Chaos, Solitons and Fractals 134 (2020) 109703

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Chaos, Solitons and Fractals

Nonlinear Science, and Nonequilibrium and Complex Phenomena
journal homepage: www.elsevier.com/locate/chaos

Frontiers

A new megastable nonlinear oscillator with infinite attractors

Gervais Dolvis Leutcho a,∗, Sajad Jafari b, Ibrahim Ismael Hamarash c,g, Jacques Kengne d,
Zeric Tabekoueng Njitacke e, Iqtadar Hussain f
a
Research Unit of Laboratory of Condensed Matter, Electronics and Signal Processing (UR-MACETS) Department of Physics, Faculty of Sciences, University of
Dschang, P.O. Box 67, Dschang, Cameroon
b
Department of Biomedical Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875-4413, Iran
c
Department of Computer Science and Engineering, University of Kurdistan-Hewler, Iraq
d
Research Unit of Laboratory of Automation and Applied Computer (LAIA), Electrical Engineering Department of IUT-FV, University of Dschang, P.O. Box 134,
Bandjoun, Cameroon
e
Department of Electrical and Electronic Engineering, College of Technology (COT), University of Buea, P.O. Box 63, Buea, Cameroon
f
Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar
g
Department of Electrical Engineering, Salahaddin University-Erbil, Kurdistan, Iraq

a r t i c l e i n f o a b s t r a c t

Article history: Dynamical systems with megastable properties are very rare in the literature. In this paper, we introduce
Received 4 November 2019 a new two-dimensional megastable dynamical system with a line of equilibria, having an infinite num-
Revised 8 February 2020
ber of stable states. By modifying this new system with temporally-periodic forcing term, a new two-
Accepted 17 February 2020
dimensional non-autonomous nonlinear oscillator capable to generate an infinite number of coexisting
limit cycle attractors, torus attractors and, strange attractors is constructed. The analog implementation
Keywords: of the new megastable oscillator is investigated to further support numerical analyses and henceforth
Forced oscillator validate the mathematical model.
Megastability
Self-excited attractors © 2020 Elsevier Ltd. All rights reserved.
Coexisting attractors

1. Introduction multistable when there are more than two coexisting attractors in
it [29,30]. Bistability represents the situation of two different coex-
Chaotic/hyperchaotic nonlinear dynamical systems have one of isting attractors. In this regard, many other works have been inves-
their main properties to be extreme sensitivity to noise or sensi- tigated in the literature on this interesting topic in the last decade
tive dependence to initial conditions [1–5]. Nonlinear dynamical [31–35]. Indeed, multistability is a very hot and ongoing research
systems have been recently classified into two major groups. That topic. In this scope, there are special cases of multistability which
is self-excited attractors systems and systems with hidden attrac- have attracted much attention in recent years. That is systems with
tors [6–11]. Self-excited attractors have a basin of attraction that extreme multistability (i.e., there is an infinite number of coexist-
is related to an unstable fixed point [3]. In contrast, when the ing attractors, and thus initial states cause bifurcations) [36].
delimitation regions don’t intersect with the small neighborhood Nonlinear oscillators with megastability are another type of ex-
of any equilibrium point [12], then that attractor is hidden. These citing multistable systems found recently in literature [37–41]. This
groups of attractors have been investigated in literature in different latter case has an infinite number of coexisting attractors, but there
systems including Hopfield neural networks [13,14], Chua’s circuit are no such bifurcations in them like systems with extreme multi-
[15], Sprott system [16], jerk [17,18], hyperjerk systems [19–23] and stability [37]. Indeed, the term megastable has been used for sys-
fractional-order chaotic systems [24–27]. In terms of their complex tems with countable infinite attractors [42–46], in contrast to ex-
and extremely rich dynamic, these chaotic systems can generate treme multistability which is related to non-countable infinite at-
different types of stable states for a specific parameter set when tractors [47]. From the point of view of applications, the coexis-
using random initial conditions. This latter characteristic of nonlin- tence of different stable states offers great flexibility in the sys-
ear dynamical systems is called multistability. Recall that multista- tem performance without major parameter changes; that can be
bility is very important in dynamical systems [1,28]. A system is exploited with the right control strategies to induce a definite
switching between different coexisting states [48–52].
Megastable property of the nonlinear dynamical system has
∗
Corresponding author. been investigated in some few cases very recently. In 2017 Sprott
E-mail address: leutchoeinstein@yahoo.com (G.D. Leutcho).

