model is then replaced with a fractional integrator.

3. Transfer function approximations

  • Using the algorithm in [11], Table 1 gives the resulting approximating transfer functions, HðsÞ for different fractional orders, in increments of 0.1, for the mth order output integrator, assuming xmax ¼ 100 and pT ¼ 0:01.
  • Notice that the order of the approximate transfer functions increases as the desired error decreases.
  • The chaotic systems can then be represented by block diagrams as shown in Fig. 1a and b. In Fig. 1c, the authors show the magnitude Bode diagrams for a fractional integrator of order m ¼ 0:5, and its linear approximating transfer function (from Table 1).
  • It can be seen that within the bandwidth of interest the two diagrams are in good agreement.

4. Simulation results

  • In their simulations, the authors have visually inspected the bifurcation diagrams to identify chaos.
  • These figures demonstrate clearly that indeed the chaotic behavior of the oscillator is not destroyed by order reduction.
  • Also, it can be noticed that the effective chaotic range of the control parameter shifts downward away from that corresponding to the integer order model.
  • In Figs. 9–11, the authors show simulation results of the fractional jerk model of total order of 2.9 and using the nonlinearities 9a, 9b, and 9c.
  • From the above results, the authors present the following conjecture: Conjecture.

5. Conclusion

  • The authors have demonstrated via numerical simulations that chaotic autonomous nonlinear systems can still exhibit chaotic behaviors when the order becomes fractional.
  • Linear approximation techniques which are based on frequency domain arguments have been used to obtain approximate linear models of the given systems.
  • The resulting order of the approximating function of the fractional integrator depends on the desired bandwidth, and error between actual and approximate Bode magnitude plots of the fractional integrator.
  • Both electronic chaotic oscillators and chaotic jerk models were studied.
  • The effective chaotic range of the control parameter for either model shrinks and moves downward as the total system order decreases and approaches 2, i.e. the chaotic range is affected by contraction and translation processes as the fractional order decreases.

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Chaos in fractional-order autonomous nonlinear systems
Wajdi M. Ahmad
a,1
, J.C. Sprott
b,
*
a
Department of Electrical and Electronics Engineering, University of Sharjah, P.O. Box 27272, Sharjah,
United Arab Emirates
b
Department of Physics, University of Wisconsin, Madison, WI 53706, USA
Accepted 19 September 2002
Abstract
We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer
function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0; 1, based on
frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered
in this study; an electronic chaotic oscillator, and a mechanical chaotic ‘‘jerk’’ model. In both models, numerical
simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control
parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that
third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2 þ e,1> e > 0,
using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is
studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by
contraction and translation. Robustness against model order reduction is demonstrated.
Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction
Chaotic systems have been a focal point of renewed interest for many researchers in the past few decades. Such
nonlinear systems can occur in various natural and man-made systems, and are known to have great sensitivity to initial
conditions. Thus, two trajectories starting at arbitrarily nearby initial conditions in such systems could evolve in
drastically different fashions, and soon become totally uncorrelated. At first glance, chaotic time trajectories look very
much like noise. In fact, chaotic signals and noise have similar broad-band frequency spectrum charactersitics.
However, there is a fundamental difference between noise and chaos, determinism: whereas chaos can be classified as
deterministic but unpredictable, noise is neither deterministic nor predictable. This unpredictability of chaotic time
signals has been utilized for secure communication applications. Basically, the useful signal is encapsulated in a chaotic
envelope (produced by a chaotic oscillator) at the transmitter end, and is transmitted over the communication channel
as a chaotic signal. At the receiver end, the information-bearing signal is recovered using various techniques, e.g.
synchronization [1–3].
According to the Poincare–Bendixon theorem [4], an integer order chaotic nonlinear system must have a minimum
order of 3 for chaos to appear. However, in fractional-order nonlinear systems, it is not the case. For example, it has
been shown that ChuaÕs circuit of order as low as 2.7 can produce a chaotic attractor [5]. Nonautonomous Duffing
systems of fractional order have been addressed in [6], where it is shown that a sinusoidally driven Duffing system of
order less than 2 can still behave in a chaotic manner.
*
Corresponding author. Tel.: +1-608-263-4449; fax: +1-608-262-7205.
E-mail addresses: wajdi@sharjah.ac.ae (W.M. Ahmad), sprott@physics.wisc.edu (J.C. Sprott).
1
Tel.: +971-6-505-0143; fax: +971-6-505-0137.
0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 960-0 7 7 9 ( 0 2 ) 0 0 4 3 8 - 1
Chaos, Solitons and Fractals 16 (2003) 339–351
www.elsevier.com/locate/chaos

