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April 23, 2021 14:45 WSPC/S0218-1274 2130014
International Journal of Bifurcation and Chaos, Vol. 31, No. 5 (2021) 2130014 (17 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0218127421300147
Coexisting Infinite Equilibria and Chaos
Chunbiao Li∗,†,§, Yuxuan Peng†,‡,¶and Ze Tao†,‡,
∗
School of Artificial Intelligence,
Nanjing University of Information Science and Technology,
Nanjing 210044, P. R. China
†
Jiangsu Collaborative Innovation Center of Atmospheric
Environment and Equipment Technology (CICAEET),
Nanjing University of Information Science and Technology,
Nanjing 210044, P. R. China
‡
Jiangsu Key Laboratory of Meteorological Observation and
Information Processing, Nanjing University of Information
Science and Technology, Nanjing 210044, P. R. China
§
goontry@126.com
§
chunbiaolee@nuist.edu.cn
¶
alexpengcn@foxmail.com
lebrontaoze@163.com
Julien Clinton Sprott
Department of Physics,
University of Wisconsin−Madison,
Madison, WI 53706, USA
sprott@physics.wisc.edu
Sajad Jafari
Biomedical Engineering Faculty,
Amirkabir University of Technology,
424 Hafez Ave, 15875-4413, Tehran, Iran
sajadjafari83@gmail.com
Received March 18, 2020; Revised October 18, 2020
Equilibria are a class of attractors that host inherent stability in a dynamic system. Infinite
number of equilibria and chaos sometimes coexist in a system with some connections. Hidden
chaotic attractors exist independent of any equilibria rather than being excited by them. How-
ever, the equilibria can modify, distort, eliminate, or even instead coexist with the chaotic
attractor depending on the distance between the equilibria and chaotic attractor. In this paper,
chaotic systems with infinitely many equilibria are considered and explored. Extra surfaces of
equilibria are introduced into the chaotic flows, showing that a chaotic system can maintain its
basic dynamics if the newly added equilibria do not intersect the original attractor. The offset-
boostable plane of equilibria rescales the frequency of the chaotic oscillation with an almost
linearly modified largest Lyapunov exponent or conversely drives the system into periodic oscil-
lation, even ending in a divergent state. Furthermore, additional infinite number of equilibria
or even a solid space of equilibria are safely nested into the chaotic system without destroying
§Author for correspondence
2130014-1
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C. Li et al.
the original dynamics, which provides an alternate permanent location for a dynamical system.
A circuit simulation agrees with the numerical calculation.
Keywords: Chaotic attractor; infinite equilibria; offset boosting.
1. Introduction
Equilibrium points play an important role in non-
linear dynamical systems, which drive the system
into various states. However, it appears that chaotic
attractors can be independent of the equilibria in
the case of a hidden attractor [Leonov & Kuznetsov,
2013a; Leonov et al.,2011,2012,2015; Zhang &
Wang ,2019a]. Chaos exists in dynamical systems
with no equilibria [Zhou et al.,2018; Jafari &
Sprott,2013a; Maaita et al.,2015; Akgul et al.,
2016; Jafari et al.,2016a], with only stable equi-
libria [Molaie & Jafari,2013; Wang & Chen,2013;
Deng & Wang,2019; Li & Sprott,2013; Yang &
Chen,2008], with line equilibria [Ma et al.,2015; Li
et al.,2014; Jafari & Sprott,2013b; Li & Sprott,
2014; Li et al.,2015b], with planes of equilibria
[Ekmekci & Rockwell,2010; Bao et al.,2017; Jafari
et al.,2016b; Li & Sprott,2017; Macbeath,1965],
or even with any number of equilibria [Wang &
Chen,2013]. Even when unstable equilibria exist,
the initial condition in the neighborhood of the
equilibria may instead lead to an alternate third
chaotic state [Li et al.,2017a; Li et al.,2015a]. All
of the chaotic attractors identified in the above cases
are hidden attractors [Leonov & Kuznetsov,2013b;
Leonov et al.,2015; Zhang & Wang,2019a]. These
hidden chaotic attractors share a common feature
in that their basins of attraction do not cover or
cross the neighborhood of the existing equilibria.
Furthermore, chaos can coexist with infinitely many
equilibrium points when a periodic function is
introduced to construct infinitely many attrac-
tors based on initial-condition-based offset boosting
[Kuznetsov et al.,2013; Bao et al.,2016; Li et al.,
2018b; Lai & Chen,2016; Zhang & Wang,2019b].
