ChapterPDF Available

Abstract and Figures

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.
Content may be subject to copyright.
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 [711]. 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 [1215]. 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=−x1ax2+5x1x3,(1)
˙x3=cx35x1x2,
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)=1(x), ϕ2(x), ϕ3(x))T,is
given by
=∇ (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 =c2a<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 c0.15. 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 c0.15.
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 c0.15. 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 c0.14 (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 c0.16. 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 c0.144. 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 c0.16. 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=(X1X2)(Y1Y2)/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=104
R1C1
,b=104
R3C1
,c=104
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=10nτ), 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
References
1. Masoller, C.: Coexistence of attractors in a laser diode with optical feedback from a large
external cavity. Phys. Rev. A 50, 2569–2578 (1994)
2. Cushing, J.M., Henson, S.M.: Blackburn: multiple mixed attractors in a competition model. J.
Biolog. Dyn. 1, 347–362 (2007)
3. Upadhyay, R.K.: Multiple attractors and crisis route to chaos in a model of food-chain. Chaos,
Solitons Fractals 16, 737–747 (2003)
4. Massoudi, A., Mahjani, M.G., Jafarian, M.: Multiple attractors in Koper-Gaspard model of
electrochemical. J Electroanal. Chem. 647, 74–86 (2010)
5. Li, C., Sprott, J.C: Coexisting hidden attractors in a 4-D simplified Lorenz system. Int. J. Bifurc.
Chaos 24, 1450034 (2014)
6. Leipnik, R.B., Newton, T.B.: Double strange attractors in rigid body motion with linear feed-
back control. Phys. Lett. A 86, 63–87 (1981)
7. Vaithianathan, V., Veijun, J.: Coexistence of four different attractors in a fundamental power
system model. IEEE Trans. Cir. Syst. I46, 405–409 (1999)
8. Kengne, J.: Coexistence of chaos with hyperchaos, period-3 doubling bifurcation, and transient
chaos in the hyperchaotic oscillator with gyrators. Int. J. Bifurc. Chaos 25(4), 1550052 (2015)
9. Pivka, L., Wu, C.W., Huang, A.: Chua’s oscillator: a compendium of chaotic phenomena.
J.Frankl. Inst. 331B(6), 705–741 (1994)
10. Kuznetsov, A.P., Kuznetsov, S.P., Mosekilde, E., Stankevich, N.V.: Co-existing hidden attrac-
tors in a radio-physical oscillator. J. Phys. A Math. Theor. 48, 125101 (2015)
11. Kengne, J., Njitacke, Z.T., Fotin, H.B.: Dynamical analysis of a simple autonomous jerk system
with multiple attractors. Nonlinear Dyn. 48, 751–765 (2016)
12. Li, C., Hu, W., Sprott, J.C., Wang, X.: Multistability in symmetric chaotic systems. Eur. Phys.
J. Spec. Top. 224, 1493–1506 (2015)
13. Letellier, C., Gilmore, R.: Symmetry groups for 3D dynamical systems. J. Phys. A. Math.
Theor. 40, 5597–5620 (2007)
14. Rosalie, M., Letellier, C.: Systematic template extraction from chaotic attractors: I. Genus-one
attractors with inversion symmetry. J. Phys. A Math. Theor. 46, 375101 (2013)
15. Rosalie, M., Letellier, C.: Systematic template extraction from chaotic attractors: II. Genus-one
attractors with unimodal folding mechanisms. J. Phys. A Math. Theor. 48, 235100 (2015)
16. Xuedi, W., Lixin, T.: Bifurcation analysis and linear control of the Newton-Leipnik system.
Chaos, Solitons Fractals 27, 31–38 (2006)
17. Lofaro, T.: A model of the dynamics of the Newton-Leipnik attractor. Int. J. Bifurc. Chaos
7(12), 2723–2733 (1997)
18. Wang, X., Gao, Y.: The inverse optimal control of chaotic system with multiple attractors. Mod.
Phys. Lett. B 21, 1199–2007 (2007)
19. Hendrik, R.: Controlling chaotic systems with multiple strange attractors. Phys. Lett. A 300,
182–188 (2002)
20. Strogatz S.H.: Nonlinear Dynamics and Chaos. Addison-Wesley, Reading
21. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational and
Experimental Methods. Wiley, New York
22. Leonov, G.A., Kuznetsov, N.V.: Hidden attractors in dynamical systems. From hidden oscilla-
tions in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in
Chua circuits. Int. J. Bifurc. Chaos 23, 1793–6551 (2013)
23. Leonov, G.A., Kuznetsov, N.V., Mokaev, T.N.: Homoclinic orbits, and self-excited and hidden
attractors in a Lorenz-like system describing convective fluid motion. Eur. Phys. J. Spec. Top.