The underlying theory behind fractional-order systems is deeply rooted in fractional calculus [7]. Basically, these
systems are characterized by characteristic equations of fractional order. From a state-space point of view, the frac-
tional order of the resulting characteristic equation results from a fractional-order integrator placed, for example, in
place of the output integrator in the system block diagram realization. From a circuit realization standpoint, a frac-
tional-order circuit is obtained using fractional capacitors. Recently, a solid-state implementation of fractal capacitors
has been reported [8]. A fractional capacitor ÔCÕ of order Ô mÕ has an Laplace domain impedance given by ZðsÞ¼1=ðCsÞ
m
.
Clearly, this device is characterized by a phase shift of mp=2, for sinusoidal excitation, and becomes 90° as expected
when the order becomes unity. Moreover, the fractional capacitor is not lossless, as can be seen from its impedance. As
more advances are brought about in the way of solid-state implementations of such devices, we are likely to see a flury
of fractional-order integrated circuits. Thus, the need to understand and be able to analyze such circuits becomes vital.
The analysis of fractional-order systems is by no means trivial. Therefore, the course of numerical simulations is
often adopted in order to study the behavior of these systems. For example, fractional-order nonchaotic Wien bridge
Fig. 1. (a) Block diagram of fractional chaotic oscillator. (b) Block diagram of fractional chaotic jerk model. (c) Fractional integrator
frequency responses: (––) actual, () approximation; m ¼ 0:5, and corner frequency of 0.01 rad/s.
340 W.M. Ahmad, J.C. Sprott / Chaos, Solitons and Fractals 16 (2003) 339–351

