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On the Dynamics of Chaotic Systems

with Multiple Attractors: A Case Study

J. Kengne, A. Nguomkam Negou, D. Tchiotsop, V. Kamdoum Tamba

and G.H. Kom

Abstract In this chapter, the dynamics of chaotic systems with multiple coexisting

attractors is addressed using the well-known Newton–Leipnik system as prototype. In

the parameters space, regions of multistability (where the system exhibits up to four

disconnected attractors) are depicted by performing forward and backward bifur-

cation analysis of the model. Basins of attraction of various coexisting attractors

are computed, showing complex basin boundaries. Owing to the fractal structure

of basin boundaries, jumps between coexisting attractors are predicted in experi-

ment. A suitable electrical circuit (i.e., analog simulator) is designed and used for

the investigations. Results of theoretical analysis are veriﬁed by laboratory experi-

mental measurements. In particular, the hysteretic behavior of the model is observed

in experiment by monitoring a single control resistor. The approach followed in

this chapter shows that by combining both numerical and experimental techniques,

one can gain deep insight into the dynamics of chaotic systems exhibiting multiple

attractor behavior.

1 Introduction

It is well known that nonlinear dynamical systems can develop various forms of com-

plexity such as bifurcation, chaos, hyperchaos, and intermittency, just to name a few.

The occurrence of two or more asymptotically stable equilibrium points or attract-

ing sets (e.g., period-nlimit cycle, torus, chaotic attractor) as the system parameters

are being monitored represents another striking and complex behavior observed in

nonlinear systems. In a system developing coexisting attractors, the trajectories selec-

J. Kengne (B

)·A. Nguomkam Negou ·D. Tchiotsop ·G.H. Kom

Laboratoire d’Automatique et Informatique Appliquée (LAIA), Department of Electrical

Engineering, IUT-FV Bandjoun, University of Dschang, Dschang, Cameroon

e-mail: kengnemozart@yahoo.fr

A. Nguomkam Negou ·V. Kamdoum Tamba

Laboratory of Electronics and Signal Processing, Department of Physics,

University of Dschang, 67, Dschang, Cameroon

© Springer International Publishing AG 2018

K. Kyamakya et al. (eds.), Recent Advances in Nonlinear Dynamics

and Synchronization, Studies in Systems, Decision and Control 109,

DOI 10.1007/978-3-319-58996-1_2

17

18 J. Kengne et al.

tively converge to either of the attracting sets depending on the initial state of the

system. Correspondingly, the basin of attraction of an attractive set is deﬁned as the

set of initial points whose trajectories converge to the given attractor. The bound-

ary separating each basin of attraction can be a smooth boundary or riddled basin

with no clear demarcation (i.e., fractal). This striking and interesting phenomenon

has been encountered in various nonlinear systems including lasers [1], biological

systems [2,3], chemical reactions [4], Lorenz systems [5], Newton–Leipnik sys-

tems [6], and electrical circuits [7–11]. Such a phenomenon is connected primarily

to the system symmetry and may be accompanied by some special effects such as

symmetry-breaking bifurcation, symmetry-restoring crisis, coexisting bifurcations,

and hysteresis [12–15]. In practice, the coexistence of multiple attractors implies

that an attractor may suddenly jump to a different attractor, the situation in which

coexisting attractors possess a fractal or intermingled basin of attraction being the

most intriguing. In this case, due to noise, the observed signal may be the result of

random switching of the system trajectory between two or more concurrent coexist-

ing attractors. This chapter deals with the dynamics of the Newton–Leipnik equation

considered as a prototypal dynamical system with multiple coexisting attractors. First

of all, let us review some interesting works related to the analysis and control of this

particular system. The mathematical model of the so-called Newton–Leipnik system

was introduced by Newton and Leipnik [6] in 1981. The Euler rigid-body equations

were modiﬁed with the addition of linear feedback. A system of three quadratic dif-

ferential equations was obtained that for certain feedback gains develops two strange

attractors. The attractor for an orbit was determined by the location of the initial point

for that orbit. In [16], Wang and Tian consider the bifurcation analysis and linear con-

trol of the Newton–Leipnik system as a prototypal dynamic system with two strange

attractors. The static and dynamic bifurcations of the model are studied. Chaos con-

trolling is performed by a linear controller, and numerical simulation of the control

is supplied. Further results on the dynamics and bifurcations of the Newton–Leipnik

equation were provided by Lofaro [17]. The authors used numerical computations

and local stability calculations to suggest that the dynamics of the Newton–Leipnik

equations are related to the dynamics and bifurcations of a family of odd symmetric

bimodal maps. The article [18] also studies the dynamical behavior of the Newton–