224, 1421–1458 (2015)
24. Wolf, A., Swift, J.B., Swinney, H.L., Wastano, J.A.: Determining Lyapunov exponents from
time series. Physica D 16, 285–317 (1985)
25. Pisarchik, A.N., Feudel, U.: Control of multistability. Phys. Rep. 540(4), 167–218 (2014)
26. Hens, C., Dana, S.K., Feudel, U.: Extreme multistability: attractors manipulation and robust-
ness. Chaos 25, 053112 (2015)
32 J. Kengne et al.
27. Chedjou, J.C., Fotsin, H.B., Woafo P., Domngang, S.: Analog simulation of the dynamics of
a van der Pol oscillator coupled to a Duffing oscillator. IEEE Trans. Circuits Syst. I: Fundam.
Theory Appl. 48, 748–756 (2001)
28. Zhao, R., Song, Y.: Circuit realization of Newton-Leipnik chaotic system via EWB. Chinese
control and decision conference, Yantai, Shandong, pp. 5111–5114 (2008)
... In addition, they have now been widely adopted to produce complex dynamics, because of their nonlinearity and memory ability. These include multistability [9][10][11][12][13][14][15] and hidden extreme multistability [16][17][18]. ...
... From equation (14), M J (P) has three constant eigenvalues namely, two zero-eigenvalue (ζ 1 = ζ 2 = 0), a negative eigenvalue (ζ 3 = −b) and three eigenvalues depending on the value of p 2 . These latter are the roots of the following equation: ...
Article
Full-text available
A simple 5D hyperchaotic system recently introduced in the literature is modified by using a charge-controlled memristor model and striking behaviors are uncovered. The resulting system is a 6D hyperchaotic system, which generates hidden attractors with the unusual feature of having plan and line equilibrium under different parameter conditions. Its dynamical behaviors are characterized through bifurcation diagrams, Lyapunov exponents, phase portraits, Poincaré sections and time series. Rich nonlinear dynamics such as limit cycles, quasi-periodicity, chaos, hyperchaos, bursting and hidden extreme multistability are found for appropriate sets of parameter values. The high complexity of the system is confirmed by its Kaplan–yorke dimension (greater than five). Additionally, an electronic circuit is designed to implement the novel system and PSpice simulation results are in good accordance with the numerical investigations. To the best of our knowledge, this system is the first with higher order presenting all those phenomena.
... [35][36][37][38][39][40][41][42][43][44][45][46][47][48] Another key remark concerns the existence of multiple solutions, i.e., the ability of the considered system to select (in an apparently random way) different steady modes of convection for a fixed aspect ratio and Rayleigh number (see Table II). As demonstrated by other studies based on numerical simulations, [8][9][10]45,47,48 these alternate states can be accessed through relevant changes in the initial conditions because they exist in the space of phases as independent solutions with disjoint basins of attraction (Yao, 51 Kengne et al. 52 ). In particular, we could observe this feature of RB convection when conducting the experiments for A ¼ 0.3 (Figs. 8 and 9) and A ¼ 0.6 ( Figs. 11 and 12). ...
Article
Full-text available
The modes of pure buoyant (thermogravitational) convection emerging in a liquid bridge of water (Pr ≅ 6.1), uniformly heated from below and cooled from above are investigated experimentally by means of a microscale facility, a related laser-cut technique (used to illuminate isodense tracers dispersed in the liquid) and a particle image velocimetry method. In particular, the following conditions are examined: aspect ratio (A = length/diameter) in the range 0.3 ≤ A ≤ 0.9, volume ratio 0.7 ≤ S ≤ 1.3, and Rayleigh number spanning the interval from the initial quiescent state up to the development of oscillatory motion. A multitude of patterns is obtained, revealing the coexistence of different branches of steady flows in the space of parameters in the form of multiple solutions. These can evolve into oscillatory states featured by disturbances with the characteristics of standing waves (a kind of rocking motion). The analysis largely relies on a novel approach where the position of the center of the main vortex of buoyant nature established in the liquid bridge is carefully monitored in space. The related trajectory is used to discern the flow spatial degrees of freedom, which are progressively enabled as the temperature difference is increased. It is shown that the effective volume of liquid held by surface tension between the hot and cold walls can have an appreciable impact on the onset of unsteadiness and the related oscillation frequency.