https://doi.org/10.1016/j.chaos.2020.109703
0960-0779/© 2020 Elsevier Ltd. All rights reserved.
2 G.D. Leutcho, S. Jafari and I.I. Hamarash et al. / Chaos, Solitons and Fractals 134 (2020) 109703

Fig. 4. A zoomed version of Fig. 3 showing the structure of strange attractor with
Fig. 1. Trajectories showing different coexisting limit cycles of system (2), for 75
omitted transition parts.
different initial conditions (located on a mesh x, y = −150 to x, y = +150 with steps
equal to 4). Each trajectory is for a duration of 20 0 0 sec.

den dynamics. The megastability exhibited by the model was high-

lighted using nonlinear analysis tools such as bifurcation diagrams,
graph of Lyapunov exponents and phase portraits plots. Tang et al.,
2018 [43] have reported several coexisting attractors with a struc-
ture similar to cabbage exhibiting the striking property of megasta-
bility. Besides, Wei and colleagues [46] have addressed the modifi-
cation of two well-known 2D oscillators and obtain a novel inter-
esting oscillator with the multistable property. By transforming the
obtained system by its forced version which can display very in-
teresting features of coexistence of attractors and fascinating phe-
nomenon characterized by the fact that some initial conditions can
escape from the gravity of nearby attractors and travel far away
before being trapped in an attractor beyond the usual access. Very
recently, a new dissipative and conservative chaotic system with
megastability is reported [53].
Fig. 2. Trajectories showing different coexisting torus attractors, limit cycles, and
strange attractors of System (5) for A = 0.55, with 75 different initial conditions
From these few works found in the literature about megastable
(located on a mesh x, y = −150to x, y = +150 with steps equal to 4). Each trajectory systems, one main motivation in this work is to enrich the lit-
is for a duration of 20 0 0 sec. erature of such types of systems by introducing a novel model
megastable oscillator with both mathematic and electronic model.
In the next section, the steps of our design are described; the new
chaotic oscillator is introduced. In Section 3 its dynamical proper-
ties are presented via state portrait of trajectories, time series, bi-
furcation diagram, Lyapunov exponents, bifurcation like sequence
and two-parameter diagrams. The discussion of the obtained re-
sults is also addressed. Circuit implementations of the model, as
well as PSpice simulations, are provided in Section 4. Finally, con-
cluding remarks are given in Section 5.

2. The new oscillator

Consider the following two-dimensional (2-D) nonlinear oscil-

lator which is investigated in [54],

Fig. 3. A zoomed version of Fig. 2 with omitted transition parts. x˙ = y
(1)
y˙ = −x + y cos (x )
and collaborators [42] describe the dynamics of a chaotic system
similar to the forced van der Pol oscillator with spatially-periodic Inspired by System (1), we introduce the following two-
damping. The authors found that the investigated system has an dimensional new nonlinear oscillator:
infinite number of nested coexisting hidden attractors of different 
types including limit cycles, tori, and strange attractors. In addi- x˙ = −y
(2)
tion, the authors found that the multiple coexisting attractors were y˙ = k1 sin (ω1 x ) + k2 sin (ω2 x ) + y cos (x )
grouped in a layered structure similar to cabbage hence the name
megastability. Wang et al., 2018 [45] introduced another 2D nonlin- System (2) has an infinite equilibrium points. Note that the triv-
ear dynamical system which can display both self-existed and hid- ial fixed point is located at the origin O(0, 0). The Jacobian matrix
 G.D. Leutcho, S. Jafari and I.I. Hamarash et al. / Chaos, Solitons and Fractals 134 (2020) 109703 3

Fig. 5. (a) Bifurcation diagrams of forced system with respect to forcing amplitude A and (b) corresponding maximum values of y versus forcing amplitude A. The initial
conditions are (1, 0) and (10, 0).