Chaotic systems with surfaces of equilibria have
various mechanisms, some of which are induced by
dimension redundancy [Bao et al.,2017], a precon-
straint [Jafari et al.,2016b], or even time rescal-
ing [Li & Sprott,2017]. In fact, function-based time
rescaling produces a mixed effect of frequency mod-
ification and bifurcation control when the surfaces
of equilibria are introduced by a function. Chaotic
systems with surfaces of equilibria have other
possibilities for hosting infinitely many equilibrium
points, including other types of surfaces, lines of
equilibria, and even a solid space of equilibria if
they do not intersect or conflict with the previ-
ously existing attractors. In this paper, from the
newly found chaotic flows with surfaces of equilibria
[Jafari et al.,2016c], additional cases with infinitely
many equilibrium points are derived and explored
when new functions are introduced. In Sec. 2, the
derived chaotic systems with surfaces of equilibria
are listed with a basic dynamical analysis. In Sec. 3,
from a systematic bifurcation analysis, it is shown
that the newly introduced functions cause a time
rescaling, and the fundamental dynamics of those
cases with infinitely many equilibria depend on the
core system rather than the newly attached equilib-
ria. Electrical circuit simulation described in Sec. 4
confirms the findings from numerical simulation. A
discussion and conclusion are presented in the last
section.
2. Chaotic Systems with Infinite
Equilibria
Based on the newly found simplest chaotic flows
with surfaces of equilibria [Jafari et al.,2016c], addi-
tional surfaces are considered to introduce extra
equilibria, as shown in Table 1. Here, to employ
a compatible differential equation, the function fi0
represents the original flows proposed in the ref-
erence [Jafari et al.,2016c], while fij are the func-
tions added to construct other chaotic systems with
new surfaces of equilibria. The Lyapunov exponents
and Kaplan–Yorke dimensions are calculated from
initial conditions close to the attractor, which are
close to those of the original chaotic systems. In the
reference [Jafari et al.,2016c], an exhaustive com-
puter search was performed to seek elegant dissipa-
tive cases for which the largest Lyapunov exponent
is greater than 0.001, where the simplest candidates
for the surface f(x, y, z) are simple planes (a single
plane, two orthogonal planes or even three orthog-
onal planes) or other standard quadrics (ellipsoids,
hyperboloids, and paraboloids). In Table 1, we show
that different equilibria can be introduced into the
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Coexisting Infinite Equilibria and Chaos
Table 1. Chaotic systems with surfaces of equilibria.
Cases System Structure Introduced Functions Surfaces of Equilibria LEs DKY (x0,y
0,z
0)
ES1
˙x=fij ×(y)
˙y=fij ×(z)
˙z=fij ×(−x+ay2−xz)
f10 =x(ES1a) (0,y,z)
0.0071
0
−1.0864
2.0065
6
0
−1
a=1.54 f11 =1+x2−y2(ES1b) y2−x2=1
0.0394
0
−6.0477
2.0065
f12 =z+x2+y2(ES1c) x2+y2=−z
0.0427
0
−6.4770
2.0066
f13 =z+x2−y2(ES1d) y2−x2=z
0.0183
0
−2.8219
2.0065
ES2
˙x=fij ×(y)
˙y=fij ×(−x+az)
˙z=fij ×(by2−xz)
f20 =x(ES2a) (0,y,z)
0.0644
0
−0.8279
2.0778
0.15
0
0.8
a=1
b=3 f21 =z(ES2b) (x, y, 0)
0.0830
0
−1.0662
2.0778
f22 =xz (ES2c) (0,y,z)
(x, y, 0)
0.045417
0
−0.58463
2.0777
f23 =1+x2−y2(ES2d) y2−x2=1
0.16203
0
−0.2083
2.0778
f24 =z+x2+y2(ES2e) x2+y2=−z
0.18284
0
−2.3481
2.0779
f25 =z+x2−y2(ES2f) y2−x2=z
0.1398
0
−1.7931
2.0779
ES3
˙x=fij ×(y2+axy)
˙y=fij ×(−z)
˙z=fij ×(b+xy)
f30 =x(ES3a) (0,y,z)
0.0661
0
−1.664
2.0397
0.87
0.4
0
a=2
b=1
f31 =34.81 −x2−y2−z2
(ES3b) x2+y2+z2=34.81
2.3995
0
−60.4616
2.0397
f32 =30.25 −x2−y2
(ES3c) x2+y2=30.25
2.1569
0
−54.3899
2.0397
f33 =1+x2−0.3460y2
(ES3d) 0.3460y2−x2=1
0.0640
0
−1.6154
2.0397
(Continued)
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C. Li et al.