Leipnik system and its trajectory-transformation control problem to multiple attrac-

tors. A simple linear state feedback controller for the Newton–Leipnik system based

on Lyapunov stability theory and application of the inverse optimal control strategy

is designed. Chaotic attractors are stabilized asymptotically to unstable equilibria of

the systems, so that the transformation of one attractor to another for the trajectory

of the Newton–Leipnik system is realized. In [19], it is shown how a chaotic system

with more than one strange attractor can be controlled. Issues in controlling multiple

(coexisting) strange attractors as stabilizing a desired motion within one attractor as

well as taking the system dynamics from one attractor to another are addressed. Real-

ization of these control objectives is demonstrated using as a numerical example the

Newton–Leipnik equation. Motivated by the above-mentioned results, this chapter

proposes a methodological analysis of the Newton–Leipnik equation considered as

a prototypal dynamical system with multiple coexisting attractors. Regions of mul-

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 19

tiple attractor behavior (i.e., hysteretic dynamics) are illustrated using bifurcation

diagrams computed based on suitable techniques. Furthermore, basins of attraction

of various coexisting attractors are also computed to visualize how the various coex-

isting attractors magnetize the state space. Owing to the fast computation speed of

analog computers, we suggest that such an apparatus can be advantageously exploited

to investigate nonlinear systems with multiple attractors such as the Newton–Leipnik

equation. The rest of the chapter is structured as follows. Section 2describes the math-

ematical model of the Newton–Leipnik system. Some basic properties of the model

are underlined with practical implications for the occurrence of multiple attractors. In

Sect. 3, the bifurcation structures of the system are investigated numerically showing

period-doubling and symmetry-recovering crisis phenomena. Regions of the para-

meters space corresponding to the occurrence of multiple coexisting attractors are

depicted. Correspondingly, basins of attraction of various coexisting solutions are

computed showing complex basin boundaries. A suitable electrical circuit (i.e., ana-

log computer) that can be exploited for the analysis of the Newton–Leipnik equation

is proposed in Sect. 4. Finally, some concluding remarks are presented in Sect.5.

2 Description and Analysis of the Model

2.1 The Model

The mathematical model of the Newton–Leipnik equation [6] considered in this

chapter is expressed by the following set of three coupled ﬁrst-order nonlinear dif-

ferential equations:

˙x1=−ax1+x2+bx2x3,

˙x2=−x1−ax2+5x1x3,(1)

˙x3=cx3−5x1x2,

where x1,x2, and x3are the state variables; a,b, and care three positive real constants.

It can be seen that the model possesses three quadratic nonlinearities in which are

involved the three state variables (x1,x2, and x3).

The presence of this nonlinearity is responsible for the complex behaviors exhib-

ited by the whole system. Obviously, system (1) is invariant under the transformation

(x1,x2,x3)⇔(−x1,−x2,x3). Therefore, if (x1,x2,x3) is a solution of system (1)

for a speciﬁc set of parameters, then (−x1,−x2,x3) is also a solution for the same

parameter set. The ﬁxed point E0(0,0,0)is a trivial symmetric static solution. Also,

attractors in state space have to be symmetric by reﬂection in the x3-axis; otherwise,

they must appear in pairs to restore the exact symmetry of the model equations. This

exact symmetry could serve to explain the presence of several coexisting attractors

in state space [20,21]. Furthermore, it represents a good way to check the scheme

used for numerical analysis. It is important to note that for typical parameters val-

ues a=0.4, b=10, c=0.175, system (1) has ﬁve equilibrium points, which are

20 J. Kengne et al.

all unstable [18,19]. Also, for these parameter values, the system experiences self-

excited oscillations [22,23].

2.2 Dissipation and Existence of Attractors

Preliminary insights related to the existence of attractors in Newton–Leipnik systems

can be gained by evaluating the volume contraction rate [20,21] of the model.