... Thereby heightening the randomness of the system. This is a favorable feature for certain applications requiring, for example, random-like sequence generation [36]. 5. ...
Article
Full-text available
This paper presents a new chaotic system that has four attractors including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial condition, Lyapunov spectrum, measure of strangeness, basin of attraction including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation are rigorously treated. It is found by numerical computations that the system has a far-reaching composite basin of attraction, which is important for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding values of electronic components such as the circuital resistances while ensuring the dynamic ranges are well contained. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to innate properties of the system such as sensitivity on initial condition and topological mixing, the robot’s path becomes unpredictable and guaranteed to scan the workspace, respectively.
... Thereby heightening the randomness of the system. This is a favorable feature for certain applications requiring, for example, random-like sequence generation [36]. 5. ...
Preprint
This paper presents a new chaotic system having four attractors, including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial conditions, Lyapunov spectrum, strangeness measure, attraction basin, including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation, are rigorously treated. Numerical computations are used to compute the basin of attraction and show that the system has a far-reaching composite basin of attraction. Such a basin of attraction is vital for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding electronic component values such as the circuital resistances while ensuring the dynamic ranges are bounded. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to the properties of sensitivity on initial conditions and topological mixing, the robot's path becomes unpredictable and guaranteed to scan the workspace, respectively.
... Appendix B. Moreover, the system possesses hysterectic chaotic dynamics [184]. More precisely, there are four states including two stable fixed points and two chaotic attractors that the system can follow depending on a previous state. ...
Thesis
This work presents a new chaotic system having four attractors, including two fixed point attractors and two symmetrical chaotic strange attractors. It also discusses dynamical properties of the system viz. sensitive dependence on initial conditions, Lyapunov spectrum, the measure of strangeness, basin of attraction, classification and quantification of the basin of attraction, the existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation. It is found by numerical computations that the system has far-reaching basins of attraction that encompass roughly 95 percent of the state space. Moreover, sensitive dependence on initial conditions implies that long-term prediction of the system is impossible and any two such systems with slightly different initial conditions will become increasingly uncorrelated as time tends to infinity. However, despite this chaotic and unpredictable behavior, we realize various control schemes, including nonlinear control, active control, adaptive control, and robust adaptive control to synchronize two such systems in a master-slave topology irrespective of their initial conditions and parametric differences. The Lyapunov stability theory is used to ensure the boundedness of the closed-loop error dynamics. A Lyapunov-like analysis, involving the use of Barbalat lemma, is used to ensure asymptotic or exponential convergence of the error signals. Besides, the Lyapunov functions used are radially unbounded. Hence, global asymptotic or exponential stability of the closed-loop error dynamics follows. Besides, a secure communication application based on adaptive synchronization of a transmitter and a receiver based on the new chaotic system is realized. This situation is nontrivial as the adaptation mechanism tends to react against the information signals to suppress its effect on the equilibrium of the closed-loop system. Despite this situation, we can retrieve the information signals at the receiver. Depending on the nature of the modulating information signal, a detection mechanism may be used to reconstruct the original information signal at the receiver. Furthermore, the communication system based on adaptive synchronization of a transmitter-receiver system is subjected to a more realistic situation by transmitting the modulated signal via an additive white Gaussian noise (AWGN) channel of various signal-to-noise-ratios (SNRs) and noise power levels. A parametric sweep of various SNRs gives margins for continued useful communication.
... From the general theory of nonlinear dynamics, stationary points or steady states play essential role in revealing preliminary information about the complexity of the system [26][27][28]. Indeed, depending on their nature, nonlinear systems can be classified as self-excited [29][30][31] or hidden [32][33][34]. It is worth to remind that a self-excited attractor has a basin of attraction that is related to an unstable fixed point, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. ...