oscillators have recently been studied in [9], where it is shown that limit cycles can be attained for any fractional order,
with the proper value of the amplifier gain which, of course, is a function of the fractional order. In our study we will
Table 1
Linear transfer function approximations of fractional integrator of order ÔmÕ, with maximum discrepancy y ¼ 2 dB
mNHðsÞ
0.1 2
1584:8932ðs þ 0:1668Þðs þ 27:83Þ
ðs þ 0:1Þðs þ 16:68Þðs þ 2783Þ
0.2 2
79:4328ðs þ 0:05623Þðs þ 1Þðs þ 17:78Þ
ðs þ 0:03162Þðs þ 0:5623Þðs þ 10Þðs þ 177:8Þ
0.3 4
39:8107ðs þ 0:0416Þðs þ 0:3728Þðs þ 3:34Þðs þ 29:94Þ
ðs þ 0:02154Þðs þ 0:1931Þðs þ 1:73Þðs þ 15:51Þðs þ 138:9Þ
0.4 5
35:4813ðs þ 0:03831Þðs þ 0:261Þðs þ 1:778Þðs þ 12:12Þðs þ 82:54Þ
ðs þ 0:01778Þðs þ 0:1212Þðs þ 0:8254Þðs þ 5:623Þðs þ 38:31Þðs þ 261Þ
0.5 5
15:8489ðs þ 0:03981Þðs þ 0:2512Þðs þ 1:585Þðs þ 10Þðs þ 63:1Þ
ðs þ 0:01585Þðs þ 0:1Þðs þ 0:631Þðs þ 3:981Þðs þ 25:12Þðs þ 158:5Þ
0.6 5
10:7978ðs þ 0:04642Þðs þ 0:3162Þðs þ 2:154Þðs þ 14:68Þðs þ 100Þ
ðs þ 0:01468Þðs þ 0:1Þðs þ 0:6813Þðs þ 4:642Þðs þ 31:62Þðs þ 215:4Þ
0.7 5
9:3633ðs þ 0:06449Þðs þ 0:578Þðs þ 5:179Þðs þ 46:42Þðs þ 416Þ
ðs þ 0:01389Þðs þ 0:1245Þðs þ 1:116Þðs þ 10Þðs þ 89:62Þðs þ 803:1Þ
0.8 4
5:3088ðs þ 0:1334Þðs þ 2:371Þðs þ 42:17Þðs þ 749:9Þ
ðs þ 0:01334Þðs þ 0:2371Þðs þ 4:217Þðs þ 74:99Þðs þ 1334Þ
0.9 2
2:2675ðs þ 1:292Þðs þ 215:4Þ
ðs þ 0:01292Þðs þ 2:154Þðs þ 359:4Þ
Table 2
Linear transfer function approximations of fractional integrator of order ÔmÕ, with maximum discrepancy y ¼ 3dB
mNHðsÞ
0.1 2
501:14ðs þ 0:6811Þ
ðs þ 0:3162Þðs þ 681:1Þ
0.2 2
141:2538ðs þ 0:1334Þðs þ 10Þ
ðs þ 0:05623Þðs þ 4:217Þðs þ 316:2Þ
0.3 4
125:8925ðs þ 0:08483Þðs þ 2:276Þðs þ 61:05Þ
ðs þ 0:03162Þðs þ 0:8483Þðs þ 22:76Þðs þ 610: 5Þ
0.4 5
26:6073ðs þ 0:07499Þðs þ 1:334Þðs þ 23:71Þ
ðs þ 0:02371Þðs þ 0:4217Þðs þ 7:499Þðs þ 133: 4Þ
0.5 5
50:1187ðs þ 0:07943Þðs þ 1:259Þðs þ 19:95Þðs þ 316:2Þ
ðs þ 0:01995Þðs þ 0:3162Þðs þ 5:012Þðs þ 79:43Þðs þ 1259Þ
0.6 5
28:1838ðs þ 0:1Þðs þ 1:778Þðs þ 31:62Þðs þ 562:3Þ
ðs þ 0:01778Þðs þ 0:3162Þðs þ 5:623Þðs þ 100Þðs þ 1778Þ
0.7 5
7:9433ðs þ 0:1638Þðs þ 4:394Þðs þ 117:9Þ
ðs þ 0:01638Þðs þ 0:4394Þðs þ 11:79Þðs þ 100Þðs þ 316:2Þ
0.8 4
8:1752ðs þ 0:487Þðs þ 36:52Þðs þ 2738Þ
ðs þ 0:0154Þðs þ 1:155Þðs þ 86:6Þðs þ 6494Þ
0.9 2
4:2987ðs þ 14:68Þðs þ 31620Þ
ðs þ 0:01468Þðs þ 31:62Þðs þ 68130Þ
W.M. Ahmad, J.C. Sprott / Chaos, Solitons and Fractals 16 (2003) 339–351 341

focus on two types of autonomous chaotic systems that are of practical interest: an electronic chaotic oscillator model
[10], and a mechanical chaotic ‘‘jerk’’ model [11,12]. We will present simulation results of the chaotic behaviors pro-
duced from these two interesting systems as they acquire fractional orders.
One way to study fractional-order systems is through linear approximations. By utilizing frequency domain tech-
niques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of
which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode
diagrams [13]. In this paper, we use this approach to study the behaviors of the fractional-order versions of our chaotic
oscillator and jerk models. We derive the approximate linear transfer functions for the fractional integrator of order
that varies from 0.1 to 0.9, and study the resulting behavior of the entire system for each case under the effect of different
types of nonlinearities. We demonstrate that chaotic attractors are obtained for total system order as low as 2.1.
In the following, we will consider the state space equations of the systems under study. The output integrator block is
assumed to have a fractional order ÔmÕ in ð0; 1. In the Laplace domain, a fractional integrator of order ÔmÕ can be
represented by the transfer function:
F ðsÞ¼
1
s
m
ð1Þ
Fig. 2. (a) Integer order chaotic oscillator, a ¼ 0:8, f ðx
1
Þ¼sgnðx
1
Þ. (b) Fractional oscillator, m ¼ 0:9, a ¼ 0:4, f ðx
1
Þ¼sgnðx
1
Þ, y ¼ 2
dB.
342 W.M. Ahmad, J.C. Sprott / Chaos, Solitons and Fractals 16 (2003) 339–351