Brieﬂy recall that the volume contraction rate of a continuous-time dynamical system

described by ˙x=ϕ(x), where x=(x1,x2,x3)Tand ϕ(x)=(ϕ1(x), ϕ2(x), ϕ3(x))T,is

given by

=∇.ϕ (x)=∂ϕ1

∂x1

+∂ϕ2

∂x2

+∂ϕ3

∂x3

.(2)

We note that if is a constant, then the time evolution in phase space is determined

by V(t)=V0eλ(t), where V0=V(t=0). Thus, a negative value of leads to a

fast exponential shrinkage (i.e., damped) of the volume in state space; the dynamical

system is dissipative and can experience or develop attractors. For =0, phase space

volume is conserved, and the dynamical system is conservative. If is positive, the

volume in phase space expands, and hence there exist only unstable ﬁxed points or

cycles or possibly chaotic repellors [20,21]; in other words, the dynamics diverge as

the system evolves (i.e., for t→∞) if the initial conditions do not coincide exactly

with one of the ﬁxed points or stationary states. Referring to the model in (1), it can

easily be shown that =c−2a<0 independently of the position (x1,x2,x3)Tin

state space; hence system (1) is dissipative, and thus can support attractors.

3 Numerical Study

3.1 Numerical Methods

In order to examine the rich variety of dynamical behaviors that can be observed

in a Newton–Leipnik system, we solve numerically system (1) using the classical

fourth-order Runge–Kutta integration algorithm. For each set of parameters used

in this chapter, the time step is always t=0.005 and the calculations are carried

out using variables and constant parameters in extended mode. For each parameter

setting, the system is integrated for a sufﬁciently long time and the transient is can-

celed. Two indicators are used to identify the type of scenario giving rise to chaos.

The bifurcation diagram represents the ﬁrst indicator, the second indicator being

the graph of the largest Lyapunov exponent (λmax). Concerning the latter case, the

dynamics of the system is classiﬁed using its Lyapunov exponent, which is com-

puted numerically using the algorithm described by Wolf and collaborators [24]. In

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 21

particular, the sign of the largest Lyapunov exponent determines the rate of almost all

small perturbations to the system’s state, and consequently, the nature of the under-

lined dynamical attractor. For λmax <0, all perturbations vanish, and trajectories

starting sufﬁciently close to each other converge to the same stable equilibrium point

in state space; for λmax =0, initially close orbits remains close but distinct, corre-

sponding to oscillatory dynamics on a limit cycle or torus; and ﬁnally, for λmax >0,

small perturbations grow exponentially, and the system evolves chaotically within

the folded space of a strange attractor.

3.2 Route to Chaos

To investigate the sensitivity of the system with respect to a single control parameter,

we ﬁx a=0.6, b=5 and vary cin the range 0.13 ≤c≤0.15. In monitoring the

control parameter, it is found that the Newton–Leipnik system under consideration

can experience very rich and striking bifurcation scenarios. Sample results show-

ing bifurcation diagrams for varying cand the corresponding spectrum of largest

Lyapunov exponent are provided in Fig. 1a and b respectively.

Fig. 1 Bifurcation diagram

ashowing local maxima of

the coordinate x1versus c

and the corresponding graph

bof largest Lyapunov

exponent (λmax) plotted in

the range 0.13 ≤c≤0.15.

A window of hysteretic

dynamics can be noticed for

lower values of c.Magenta

and blue diagrams

correspond respectively to

increasing and decreasing

values of c. The positive

value of λmax is the signature

of chaotic motion: Parameter

values (a,b)=(0.6,5)

22 J. Kengne et al.

The bifurcation diagram is obtained by plotting local maxima of the coordinate x1

in terms of the control parameter, which is increased (or decreased) in tiny steps in

the range 0.13 ≤c≤0.15. The ﬁnal state at each iteration of the control parameter

serves as the initial state for the next iteration. In the graph of Fig.1a, two sets of data

corresponding to increasing (blue) and decreasing (magenta) values of care super-

imposed. This strategy, known as forward and backward continuation, represents a

simple way to localize the window in which the system develops multiple coexisting

attractor behaviors (see Sect.4). In light of Fig. 1a and b, the following bifurcation

sequence emerges when the control parameter cis slowly increased. For values of

cunder the critical value cc=0.11, the system exhibits periodic oscillations (i.e.,

period-3 limit cycle). On increasing the control parameter cpast this critical value,