Article
Full-text available
This article focuses on the dynamics of a modified van der Pol–Duffing circuit (MVDPD hereafter) (Fotsin and Woafo in Chaos Solitons and Fractals 24(5):1363–1371, 2005) whose symmetry is explicitly broken with the presence an offset term. When ignoring offset terms, the system displays an exact symmetry which is reflected in the location of the equilibrium points, the attractor topologies and the attraction basins shapes as well. In this mode of operation, the system displays typical behaviors such as period doubling sequences; spontaneous symmetry breaking, symmetry recovering, and multistability involving several pairs of mutually symmetric attractors. In the presence of offset terms, the MVDPD circuit is non-symmetric and more complex nonlinear phenomena arise such as parallel bifurcation branches, coexisting multiple (i.e. two, three, four or five) asymmetric attractors, and crises. It should be noted that for each case of multistability discussed in this work, a hidden attractor (period-1 limit cycle) coexists with self-excited others. To the best of our knowledge, the coexistence of five attractors (symmetrical or asymmetrical), one of which is hidden has not yet been reported in the MVDPD circuit and thus deserves dissemination. PSpice simulation investigations based on the implementation of the MVDPD confirm the theoretical predictions.
... From the general theory of nonlinear dynamics, stationary points or steady states play essential role in revealing preliminary information about the complexity of the system [43][44][45]. Indeed, depending on their nature, nonlinear systems can be classified as self-excited [46][47][48] or hidden [49][50][51]. It is worth to remind that a self-excited attractor has a basin of attraction that is related to an unstable fixed point, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. ...
Article
Full-text available
It is well known that the symmetry break deliberately induced in a nonlinear system may help to discover new nonlinear patterns. In this work, we investigate the impact of an explicit symmetry break on the dynamics of a recently introduced chaotic system with a curve of equilibriums (Pham et al. in Circuits Syst Signal Process 37(3):1028-1043, 2018). We demonstrate that the symmetry break engenders rich and striking nonlinear phenomena including the coexistence of multiple asymmetric stable states, the presence of parallel bifurcation branches, hysteresis, and critical transitions as well. A simple control strategy based on linear augmentation technique that enables to drive the system from the state of four coexisting self-excited attractors to a monostable state is successfully adapted. To the best of the authors' knowledge, this work represents the first report on symmetry breaking for a chaotic system with infinite equilibriums and thus deserves dissemination.
... From the general theory of nonlinear dynamics, stationary points or steady states play an essential role in revealing preliminary information about the complexity of the system. [37][38][39] Indeed, depending on their nature, systems can be classi ed as self-excited [40][41][42] or hidden. [43][44][45] It is worth recalling that a self-excited attractor has a basin of attraction that is related to an unstable xed point, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. ...
Article
A simplified hyperchaotic canonical Chua’s oscillator (referred as SHCCO hereafter) made of only seven terms and one nonlinear function of type hyperbolic sine is analyzed. The system is found to be self-excited, and bifurcation tools associated with the spectrum of Lyapunov exponents reveal the rich dynamical behaviors of the system including hyperchaos, torus, period-doubling route to chaos, and hysteresis when turning the system control parameters. Wide ranges of hyperchaotic dynamics are highlighted in various two-parameter spaces based on two-parameter Lyapunov diagrams. The analysis of the hysteretic window using a basin of attraction as argument reveals that the SHCCO exhibits three coexisting attractors. Laboratory measurements further confirm the performed numerical investigations and henceforth validate the mathematical model. Of most/particular interest, multistability observed in the SHCCO is further controlled based on a linear augmentation scheme. Numerical results show the effectiveness of the control strategy through annihilation of the asymmetric pair of coexisting attractors. For higher values of the coupling strength, only a unique symmetric periodic attractor survives.
... More precisely, in addition to the "standard" initial conditions corresponding to a quiescent state, a "forward and backward continuation" method has been used, i.e., the final state at each iteration of the control parameter θ has been set as the initial condition for the next iteration (Ref. 57). ...