The transfer function in (1) has a Bode diagram characterized by a slope of 20m dB/decade. We will follow the
algorithm in [13] to calculate linear transfer function approximations of (1). This approximation is based on approx-
imating the 20m dB/decade line with a number of zig-zag lines connected together with alternate slopes of 0 dB/decade
and 20 dB/decade. According to [13], if the discrepancy between the actual and approximate lines is specified as ÔyÕ dB
over a frequency range of x
max
, and for a corner frequency p
T
, then Eq. (1) can be approximated as
F ðsÞ¼
1
s
m
1
1 þ
s
p
T

m
Q
N1
i¼0
1 þ
s
z
i

Q
N
i¼0
1 þ
s
p
i

ð2Þ
In other words, the fractional integrator is approximated by a linear transfer function of order N þ 1, where N is given
by
N ¼ 1 þ Integer
log
x
max
p
0

logðabÞ
0
@
1
A
ð3Þ
Fig. 3. Fractional oscillator, m ¼ 0:9, a ¼ 0:5, f ðx
1
Þ¼sgnðx
1
Þ: (left) y ¼ 2 dB, (right) 3 dB.
Fig. 4. Fractional oscillator, m ¼ 0:9, a ¼ 0:05, nonlinearity 9b: (left) y ¼ 2 dB, (right) 3 dB.
W.M. Ahmad, J.C. Sprott / Chaos, Solitons and Fractals 16 (2003) 339–351 343

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References
More filters
Book
12 May 1974
TL;DR: In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.
Abstract: This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.

2,891 citations


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  • ...According to the Poincare–Bendixon theorem [4], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear....

    [...]

Journal ArticleDOI
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Abstract: A circuit implementation of the chaotic Lorenz system is described. The chaotic behavior of the circuit closely matches the results predicted by numerical experiments. Using the concept of synchronized chaotic systems (SCS's), two possible approaches to secure communications are demonstrated with the Lorenz circuit implemented in both the transmitter and receiver. In the first approach, a chaotic masking signal is added at the transmitter to the message, and at the receiver, the masking is regenerated and subtracted from the received signal. The second approach utilizes modulation of the coefficients of the chaotic system in the transmitter and corresponding detection of synchronization error in the receiver to transmit binary-valued bit streams. The use of SCS's for communications relies on the robustness of the synchronization to perturbations in the drive signal. As a step toward further understanding the inherent robustness, we establish an analogy between synchronization in chaotic systems, nonlinear observers for deterministic systems, and state estimation in probabilistic systems. This analogy exists because SCS's can be viewed as performing the role of a nonlinear state space observer. To calibrate the robustness of the Lorenz SCS as a nonlinear state estimator, we compare the performance of the Lorenz SCS to an extended Kalman filter for providing state estimates when the measurement consists of a single noisy transmitter component. >

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TL;DR: In this article, the effects of fractional dynamics in chaotic systems were studied and it was demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior.
Abstract: This brief studies the effects of fractional dynamics in chaotic systems. In particular, Chua's system is modified to include fractional order elements. By varying the total system order incrementally from 3.6 to 3.7, it is demonstrated that systems of "order" less than three can exhibit chaos as well as other nonlinear behavior. This effectively forces a clarification of the definition of order which can no longer be considered only by the total number of differentiations or by the highest power of the Laplace variable. >

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Abstract: A fractional slope on the log log Bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole. In order to represent and study its dynamical behavior, a singularity function method is presented which consists of cascaded branches of a number of pole-zero (negative real) pairs or simple RC section. The distribution spectrum of the system can also be easily calculated, and its accuracy depends on a prescribed error specified in the beginning. The method is then extended to a multiple-fractal system which consists of a number of fractional power poles. The result can be simulated by a combination of singularity functions, each representing a single-fractal system. >

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    [...]