Tab le 1 . ( Continued)
Cases System Structure Introduced Functions Surfaces of Equilibria LEs DKY (x0,y
0,z
0)
ES4
˙x=fij ×(−y)
˙y=fij ×(x+z)
˙z=fij ×(ay2+xz −b)
f40 =z(ES4a) (x, y, 0)
0.0560
0
−1.0855
2.0516
0
0.46
0.7
a=2
b=0.35
f41 =z+x2+y2
(ES4b) x2+y2=−z
0.2330
0
−4.3048
2.0518
ES5
˙x=fij ×(−az)
˙y=fij ×(b+z2−xy)
˙z=fij ×(x2−xy)
f50 =xy (ES5a) (0,y,z)
(x, 0,z)
0.1242
0
−1.8356
2.0677
1
1.44
0
a=0.4
b=1 f51 =x(ES5b) (0,y,z)
0.0987
0
−1.4578
2.0677
f52 =y(ES5c) (x, 0,z)
0.2613
0
−3.8683
2.0675
f53 =z+x2+y2
(ES5d) x2+y2=−z
0.2613
0
−3.8683
2.0675
f54 =x2+y2+z2−1
(ES5e) x2+y2+z2=1
0.1819
0
−2.6959
2.0675
f55 =x2+y2−1
(ES5f) x2+y2=1
0.1642
0
−2.4219
2.0678
ES6
˙x=fij ×(y+ayz)
˙y=fij ×(bz +y2+cz2)
˙z=fij ×(x2−y2)
f60 =xyz (ES6a)
(0,y,z)
(x, 0,z)
(x, y, 0)
0.0294
0
−0.4051
2.0725
1
−1.3
−1
a=2
b=8
c=7
f61 =x(ES6b) (0,y,z)
0.1867
0
−2.5693
2.0727
f62 =−y(ES6c) (x, 0,z)
0.2327
0
−3.2026
2.0727
f63 =−z(ES6d) (x, y, 0)
0.0385
0
−0.5311
2.0725
f64 =−xy (ES6e) (0,y,z)
(x, 0,z)
0.2080
0
−2.8649
2.0726
f65 =z+x2+y2
(ES6f) x2+y2=−z
2.3535
0
−1.3994
2.0726
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Coexisting Infinite Equilibria and Chaos
Tab le 1 . ( Continued)
Cases System Structure Introduced Functions Surfaces of Equilibria LE s DKY (x0,y
0,z
0)
ES7
˙x=fij ×(ay)
˙y=fij ×(xz)
˙z=fij ×(−z−x2−byz)
f70 =1−x2−y2−z2
(ES7a) x2+y2+z2=1
0.0113
0
−0.9501
2.0119
0
0.1
0
a=0.4
b=6
f71 =1−x2−y2
(ES7b) x2+y2=1
0.0115
0
−0.9654
2.0119
f72 =1+x2−y2
(ES7c) y2−x2=1
0.0124
0
−1.0416
2.0119
ES8
˙x=fij ×(az +y2)
˙y=fij ×(−y+bx2)
˙z=fij ×(−xy)
f80 =1−x2−y2−z2
(ES8a) x2+y2+z2=1
0.0323
0
−0.9552
2.0338
0.24
0.2
0
a=1
b=5
f81 =1−x2−y2
(ES8b) x2+y2=1
0.0330
0
−0.9733
2.0339
f82 =1+x2−y2
(ES8c) y2−x2=1
0.0348
0
−1.0261
2.0339
ES9
˙x=fij ×(y2−axy)
˙y=fij ×(xz)
˙z=fij ×(1 −by2)
f90 =1−x2−y2
(ES9a) x2+y2=1
0.0388
0
−1.2078
2.0321
0.06
0
1
a=5
b=7
f91 =x
(ES9b) (0,y,z)
0.0028
0
−0.0875
2.0318
f92 =1+x2−y2
(ES9c) y2−x2=1
0.0399
0
−1.2485
2.0320
ES10
˙x=fij ×(a−z2)
˙y=fij ×(xz)
˙z=fij ×(y+bxz)
f10,0=1+x2−y2
(ES10a) y2−x2=1
0.0420
0
−0.2330
2.1883
0
−0.08
0
a=0.1
b=1
f10,1=1−x2−y2−z2
(ES10b) x2+y2+z2=1
0.0316
0
−0.1671
2.1891
f10,2=1−x2−y2
(ES10c) x2+y2=1
0.0364
0
−0.1938
2.1879
(Continued)
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C. Li et al.