for this system in its steady point is an infinite number of stable states around x-axis, which could be
  identified under other appropriate initial points.
0 −1
J=
k1 ω1 cos (ω1 x ) + k2 ω2 cos (ω2 x ) − y sin (x ) cos (x )
  (3)
0 −1
(x, y ) = (0, 0) k ω + k ω 1
3. The forced chaotic oscillator
−−−−−−−−−−→ 1 1 2 2

Like in the van der Pol equation, a temporally-periodic forcing

So, the eigenvalues can be calculated accordingly as: term (Asin (ωt)) can be added to the system (2) and therefore, we
  obtain the following equations describe the forced version of the
λ −1 system (2).
|λI − J| = 0 → k ω + k ω λ−1
=0
1 1 2 2
(4) 
→ λ2 − λ √ + k1 ω1 + k2 ω2 = 0 → x˙ = −y
−(1−4(k1 ω1 +k2 ω2 ) ) (5)
λ1,2 = 1± j
y˙ = k1 sin (ω1 x ) + k2 sin (ω2 x ) + y cos (x ) + A sin (ωt )
2

Provide that the characteristic equation has coefficient with a With the forcing term introduced in system (5), this latter sys-
different sign [55–57], the equilibrium point O(0, 0) is unstable. For tem can display infinite coexisting chaotic trajectories for an ap-
our simulation, we have selected the following system parameters propriate choice of system parameters and initial conditions as
k1 = 1, k2 = 2, ω1 = √1 , and ω2 = 0.05. Some exciting trajectories well. In this regard, we have selected ω = 0.7and take some dis-
2
in this system are depicted in Fig. 1. This graph represents a phase tinct value of parameter A in the chaotic region in order to re-
portrait in System (2) for 75 distinct initial conditions located on veal these interesting features. In particular, For A = 0.55, system
the x-axis. We can remark that 30 stable states are identified (by (5) can effectively display an infinite number of coexisting chaotic
using those 75 initial conditions). That is, 3 packs, each with 10 and periodic stable states as presented in Fig. 2. These state space
limit cycles are uncovered. Note that, they are just examples in plots are obtained with the same initial conditions employed in
4 G.D. Leutcho, S. Jafari and I.I. Hamarash et al. / Chaos, Solitons and Fractals 134 (2020) 109703

Fig. 6. (a) Phase portrait showing the coexistence of eight stable states with their corresponding time series (b–c) using initial conditions (5, 0), (15, 0), (25, 0), (50, 0), (90,
0), (105, 0), (115, 0), (70, 0) for black, green, cyan, red, yellow, magenta and violet attractor. (d) Represents bifurcation like sequence showing local maxima of the coordinate
y versus x(0) for A = 0.3 and ω = 0.7. (For interpretation of references to color in this figure legend, the reader is referred to the web version of this article)