Article
Full-text available
Through numerical solution of the governing time-dependent and non-linear Navier-Stokes equations cast in the framework of the Oldroyd-B model, the supercritical states of thermal Marangoni-Bénard convection in a viscoelastic fluid are investigated for increasing values of the relaxation time while keeping fixed other parameters (the total viscosity of the fluid, the Prandtl number and the intensity of the driving force, Ma=300). A kaleidoscope of patterns is obtained revealing the coexistence of different branches of steady and oscillatory states in the space of parameters in the form of multiple solutions. In particular, two main families of well-defined attractors are identified, i.e. multicellular steady states and oscillatory solutions. While the former are similar for appearance and dynamics to those typically produced by thermogravitational hydrodynamic disturbances in layers of liquid metals, the latter display waveforms ranging from pervasive standing waves to different types of spatially localised oscillatory structures (oscillons). On the one hand, these localised phenomena contribute to increase the multiplicity of solutions and, on the other hand, give rise to interesting events, including transition to chaos and phenomena of intermittency. In some intervals of the elasticity number, the interference among states corresponding to different branches produces strange attractors for which we estimate the correlation dimension by means of the algorithm originally proposed by Grassberger and Procaccia.
Chapter
This chapter proposes a three-dimensional autonomous Van der Pol-Duffing (VdPD) type oscillator which is designed from a nonautonomous VdPD two-dimensional chaotic oscillator driven by an external periodic source through replacing the external periodic drive source with a direct positive feedback loop. The dynamical behavior of the proposed autonomous VdPD type oscillator is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, phase portraits and basin of attraction plots. Some interesting phenomena are found including for instance, period-doubling bifurcation, symmetry recovering and breaking bifurcation, double scroll chaos, bistable one scroll chaos and coexisting attractors. Basin of attraction of coexisting attractors is computed showing that is associated with an unstable equilibrium. So the proposed autonomous VdPD type oscillator belongs to chaotic systems with self-excited attractors. A suitable electronic circuit of the proposed autonomous VdPD type oscillator is designed and its investigations are performed using ORCAD-PSpice software. Orcard-PSpice results show a good agreement with the numerical simulations. Finally, synchronization of identical coupled proposed autonomous VdPD type oscillators in bistable regime is studied using the unidirectional linear feedback methods. It is found from the numerical simulations that the quality of synchronization depends on the coupling coefficient as well as the selection of coupling variables.
Article
Full-text available
Asymmetric and symmetric chaotic attractors produced by the simplest jerk equivariant system are topologically characterized. In the case of this system with an inversion symmetry, it is shown that symmetric attractors bounded by genus-one tori are conveniently analyzed using a two-components Poincaré section. Resulting from a merging attractor crisis, these attractors can be easily described as being made of two folding mechanisms (here described as mixers), one for each of the two attractors co-existing before the crisis: symmetric attractors are thus described by a template made of two mixers. We thus developed a procedure for concatenating two mixers (here associated with unimodal maps) into a single one, allowing the description of a reduced template, that is, a template simplified under an isotopy. The so-obtained reduced template is associated with a description of symmetric attractors based on one-component Poincaré section as suggested by the corresponding genus-one bounding torus. It is shown that several reduced templates can be obtained depending on the choice of the retained one-component Poincaré section.
Article
Full-text available
In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.
Article
Full-text available
In this paper, the dynamics of the paradigmatic hyperchaotic oscillator with gyrators introduced by Tamasevicius and co-workers (referred to as the TCMNL oscillator hereafter) is considered. This well known hyperchaotic oscillator with active RC realization of inductors is suitable for integrated circuit implementation. Unlike previous literature based on piecewise-linear approximation methods, I derive a new (smooth) mathematical model based on the Shockley diode equation to explore the dynamics of the oscillator. Various tools for detecting chaos including bifurcation diagrams, Lyapunov exponents, frequency spectra, phase portraits and Poincaré sections are exploited to establish the connection between the system parameters and various complex dynamic regimes (e.g. hyperchaos, period-3 doubling bifurcation, coexistence of attractors, transient chaos) of the hyperchaotic oscillator. One of the most interesting and striking features of this oscillator discovered/revealed in this work is the coexistence of a hyperchaotic attractor with a chaotic one over a broad range of system parameters. This phenomenon was not reported previously and therefore represents a meaningful contribution to the understanding of the behavior of nonlinear dynamical systems in general. A close agreement is observed between theoretical and experimental analyses.