Tab le 1 . ( Continued)
Cases System Structure Introduced Functions Surfaces of Equilibria LEs DKY (x0,y
0,z
0)
ES11
˙x=fij ×(yz)
˙y=fij ×(x−axz)
˙z=fij ×(x−bz2)
f11,0=z+x2+y2
(ES11a) x2+y2=−z
0.0283
0
−0.6171
2.0458
0.46
0
0.8
a=1
b=0.6
f11,1=4.84 −x2−y2−z2
(ES11b) x2+y2+z2=4.84
0.1033
0
−2.2565
2.0458
f11,2=4−x2−y2
(ES11c) x2+y2=4
0.1047
0
−2.2891
2.0457
f11,3=1+x2−0.25y2
(ES11d) 0.25y2−x2=1
0.0375
0
−0.8180
2.0458
ES12
˙x=fij ×(yz)
˙y=fij ×(−ax)
˙z=fij ×(−z+by2+xz)
f12,0=z+x2−y2
(ES11a) y2−x2=z
0.0068
0
−0.4998
2.0135
1
0
1
a=0.1
b=6
f12,1=1+x2−y2
(ES12b) y2−x2=1
0.0165
0
−1.2226
2.0135
f12,2=z+x2+y2
(ES12c) x2+y2=−z
0.0080
0
−0.5907
2.0136
same core structure, including simple planes, a
sphere, a circular cylinder, a hyperbolic cylinder,
a paraboloid, and a saddle surface. All of the intro-
duced equilibria exist outside the attractor without
influencing the basic dynamics. For systems ES3b,
ES3c, ES11b, and ES11c, the minimum radii are
5.9, 5.5, 2.2 and 2, respectively. The corresponding
typical attractors with their corresponding equi-
libria are solved by the fourth-order Runge–Kutta
method based on Matlab and are shown in Fig. 1.
The time step is 0.005, and the data is 64-bit float-
ing p oint number.
In fact, different equilibria can coexist in the
same system if they do not intersect the attractor.
For example, for the case of ES5, the newly intro-
duced functions can be:
f56 =x(z+x2+y2),
f57 =y(z+x2+y2),
f58 =x(1 + x2−0.3906y2),
f59 =y(1 + x2−0.3906y2),
f5,10 =(z+x2+y2)(1 + x2−0.3906y2),
f5,11 =xy(z+x2+y2),
f5,12 =xy(1 + x2−0.3906y2),
f5,13 =x(z+x2+y2)(1 + x2−0.3906y2),
f5,14 =y(z+x2+y2)(1 + x2−0.3906y2)and
f5,15 =xy(z+x2+y2)(1 + x2−0.3906y2);
correspondingly, the derived systems share more
than one surface of equilibria, as shown in Fig. 2.
Offset boosting [Li et al.,2019; Li et al.,2018a; Li
et al.,2017b] can be applied to the original system
to shift the attractor in case it touches the surface
of equilibria.
In an extreme scenario, the expanded equilibria
can form a solid space. Taking ES8 as an example;
except for the isolated original equilibrium point,
new functions can be introduced to embed infinitely
many surfaces of equilibria and even a solid space.
As shown in Table 2, the new function ϕ(x)can
be defined to introduce the equilibria of the solid
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Coexisting Infinite Equilibria and Chaos
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
Fig. 1. Typical phase trajectories of the cases in Table 1. (a) ES1b, (b) ES2f, (c) ES3c, (d) ES4b, (e) ES5d, (f ) ES6e, (g) ES7b,
(h) ES8c, (i) ES9c, (j) ES10b, (k) ES11d, and (l) ES12b.
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C. Li et al.
(a) (b)
(c) (d)
(e) (f)
Fig. 2. Coexisting surfaces of equilibria and chaos in the case of ES5: (a) f5,10,(b)f5,11,(c) f5,12 ,(d) f5,13,(e) f5,14 and
(f) f5,15 .
Table 2. Chaotic flows induced from ES8 with a solid space of equilibria.
Case System Structure Introduced Functions Surfaces of Equilibria (x0,y
0,z
0)
ES8 ˙x=fij ×(az +y2)
˙y=fij ×(−y+bx2)f83 =ϕ[−(x+0.6)] (ES8d) x≤−0.60.24
0.2
˙z=fij ×(−xy)f84 =ϕ[0.04 −(x+0.6)2−y2−(z+0.6)2]
(ES8e)
(x+0.6)2+y2+(z+0.6)2
≤0.04
0
a=1
b=5 f85 =ϕ[0.04 −(x+0.6)2−y2](ES8f) (x+0.6)2+y2≤0.04
f86 =ϕ[−(1 + x2−y2)] (ES8g) y2−x2≥1
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Coexisting Infinite Equilibria and Chaos
(a) (b)
(c) (d)
Fig. 3. Coexisting solid space of equilibria and chaos in the case of ES8: (a) ES8d, (b) ES8e, (c) ES8f and (d) ES8g.
space,
ϕ(x)=0,x≥0,
1,x<0.(1)
Therefore, a cubic, sphere, cylinder, or even a hyper-
boloid space of equilibria can be planted into the
body of chaotic systems, whose phase trajectories
and equilibria are shown in Fig. 3. Furthermore,
if fij = 1, all of the core systems share the same
shapes of the chaotic attractor.