a stable period-3 limit cycle born from the Hopf bifurcation undergoes a series of

period-doubling bifurcations, culminating in a single-band spiraling chaotic attrac-

tor. On further increasing cup to ccr1=0.1402, a periodic window suddenly appears

in which the system displays a period-8 and then a single-band chaotic attractor. Past

the critical value ccr2=0.143, the single-band chaotic attractor suddenly changes to

a double-band chaotic attractor following a symmetry-recovering crisis. Also note

the presence of many tiny windows of periodic motions sandwiched in the chaotic

bands. It can be seen that the bifurcation diagram coincides well with the spec-

trum of the Lyapunov exponent. With the same parameter settings in Fig.1, various

Fig. 2 Two-dimensional views of the attractor projected onto the (x1,x2)-plane (left) of the system,

showing routes to chaos (in terms of the control parameter c) and corresponding power spectra

(right): aperiod-1 for c=0.13, bperiod-2 for c=0.135, cperiod-4 for c=0.1373, dsingle-

band chaos for c=0.1388, esingle-band chaos for c=0.1328, fdouble-band for c=0.145. The

parameters are those of Fig. 2

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 23

numerical phase portraits and corresponding frequency spectra were obtained, con-

ﬁrming different bifurcation sequences depicted previously (see Fig.2).

It should be noted that for periodic motion, all spikes in the power spectrum are

harmonically related to the fundamental, whereas a broadband power spectrum is

characteristic of a chaotic mode of oscillations. Brieﬂy recall that the periodicity of

an attractor is deduced by counting the number of spikes located at the left-hand side

of the highest spike of the spectrum (the latter being included).

3.3 Occurrence of Multiple Attractors

To observe the phenomenon of multiple attractors, two prototypes were considered.

The ﬁrst is obtained by setting (a,b)=(0.6,5). With reference to the bifurcation

diagram of Fig. 1, a window of hysteretic dynamics (i.e., multiple stability) can be

identiﬁed in the range 0.13 ≤c≤0.14 (see Fig. 3).

For values of cwithin this range, the long-term behavior of the system depends on

the initial state, thus leading to the interesting and striking phenomenon of coexisting

multiple attractor behaviors. Up to four different attractors can be obtained depending

solely on the selection of initial conditions. For c=0.142, four different chaotic

attractors are presented in a phase portrait of (Fig. 4) using different initial conditions

(x1(0), x2(0), x3(0)) =(0,±0.1,±0.01).

The second approach is obtained using (a,b)=(0.73,10). We plot the corre-

sponding bifurcation diagram versus the control parameter c, as well as the corre-

sponding graph of the Lyapunov exponent (see Fig. 5a and b).

In the diagram of Fig. 5a, a window of hysteretic dynamics (and thus multistability)

can be identiﬁed in the range 0.13 ≤c≤0.16. For values of cwithin this range, the

long-term behavior of the system depends on the initial state; hence the system

develops the interesting phenomenon of coexisting multiple attractor behaviors. For

Fig. 3 Enlargement of the

bifurcation diagram of Fig. 2

showing the region in which

the system exhibits multiple

coexisting attractors. This

region corresponds to values

of cin the range

0.13 ≤c≤0.144. Two sets

of data corresponding to

increasing (magenta)and

decreasing (blue)valuesof

the control parameter are

superimposed

24 J. Kengne et al.

Fig. 4 Coexistence of four

different attractors consisting

of two pairs of single-band

chaotic attractors projected

onto the (x1,x2)-plane for

(a,b)=(0.6,5)and

c=0.142. Initial conditions

are (x1(0), x2(0), x3(0)) =

(0,±0.1,±0.01)

Fig. 5 Bifurcation diagram

ashowing local maxima of

the coordinate x1versus c

and the corresponding graph

bof the largest Lyapunov

exponent λmax plotted in the

range 0.13 ≤c≤0.16. A

window of hysteretic

dynamics can be noticed for

some values of c.Magenta

and blue diagrams

correspond respectively to

increasing and decreasing

values of c. The positive

value of λmax is the signature

of chaotic motion: parameter

values (a,b)=(0.73,10)

c=0.151, four different attractors (see Fig. 6d) can be obtained depending only on

the selection of initial conditions.