Article
Full-text available
The coexistence of infinitely many attractors is called extreme multistability in dynamical systems. In coupled systems, this phenomenon is closely related to partial synchrony and characterized by the emergence of a conserved quantity. We propose a general design of coupling that leads to partial synchronization, which may be a partial complete synchronization or partial antisynchronization and even a mixed state of complete synchronization and antisynchronization in two coupled systems and, thereby reveal the emergence of extreme multistability. The proposed design of coupling has wider options and allows amplification or attenuation of the amplitude of the attractors whenever it is necessary. We demonstrate that this phenomenon is robust to parameter mismatch of the coupled oscillators.
Article
Full-text available
The term 'hidden attractor' relates to a stable periodic, quasiperiodic or chaotic state whose basin of attraction does not overlap with the neighborhood of an unstable equilibrium point. Considering a three-dimensional oscillator system that does not allow for the existence of an equilibrium point, this paper describes the formation of several different coexisting sets of hidden attractors, including the simultaneous presence of a pair of coinciding quasiperiodic attractors and of two mutually symmetric chaotic attractors. We follow the dynamics of the system as a function of the basic oscillator frequency, describe the bifurcations through which hidden attractors of different type arise and disappear, and illustrate the form of the basins of attraction.
Article
Full-text available
A new simple four-dimensional equilibrium-free autonomous ODE system is described. The system has seven terms, two quadratic nonlinearities, and only two parameters. Its Jacobian matrix everywhere has rank less than 4. It is hyperchaotic in some regions of parameter space, while in other regions it has an attracting torus that coexists with either a symmetric pair of strange attractors or with a symmetric pair of limit cycles whose basin boundaries have an intricate fractal structure. In other regions of parameter space, it has three coexisting limit cycles and Arnold tongues. Since there are no equilibria, all the attractors are hidden. This combination of features has not been previously reported in any other system, especially one as simple as this.
Article
This paper studies the dynamical behavior of the Newton-Leipnik system and its trajectory-transformation control problem to multiple attractors. A simple linear state feedback controller for the Newton-Leipnik system based on the Lyapunov stability theory and applying the inverse optimal control method is designed. We stabilize asymptotically the chaotic attractors to unstable equilibriums of the system, so that the transformation of one attractor to another for the trajectory of the Newton-Leipnik system is realized. Theoretical analyses and numerical simulations both indicate the effectiveness of the controller. At last, the inverse optimal control method is proven effective for the chaotic systems with multiple attractors by the example on the unified chaotic system.
Article
In recent years, tremendous research efforts have been devoted to simple chaotic oscillators based on jerk equation that involves a third-time derivative of a single variable. In the present paper, we perform a systematic analysis of a simple autonomous jerk system with cubic nonlinearity. The system is a linear transformation of Model MO5 first introduced in Sprott (Elegant chaos: algebraically simple flow. World Scientific Publishing, Singapore, 2010) prior to the more detailed study by Louodop et al. (Nonlinear Dyn 78:597–607, 2014). The basic dynamical properties of the model are investigated including equilibria and stability, phase portraits, frequency spectra, bifurcation diagrams, and Lyapunov exponent plots. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry-restoring crisis scenarios. One of the key contributions of this work is the finding of a window in the parameter space in which the jerk system experiences the unusual and striking feature of multiple attractors (e.g. coexistence of four disconnected periodic and chaotic attractors). Basins of attraction of various coexisting attractors are computed showing complex basin boundaries. Among the very few cases of lower-dimensional systems (e.g. Newton–Leipnik system) capable of displaying such type of behaviour reported to date, the jerk system with cubic nonlinearity considered in this work represents the simplest and the most ‘elegant’ prototype. An appropriate electronic circuit describing the jerk system is designed and used for the investigations. Results of theoretical analyses are perfectly traced by laboratory experimental measurements.
Article
Chaotic dynamical systems that are symmetric provide the possibility of multistability as well as an independent amplitude control parameter.The Rössler system is used as a candidate for demonstrating the symmetry construction since it is an asymmetric system with a single-scroll attractor. Through the design of symmetric Rössler systems, a symmetric pair of coexisting strange attractors are produced, along with the desired partial or total amplitude control.
Article
The dynamics and bifurcations of the Newton–Leipnik equations are presented. Numerical computations and local stability calculations 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 numerically computed dynamics and bifurcations of the Newton–Leipnik equations are compared with the dynamics and bifurcations of a family of odd, symmetric, bimodal maps to motivate the connection between the two systems.