Similarly, lines of equilibria can be introduced
in chaotic systems. Taking ES5 as an example,
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x=x×(−az)
˙y=y×(b+z2−xy)
˙z=y×(x2−xy)
(2)
(a) (b)
Fig. 4. Coexisting lines of equilibria and chaos in the derived system of ES5: (a) System (2) and (b) system (3).
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x=sgn[x(x+1)]×(−az)
˙y=sgn[y(y+1)]×(b+z2−xy)
˙z=sgn[y(y+1)]×(x2−xy).
(3)
The corresponding phase trajectories are shown in
Fig. 4. Here, in the above systems (2) and (3), the
newly introduced functions in three dimensions are
different, and usually the signum function is used
to remove the amplitude information [Li & Sprott,
2017]. Surfaces of equilibria can be introduced to
the chaotic flows with a line of equilibria [Jafari &
Sprott,2013b]. Taking LE1 as an example,
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x=gi×(y),
˙y=gi×(−x+yz),
˙z=gi×(−x−axy −bxz).
(4)
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C. Li et al.
(a) (b)
(c) (d)
Fig. 5. Coexisting line and surface of equilibria and chaos of system (4): (a) Surface x=1,(b)surfacex=±1, (c) surface
x2+y2= 1 and (d) surface y2−x2=1.
When gi=1−x, 1−x2,x
2+y2−1,1+x2−y2,
coexisting surfaces of equilibria are introduced in
the chaotic flow with a line of equilibria, as shown
in Fig. 5.
3. Frequency Control and
Dynamical Analysis
As conjectured, the derived flows with different
types of equilibria exhibit almost the same funda-
mental dynamical behavior. The introduced equi-
libria result from the unified functions in all dimen-
sions. Generally, for a dynamical system such as
˙
X=F(X),X=(x1,x
2,...,x
n)T,
F=(f1,f
2,...,f
n)T,
˙
X=(˙x1,˙x2,..., ˙xn)T
=dx1
dt ,dx2
dt ,..., dxn
dt T
,
a positive real constant cin the differential equa-
tion ˙
X=cF (X) only introduces a time scaling
since the transformation t→t
ccan restore the
equation to its original form. The function in the
equation ˙
X=g(X)F(X) produces a surface of
equilibria g(X) = 0 and simultaneously modifies
the frequency and revises the dynamics accordingly.
Taking system ES5 as an example, the projection of
the chaotic attractor is located in the first quadrant
in the x–yplane; therefore, we introduce a flexible
offset-boostable plane in the core structure ES5 as
follows:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x=(x−d)×(−az)
˙y=(x−d)×(b+z2−xy)
˙z=(x−d)×(x2−xy).
(5)
When the distance between the attractor and the
plane x=dvaries, the frequency of the chaotic
oscillation is nearly linearly rescaled, and finally, the
system ends in a limit cycle as shown in Fig. 6.
Another plane y=dproduces the same influence
as shown in Fig. 7 when it is introduced in the core
system ES5:
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙x=(y−d)×(−az),
˙y=(y−d)×(b+z2−xy),
˙z=(y−d)×(x2−xy).
(6)
The effect of the surface of equilibria can be clearly
identified by the offset din the function f=x−
d. When the offset dincreases from −2to0.8,
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Coexisting Infinite Equilibria and Chaos
Fig. 6. Dynamical evolution of system (5) under the plane of x=d: (a) Lyapunov exponents and (b) bifurcation diagram.
Fig. 7. Dynamical evolution of system (6) under the plane of y=d: (a) Lyapunov exponents and (b) bifurcation diagram.
systems (5) and (6) remain dominantly chaotic with
linearly rescaled Lyapunov exponents and finally
end in a periodic oscillation. A larger Lyapunov
exponent indicates a high frequency as shown in
Fig. 8. The effect of the frequency rescaling can be
enlarged by the power function. When f=x, x2,x
3,
the frequency is modified in the positive correla-
tion shown in Fig. 9. The offset in the z-dimension
Fig. 8. Phase trajectory of system (5) with a=0.4andb=1inthex−yplane: (a) d=−10 and (b) d=0.1.
Fig. 9. Frequency rescaled by power functions in system ES5 with a=0.4andb= 1: (a) Chaotic signal of x(t)and
(b) frequency spectrum of x(t).
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C. Li et al.
Fig. 10. Frequency rescaled by other offset-boosted planes in system ES5 with a=0.4andb= 1: (a) Chaotic signal of x(t)
and (b) frequency spectrum of x(t).
(a) (b)
Fig. 11. Lyapunov exponents and bifurcation diagram of chaotic system ES5a.
(a) (b)
(c) (d)
Fig. 12. Simultaneously appearing periodic oscillations in systems ES5b and ES5d.