  • ...We will follow the algorithm in [13] to calculate linear transfer function approximations of (1)....

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Frequently Asked Questions (18)
Q1. What have the authors contributed in "Chaos in fractional-order autonomous nonlinear systems" ?

The authors numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0 ; 1, based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study ; an electronic chaotic oscillator, and a mechanical chaotic ‘ ‘ jerk ’ ’ model. 1. Consequently, the authors present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2þ e, 1 > e > 0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. 

The effective chaotic range of the control parameter for either model shrinks and moves downward as the total system order decreases and approaches 2, i.e. the chaotic range is affected by contraction and translation processes as the fractional order decreases. 

By utilizing frequency domain techniques based on Bode diagrams, one can obtain a linear approximation for the fractional-order integrator, the order of which depends on the desired bandwidth and discrepancy between the actual and the approximate magnitude Bode diagrams [13]. 

The resulting order of the approximating function of the fractional integrator depends on the desired bandwidth, and error between actual and approximate Bode magnitude plots of the fractional integrator. 

Wien bridgeoscillators have recently been studied in [9], where it is shown that limit cycles can be attained for any fractional order, with the proper value of the amplifier gain which, of course, is a function of the fractional order. 

as the fractional order decreases below m ¼ 0:5, the chaotic range shrinks significantly, and becomes much less pronounced than that for the chaotic oscillator as shown in the figure. 

The integer order canonical chaotic oscillator with nonlinearity given by f ðx1Þ ¼ sgnðx1Þ is known to give a doublescroll-like chaotic attractor for control parameter values in the range 1:0 > a > 0:49. 

it can be noticed that the effective chaotic range of the control parameter shifts downward away from that corresponding to the integer order model. 

A third-order chaotic autonomous nonlinear system, with the appropriate nonlinearity and control parameters, is chaotic for any fractional order 2þ e, 1 > e > 0. 

The chaotic behaviors of the fractional-order models exhibit some degree of qualitative robustness against model order reduction as would be caused by modeling errors. 

the authors have conducted several numerical simulations on fractional systems with total system order less than 2, as would be obtained by replacing all three integrators by fractional ones. 

According to the Poincare–Bendixon theorem [4], an integer order chaotic nonlinear system must have a minimum order of 3 for chaos to appear. 

From a state-space point of view, the fractional order of the resulting characteristic equation results from a fractional-order integrator placed, for example, in place of the output integrator in the system block diagram realization. 

The authors consider the one-parameter, third-order chaotic oscillator of canonical structure reported in [10] as being the simplest possible structure for a chaotic oscillator. 

The resulting fractional chaotic oscillator model is then given in state space asdxm1 dtm¼ x2 dx2 dt¼ x3 dx3 dt ¼ aðx1 þ x2 þ x3 þ f ðx1ÞÞð7Þwhere a is the control parameter for this oscillator, and f ðx1Þ ¼ sgnðx1Þ is the model nonlinearity. 

This model, shown in Fig. 1b, is used to determine the time derivative of acceleration of an object, referred to as jerk[3,11,12]. 

In Figs. 12 and 13, the authors show simulations of the fractional jerk model of total orders 2.5 and 2.1, using nonlinearity 9c, and control values of 0.3 and 0.05, respectively. 

The authors derive the approximate linear transfer functions for the fractional integrator of order that varies from 0.1 to 0.9, and study the resulting behavior of the entire system for each case under the effect of different types of nonlinearities.