For instance, the pair of period-1 phase portraits and a pair of chaotic attrac-

tors of Fig. (6d) can be obtained under the initial conditions (x1(0), x2(0), x3(0)) =

(0,±0.1,±0.01). We also see that for c=152, three different attractors (see Fig. 7d)

can be obtained depending only on the selection of initial conditions, a pair of

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 25

Fig. 6 Cross sections of the basin of attraction a,b,cfor x1(0)=0, x2(0)=0, and x3(0)=0, and

x3(0)=0 respectively and two-dimensional views of four different attractors consisting of a pair

of single-band chaotic attractors with a pair of asymmetric period-1 attractors dprojected onto the

(x1,x2)-plane for (a,b)=(0.73,10)and c=0.151. Initial conditions are (x1(0), x2(0), x3(0)) =

(0,±0.1,±0.01).Yell ow and green regions correspond to the period-1 pair of attractors, while

the magenta and blue regions are associated with the chaotic solutions obtained for (a,b,c)=

(0.73,5,0.151)

period-1 phases portraits and a double-band chaotic attractor for (x1(0), x2(0), x3(0)) =

(0,±0.1,±0.01). Therefore, considering the parameters in Fig. 6d and carrying out

a scan of initial conditions (see Fig. 6a–c), we have deﬁned the domain of initial

conditions in which each attractor can be found. Figure 6a–c show cross sections

of the basin of attraction respectively for x1(0)=0, x2(0)=0, and x3(0)=0 cor-

responding to the symmetric pair of limit cycles (blue and green) and the pair of

chaotic attractors (magenta and yellow). Likewise, considering the parameter setting

in Fig. 7d, we have deﬁned the domain of initial conditions in which each attractor

can be found.

Figure 7a–c show the structure of the sections of the basin of attraction respec-

tively for x1(0)=0, x2(0)=0, and x3(0)=0. Green and yellow lead to a pair of

period-1 limit cycle while magenta regions are associated to the double-band chaotic

solution. It should be mentioned that multiple attractor behavior (involving at least

26 J. Kengne et al.

Fig. 7 Cross sections of the basin of attraction a,b,cfor x1(0)=0, x2(0)=0, and x3(0)=0

respectively and two-dimensional views of three different attractors consisting of a double-band

chaotic attractor with a pair of asymmetric period-1 ones dprojected onto the (x1,x2)-plane. Initial

conditions are (x1(0), x2(0), x3(0)) =(0,±0.1,±0.01).Green and yellow regions correspond to

the period-1 pair of attractors, while magenta regions are associated with the chaotic solutions

obtained for (a,b,c)=(0.73,10,0.152)

four nonstatic disconnected attractors) is common in various nonlinear systems (see,

e.g., Sect. 1). Very recently, Hens and collaborators considered the case of coex-

istence of inﬁnitely many attractors, also referred to as extreme multistability, in

coupled dynamical systems [25]. It is obvious that the occurrence of multiple attrac-

tors is an additional source of randomness in chaotic systems that may be exploited

for chaos-based secure communication. However, in many other cases, this singular

type of behavior is not desirable and it justiﬁes the need for control. Detailed study

along this line is beyond the scope of this work; also, interested readers are referred

to the review work on control of multistability by Pisarshik and collaborators [26].

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 27

4 The Analog Computer Approach

It is well known that even with a very fast computer, scanning the parameter space

can turn out to be very time-consuming. Furthermore, there is no rigorous method for

selecting the integration step used for the numerical integration as well as the duration

of the transient time. These difﬁculties (as well as many others) faced in performing

numerical computation can be overcome by adopting the analog computer approach

[27,28]. One of the merits of the analog computer is the possibility of exploring wide

ranges of dynamic behaviors by simply monitoring, for instance, a single control

resistor. Nevertheless, the accuracy of the results of analog computation strongly

depends on the quality of the electronic components used in the construction of

the analog computer. Also, by combining the fast computation speed of an analog

simulator and the precision of a digital computer, one can gain deep insight into

the dynamics of a given nonlinear process such as the Newton–Leipnik system.

Our goal in this section is to design and implement an appropriate analog simulator

that can be exploited for the analysis of the model deﬁned in system (1). A circuit

diagram of the proposed electronic simulator is provided in Fig. 8. Compared to the

Fig. 8 Electronic circuit realization of a Newton–Leipnik system with three quadratic

interactions using R1=R7=16666, R2=R4=R5=R6=R11 =R12 =10 K, R10 =0−

100K,andC=10nF

28 J. Kengne et al.

circuit proposed in [28] (utilizing up to twenty resistors and nine operational (op)

ampliﬁers), the analog simulator shown in Fig.8involves a minimum number of

electronic components.