2130014-12
April 23, 2021 14:45 WSPC/S0218-1274 2130014
Coexisting Infinite Equilibria and Chaos
can also produce surfaces of equilibria when f=
|z|+3,z+3,(z+3)
2, providing frequency rescaling
as shown in Fig. 10.
For the fixed surfaces of equilibria, taking ES5
as an example, all of the systems exhibit a simi-
lar period-doubling bifurcation as shown in Fig. 11.
Specifically, limit cycles appear almost in the same
periodic windows. Period-1, period-2, period-4, and
period-3 cycles appear simultaneously in systems
ES5b and ES5d as shown in Fig. 12. Furthermore,
it is found that many of these cases with surfaces of
equilibria exhibit the same fundamental dynamics
when the equilibria are removed.
4. Circuit Implementation
To verify the chaotic flows with surfaces of equilib-
ria, electrical circuit simulations were carried out as
follows. Taking system ES5d as an example, a cir-
cuit schematic is designed as shown in Fig. 13, and
the circuit equation is
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙x=f×R2
R4C1−1
R3
z,
˙y=f×R7
R9C21
R11
+1
R8
z2−1
R12
xy,
˙z=f×R14
R16C31
R15
x2−1
R18
xy,
f=R20 R1
R5R22
x2+R6
R10R21
y2+R13
R17R19
z.
(7)
Here, the parameters are set as R1=R3=R4=
R5=R6=R7=R8=R9=R10 =R11 =
R12 =R13 =R14 =R15 =R16 =R17 =R18 =
R19 =R20 =R21 =R22 =10kΩ,R2=4kΩand
C1=C2=C3= 100 nF. Small capacitor values
are selected for suitable time rescaling. The initial
values are 1 V, 1.44 V, and 0 V, respectively. All the
operational amplifiers are LM741H, which are pow-
ered with VCC =15VandVEE =−15 V. For the
Fig. 13. Circuit schematic of system ES5d.
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April 23, 2021 14:45 WSPC/S0218-1274 2130014
C. Li et al.
Fig. 14. Phase trajectory of system ES5d on the oscilloscope: (a) x–yplane and (b) y–zplane.
multiplier, the scaling parameters are A1=A2=
A3=A4=A5=1V/V. The newly introduced
equilibria become a “wall” of multipliers, which
introduces feedback into the core structure. In fact,
this wall of equilibria does not influence the attrac-
tor while providing a new parasitic position for the
system when the initial conditions are present. The
phase trajectory of system ES5d on the oscilloscope
isshowninFig.14.
To observe the influence of the distance between
the surface of equilibria and the attractor, an
offset-boostable plane is introduced in the core
system ES5. The corresponding circuit schematic
is designed as shown in Fig. 15, and the circuit
Fig. 15. Circuit schematic of system (5).
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Coexisting Infinite Equilibria and Chaos
Fig. 16. Phase trajectory of system (5) on the simulated oscilloscope in the x–yplane: (a) R19 =R20 =R22 =1kΩ,and
Vdd =−10 and (b) R19 =R20 =R22 = 100 Ω, and Vdd =0.1.
equation is
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙x=f×R2
R4C1−1
R3
z,
˙y=f×R7
R9C21
R11
+1
R8
z2−1
R12
xy,
˙z=f×R14
R16C31
R15
x2−1
R18
xy,
f=R20 R1
R5R22
x−R1
R5R19
Vdd.
(8)
Here, the parameters are set as R1=R3=R4=
R5=R6=R7=R8=R9=R10 =R11 =
R12 =R13 =R14 =R15 =R16 =R17 =
R18 =R19 =R22 =10kΩ,R
2=4kΩand
C1=C2=C3= 100 nF. Select R19 =R20 =R22 =
100 Ω and R19 =R20 =R22 =1kΩfordifferent
offset-boosting. As shown in Fig. 8, it is clear that
the frequency of the chaotic attractor of system (5)
is modified accordingly. A larger offset produces a
higher frequency as shown in Fig. 16.
5. Conclusion and Discussion
Chaotic systems with infinitely many equilibria are
discussed in this paper. Specifically, extra functions
are introduced in the differential equation in a uni-
fied or slightly different manner to construct lines,
surfaces, or even a solid space of equilibria. Follow-
ing this method, additional cases of chaotic sys-
tems with surfaces, lines or even a solid space of
equilibria are constructed based on the previously
found elegant flows. In fact, if the surface does not
intersect the existing attractor in the core struc-
ture, any desired equilibria of a particular type can
be introduced into the core structure. Note that for
considering dynamical systems generated by ODEs,
their limit objects (like attractors) and limit quan-
tities (like LEs and DKY ) should be investigated,
and one should study the existence and uniqueness
of ODE solutions and extensibility of these solu-
tions over time to positive infinity [Kuznetsov et al.,
2018; Leonov et al.,2015]. In this work many of
those induced systems are artificially constructed
with continuous yet nonsmooth functions in their
right-hand side (such as the function in Eq. (1), or
the absolute value function) and even discontinu-
ous signum functions, further investigation on Lip-
schitz condition or the boundedness of solutions is
suggested in the future for ensuring that the ODEs
indeed generate dynamical systems.
