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

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 bifurcation 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 experiment. A suitable electrical circuit (i.e., analog simulator) is designed and used for the investigations. Results of theoretical analysis are verified by laboratory experimental 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.

Full text

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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