Fig. 1. One can remark in Fig. 2 that some of the limit cycles which torus attractors can be observed. the results of the investigation of
existed in Fig. 1, have lost their own delimitation regions. Indeed, the new megastable system are also provided in Fig. 7 through the
we have presented in Fig. 3 and Fig. 4, the main pack of coexist- two-parameter Lyapunov exponent diagrams in the same range of
ing solutions around the origin. That is a zoom version of trajecto- parameter space (A, k2 ). These diagrams are very important to bet-
ries shown in Fig. 2. The situation depicted in Fig. 4 is typically for ter localize the regions of multiple coexisting solutions in the sys-
chaotic trajectories. To further highlight these coexisting solutions tem. From the graphs in Fig. 7, it can be seen that for two different
for values of the control parameter A, we provide the bifurcation initial conditions, the coexisting solutions regions are materialized
of Fig. 5(a and b) obtained when monitoring this parameter (i.e., by the simultaneous appearance of different colors on figures in
forcing amplitude) for two distinct initial conditions in the range the same range of parameter space.
A ∈ [0, 0.6].
As shown in Fig. 5, the bifurcation points differ in some range
4. The circuit implementation
of the control parameter for these initial conditions. To analyze the
new megastable system, we also calculate the maximum Lyapunov
The analog implementation of the new megastable oscillator in-
exponents (MLEs) related to the bifurcation diagrams (see Fig. 5(c
vestigated in this work is addressed in this section. The introduced
and d)) using Wolf algorithm [58]. It is easy to observe that on the
two-dimensional system is constructed using electronic component
graph of the Lyapunov exponent diagram, the dynamical behavior
of the Pspice environment [59,60]. This electronic implementation
of the new megastable system alternates between torus, limit cy-
of the model is very singular because the model is build based
cles, and chaotic attractors when A is increased. One can remark
only on linear and trigonometric terms. The circuit of the new
that those regions are torus. They may look like chaos in the bi-
megastable oscillator is designed and shown in Fig. 8. The circuit of
furcation diagram, in the state-space plot, or even in time-series.
the new system is designed using two capacitors (C1 , C2 ), thirteen
However, the largest LE in them is zero (they have two zero LEs
resistors (R1 , ..., R13 ) including, six op-amps TL082CD, one multi-
and one negative LE). Fig. 6 shows some very interesting features
plier which can be implemented practically using AD633JN ver-
in this system. It is a plot of trajectories in the system (5), for
sions of the AD633 four-quadrant voltage multipliers chips used to
8 different initial conditions. The coexistence of limit cycles with
implement the nonlinear terms of our model. The signal (W) at the
 G.D. Leutcho, S. Jafari and I.I. Hamarash et al. / Chaos, Solitons and Fractals 134 (2020) 109703 5

output is related to those at inputs X1 (+ ), X2 (− ), Y1 (+ ), Y2 (− ), and

Z(W = (X1 − X2 )(Y1 − Y2 )/10 + Z ). And a symmetric power supply.
The circuit equation using Kirchhoff’s electrical circuit laws can be
obtained as:

C1 dX = 1
Y
dt R1
 R3   R4 
C2 dY
dt
= 1
R11
sin R2
X + 1
R10
sin R5
X + 1
10R12
Y cos (X ) + Vmax
R13
sin (W t )
(6)
setting C1 = C2 = C = 10nF ,Vmax = 15V ,R = Ri = 10k except R2 , R5 ,
R10 , R12 , and R13 ; adopting the rescale of time t = τ RCand vari-
ables: X = 1V × x, Y = 1V × y.
System (6) is same to the introduced system (5) with the fol-
lowing expression of parameters:
R R
R11 = KR = 10k, R2 = W3 = 14.14k, R5 = W4 = 200k,
1 1 2
R10 = R
K2 = 5k, R12 = R
10 = 1k, R13 = Vmax R
A
= 272.73k, and
W = ω0 ω = 70 0 0 with ω0 = RC 1

The dynamics of the model obtained in Fig. 9 is simulated in

PSpice with the values parameters given in the caption of the same
Figure. This obtained result agrees with the one obtained based on
Pascal simulation of Fig. 2.

5. Conclusion

Constructing a chaotic system capable to exhibit some particu-

lar behavior has received intensive attention recently. In this pa-
per, a detailed analysis of a new two-dimensional megastable os-
cillator is presented. It is found that the introduced model pos-
sesses a line of equilibria including the origin whose stability is
addressed. We demonstrate that it has an infinite number of co-
existing stable states for specific parameters. Besides, the analog
implementation of the new megastable oscillator is investigated to
further confirm the performed numerical investigations and hence-
forth validate the mathematical model.

Fig. 7. Coexisting of two-parameters sweep diagrams plotted in the same range of

parameters with just two different initial conditions. The coexisting solutions re-
gions are synthesized by the simultaneous appearance of different colors on figures
for the same range of parameters. The rest of system parameters are fixed as in the
text and the initial conditions are (5, 0) and (0.1, 0).

Fig. 8. The circuit schematic of the novel oscillator with megastable behavior.
6 G.D. Leutcho, S. Jafari and I.I. Hamarash et al. / Chaos, Solitons and Fractals 134 (2020) 109703

Fig. 9. Two dimensional projection of the attractor obtained from circuit realization of the new chaotic circuit and their corresponding time series for three different initial
conditions(5, 0)V, (10, 0)Vand (15, 0)V respectively.

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