Offset-boosting makes the surface farther from
or closer to the attractor in the core system, pro-
viding time rescaling, which influences the fre-
quency without changing the basic dynamics.
Newly equipped surfaces of equilibria can be intro-
duced and superimposed, which provides an addi-
tional parameter to vary. A bifurcation analysis
shows that the fundamental dynamics are almost
independent of the newly introduced infinite equi-
libria in some circumstances. Dynamical maps
or basins of attraction analysis are suggested to
enhance this claim for further exploration. The cir-
cuit simulations agree with the numerical simula-
tion. The possible practical application of the devel-
oped systems is that in such systems those newly
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April 23, 2021 14:45 WSPC/S0218-1274 2130014
C. Li et al.
equipped surfaces of equilibria can be applied for
controlling the system to various stabilities.
Acknowledgments
This work was supported financially by the National
Natural Science Foundation of China (Grant
Nos. 61871230 and 61971228), the Natural Sci-
ence Foundation of Jiangsu Province (Grant No.
BK20181410), and a project funded by the Priority
Academic Program Development of Jiangsu Higher
Education Institutions (PAPD).
Conflict of Interest
The authors declare that they have no conflict of
interest.
References
Akgul, A., Calgan, H., Koyuncu, I., Pehlivan, I. & Istan-
bullu, A. [2016] “Chaos-based engineering applica-
tions with a 3D chaotic system without equilibrium
points,” Nonlin. Dyn. 84, 481–495.
Bao, B., Jiang, T., Xu, Q., Chen, M., Wu, H. & Hu, Y.
[2016] “Coexisting infinitely many attractors in active
band-pass filter-based memristive circuit,” Nonlin.
Dyn. 86, 1711–1723.
Bao, B., Jiang, T., Wang, G., Jin, P., Bao, H. & Chen,
M. [2017] “Two-memristor-based Chua’s hyperchaotic
circuit with plane equilibrium and its extreme multi-
stability,” Nonlin. Dyn. 89, 1–15.
Deng, Q. & Wang, C. [2019] “Multi-scroll hidden attrac-
tors with two stable equilibrium points,” Chaos 29,
093112.
Ekmekci, A. & Rockwell, D. [2010] “Effects of a geo-
metrical surface disturbance on flow past a circular
cylinder: A large-scale spanwise wire,” J. Fluid Mech.
665, 120–157.
Jafari, S. & Sprott, J. C. [2013a] “Elementary quadratic
chaotic flows with no equilibria,” Phys. Lett. A 377,
699–702.
Jafari, S. & Sprott, J. C. [2013b] “Simple chaotic flows
with a line equilibrium,” Chaos Solit. Fract. 57,
79–84.
Jafari, S., Pham, V. T. & Kapitaniak, T. [2016a] “Mul-
tiscroll chaotic sea obtained from a simple 3D system
without equilibrium,” Int. J. Bifurcation and Chaos
26, 1650031-1–7.
Jafari, S., Sprott, J. C. & Molaie, M. [2016b] “A sim-
ple chaotic flow with a plane of equilibria,” Int. J.
Bifurcation and Chaos 26, 1650098-1–6.
Jafari, S., Sprott, J. C., Pham, V. T., Volos, C. & Li,
C. [2016c] “Simple chaotic 3D flows with surfaces of
equilibria,” Nonlin. Dyn. 86, 1–10.
Kuznetsov, A. P., Kuznetsov, S. P., Mosekilde, E. &
Stankevich, N. V. [2013] “Generators of quasiperiodic
oscillations with three-dimensional phase space,” Eur.
Phys. J. Special Topics 222, 2391–2398.
Kuznetsov, N. V., Leonov, G. A., Mokaev, T. N., Prasad,
A. & Shrimali, M. D. [2018] “Finite-time Lyapunov
dimension and hidden attractor of the Rabinovich sys-
tem,” Nonlin. Dyn. 92, 267–285.
Lai, Q. & Chen, S. [2016] “Generating multiple chaotic
attractors from Sprott B system,” Int. J. Bifurcation
and Chaos 26, 1650177-1–13.
Leonov, G. A., Vagaitsev, V. I. & Kuznetsov, N. V.
[2011] “Localization of hidden Chua’s attractors,”
Phys. Lett. A 375, 2230.
Leonov, G. A., Vagaitsev, V. I. & Kuznetsov, N. V. [2012]
“Hidden attractor in smooth Chua systems,” Phys-
ica D 241, 1482.