The electronic multipliers are the analog devices AD633JN, versions of the

AD633 four-quadrant voltage multipliers chips used to implement the nonlinear

terms of our model. They operate over a dynamic range of ±1 V with a typical

error less than 1%. They also have a built-in divide-by-ten feature. The signal W

at the output depends on the signals at inputs X1(+),X2(−),Y1(+),Y2(−), and

W=(X1−X2)(Y1−Y2)/10 +Z. The operational ampliﬁers and associated cir-

cuitry implement the basic operations of addition, subtraction, and integration. By

adopting an appropriate time scaling, the simulator outputs can be displayed directly

on an oscilloscope by connecting the output voltage of X1to the Xinput and the

output voltage of X2to the Yinput. By applying the Kirchhoff current and voltage

laws to the circuit in Fig. 8, it can be shown that the voltages X1,X2, and X3satisfy

the set of three coupled ﬁrst-order nonlinear differential equations

dX1

dt =− X1

R1C1

+X2

R2C1

+X2X3

10R3C1

,

dX2

dt =− X2

R7C2

−X1

R6C2

+X1X3

10R8C2

,(3)

dX3

dt =X3

R10C3

−X1X2

10R9C3

.

Fig. 9 The experimental Newton–Leipnik simulator in operation

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 29

Fig. 10 Experimental phase portraits (left) obtained from the circuit using a dual trace oscilloscope

in XY mode; the corresponding numerical phase portraits are shown on the right, obtained by a

direct integration of the system (1) with (a,b,c)=(0.6,5,0.145):aprojection onto the (X3,X2)-

plane, bprojection onto the (X1,X2)-plane, and cprojection onto (X1,X3)-plane. The scales are

X=1V/divand Y=0.5V/divfor all pictures

30 J. Kengne et al.

With a time unit of 104, the parameters of system (1) are expressed in terms of the

values of capacitors and resistors as follows (provided that the critical relationships

104R2C1=1, 104R6C2=1, 5.105R8C2=1, 5.105R9C3=1 are satisﬁed):

a=10−4

R1C1

,b=10−4

R3C1

,c=10−4

R10C3

.(4)

We brieﬂy recall that the time scaling process offers to analog devices (operational

ampliﬁers and analog multipliers) the possibility of operating under their bandwidth.

Furthermore, time scaling offers the possibility of simulating the behavior of the

system at any given frequency by expressing the real time variable τversus the

analog computation time variable t(t=10−nτ), allowing the simulation frequency

to be less than the real frequency by a factor of order 10+n. In the latter expression,

the positive integer depends on the values of resistors and capacitors used in the

analog simulator. A photograph of the experimental analog simulator in operation

is shown in Fig. 9, while a comparison between numerical and experimental phase

portraits is provided in Fig. 10.

From the graphs in Fig. 10, it clearly appears that the dynamics of the Newton–

Leipnik system is well reproduced by the analog simulator.

5 Concluding Remarks

This chapter has focused on a methodological analysis of the Newton–Leipnik sys-

tem considered as a prototypal dynamical system with multiple attractors. Regions of

multiple attractors in the parameter space were depicted using bifurcation diagrams

based on appropriate techniques (e.g., forward and backward bifurcation diagrams).

Furthermore, basins of attraction of various competing attractors were computed,

showing nontrivial basin boundaries, thus suggesting possible jumps between dif-

ferent coexisting attractors in experiment. Moreover, one piece of interesting infor-

mation that can be gained from basins of attraction is the chance of the appearance

of attractors in a real system. A suitable electrical circuit (i.e., analog simulator)

was designed that was shown to reproduce the Newton–Leipnik attractor. Combined

with numerical techniques, the proposed analog computer may be particularly use-

ful for exploring the parameter space in view of tracking further regions of multiple

attractors in the Newton–Leipnik model. We stress that the approach followed in

this chapter may be exploited advantageously in the investigation of other nonlinear

dynamical systems exhibiting multiple attractors.

On the Dynamics of Chaotic Systems with Multiple Attractors: A Case Study 31

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