Leonov, G. A. & Kuznetsov, N. V. [2013a] “Hidden
attractors in dynamical systems from hidden oscilla-
tions in Hilbert–Kolmogov, Aizerman, and Kalman
problems to hidden chaotic attractors in Chua cir-
cuits,” Int. J. Bifurcation and Chaos 23, 1330002-1–
69.
Leonov, G. A. & Kuznetsov, N. V. [2013b] Prediction of
Hidden Oscillations Existence in Nonlinear Dynam-
ical Systems:Analytics and Simulation (Springer
International Publishing).
Leonov, G. A., Kuznetsov, N. V. & Mokaev, T. N.
[2015] “Homoclinic orbits, and self-excited and hidden
attractors in a Lorenz-like system describing convec-
tive fluid motion,” Eur. Phys. J. Special Topics 224,
1421.
Li, C. & Sprott, J. C. [2013] “Multistability in a butterfly
flow,” Int. J. Bifurcation and Chaos 23, 1350199-1–
10.
Li, C. & Sprott, J. C. [2014] “Chaotic flows with a single
nonquadratic term,” Phys. Lett. A 378, 178–183.
Li, C., Sprott, J. C. & Thio, W. [2014] “Bistability in a
hyperchaotic system with a line equilibrium,” J. Exp.
Theor. Phys. 118, 494–500.
Li, C., Sprott, J. C. & Thio, W. [2015a] “Linearization
of the Lorenz system,” Phys. Lett. A 379, 888–893.
Li, C., Sprott, J. C., Yuan, Z. & Li, H. [2015b] “Con-
structing chaotic systems with total amplitude con-
trol,” Int. J. Bifurcation and Chaos 25, 1530025-
1–14.
Li, C. & Sprott, J. C. [2017] “How to bridge attrac-
tors and repellors,” Int. J. Bifurcation and Chaos 27,
1750149-1–11.
Li, C., Sprott, J. C. & Xing, H. [2017a] “Constructing
chaotic systems with conditional symmetry,” Nonlin.
Dyn. 87, 1351–1358.
Li, C., Wang, X. & Chen, G. [2017b] “Diagnosing multi-
stability by offset boosting,” Nonlin. Dyn. 90, 1335–
1341.
2130014-16
April 23, 2021 14:45 WSPC/S0218-1274 2130014
Coexisting Infinite Equilibria and Chaos
Li, C., Sprott, J. C., Liu, Y., Gu, Z. & Zhang, J. [2018a]
“Offset boosting for breeding conditional symmetry,”
Int. J. Bifurcation and Chaos 28, 1850163-1–13.
Li, C., Thio, J. C., Sprott, J. C., Iu, H. H. C. & Xu,
Y. [2018b] “Constructing infinitely many attractors
in a programmable chaotic circuit,” IEEE Access 6,
29003–29012.
Li, C., Lu, T., Chen, G. & Xing, H. [2019] “Doubling the
coexisting attractors,” Chaos 29, 051102.
Ma, J., Chen, Z., Wang, Z. & Zhang, Q. [2015] “A four-
wing hyper-chaotic attractor generated from a 4D
memristive system with a line equilibrium,” Nonlin.
Dyn. 81, 1275–1288.
Maaita, J., Volos, C. K., Kyprianidis, I. & Stouboulos,
I. [2015] “The dynamics of a cubic nonlinear system
with no equilibrium point,” J. Nonlin. Dyn. 2015,
257923.
Macbeath, A. M. [1965] “Geometrical realization of iso-
morphisms between plane groups,” Bull. Amer. Math.
Soc. 71, 629–630.
Molaie, M. & Jafari, S. [2013] “Simple chaotic flows with
one stable equilibrium,” Int. J. Bifurcation and Chaos
23, 1350188-1–7.
Wang, X. & Chen, G. [2013] “Constructing a chaotic
system with any number of equilibria,” Nonlin. Dyn.
71, 429–436.
Yang, Q. & Chen, G. [2008] “A chaotic system with one
saddle and two stable node foci,” Int. J. Bifurcation
and Chaos 18, 1393–1414.
Zhang, X. & Wang, C. [2019a] “Multiscroll hyper-
chaotic system with hidden attractors and its circuit
implementation,” Int. J. Bifurcation and Chaos 29,
1950117-1–14.
Zhang, X. & Wang, C. [2019b] “A novel multi-attractor
period multi-scroll chaotic integrated circuit based
on CMOS wide adjustable CCCII,” IEEE Access 7,
16336–16350.
Zhou, L., Wang, C. & Zhou, L. [2018] “A novel no-
equilibrium hyperchaotic multi-wing system via intro-
ducing memristor,” Int. J. Circ. Th. Appl. 46, 84–98.
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