# Study on Amplitude Modulation Principle of Chaotic System

Authors: Jian-Rong Du; Chun-Lai Li; Kun Qian; Zhao-Yu Li; Wen Li
Journal: Complexity
 (2020)

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## Abstract

Exploring the amplitude modulation phenomenon of chaotic signal has become a subject of great concern in recent years. This paper mainly concentrates on the preliminary study on amplitude modulation principle of a chaotic system. First, two 3D chaotic systems with quadratic product terms are introduced for studying the amplitude modulation phenomenon of chaotic signal. It is found that the signal amplitude of the first system can be controlled by partial quadratic coefficient. But for the second system, none of nonlinear coefficient can be employed to control the signal amplitude. Then, the amplitude modulation principle of chaotic system is preliminarily studied by exploring the intrinsic relationship between nonzero equilibrium point and phase space trajectory, and it is further validated by introducing unified parameter to the two 3D chaotic systems. As a necessary condition, the principle provides a feasible and simple method for constructing and analyzing an amplitude modulation chaotic system.


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## Body

## 1. Introduction

Chaos has evoked much attention in many scientific fields due to its unique characteristics, such as sensitivity to initial conditions and parameter deviations, strange attractor with locally unbounded but globally bounded trajectory, unpredictability of future behavior, and so on [1–4]. In the past few decades, the issue of construction, analysis, and application of chaotic systems has become a very active topic [5–10].The signal amplitude of a chaotic system can be often modulated by controlling one or more coefficients in the dynamical equations, while the Lyapunov exponents and power spectral density remain invariable [11–14]. Since the variables can be rescaled by directly controlling the amplitude parameter to avoid the exceeding limitations of bandwidth and amplitude, this kind of system can resolve the contradiction of signal processing and information retention in practical amplification circuit, so it is a promising type of system for the reality of chaotic encryption, chaotic radar, chaotic communication, and chaotic signal processing.Generally, for the presented amplitude modulation systems with quadratic nonlinearity, the amplitude parameters are the coefficients of quadratic terms, which can nonlinearly modulate the signal amplitude of partial or total state variables [11–15]. In fact, while chaotic system holds the only nonlinear term, the corresponding coefficient can control the signal amplitude since it uniquely determines the scale of the variables [16–18]. As an example, a simple chaotic system with a single nonlinearity x13 is recalled as x˙1=x2, x˙2=x3, x˙3=bx1−x2−ax3−bx13 [16]. Accordingly, it holds the resulting system x˙1=x2, x˙2=x3, x˙3=bx1−x2−ax3−bpx13 with the substitution x1⟶px1, x2⟶px2, x3⟶px3. Thereby, the coefficient of the nonlinear term x13 can control the amplitude of x1,x2,x3 according to 1/p. However, it is found in the recent work that for the dynamical system with exponential nonlinearity, the coefficient of quadratic term cannot provide amplitude modulation [19]. Thus, naturally, an interesting question is raised as “whether each coefficient of nonlinear term in chaotic system can control the signal amplitude.” Another more important question is “what is the principle of constructing and analyzing amplitude modulation chaotic system.” However, as far as we know, there is little information about this kind of system in the literature so far, so it still remains open and challenging.This paper attempts to provide some answers to these questions by introducing two carefully screened chaotic systems. Differing from the system with a single linear or a single nonlinear term, the present systems have nonunique linear terms and three quadratic cross-product terms. Basic dynamics of these two nonlinear systems are analyzed theoretically and numerically. Somewhat surprisingly, only one coefficient of the quadratic nonlinearity in the first system can be employed to control the amplitude of chaotic signal. But for the second system, none of the coefficient of quadratic nonlinearity can be employed to control the amplitude of chaotic signal. The discovery that not all coefficients of nonlinear term can provide amplitude control is of interest and inspiration. As a further concern in this work, the amplitude modulation principle of chaotic system is addressed, based on the analysis of the intrinsic relationship between nonzero equilibrium point and phase space trajectory. Although it is not a sufficient and necessary condition for amplitude modulation, the proposed principle provides a feasible method for constructing and analyzing amplitude modulation chaotic system. Furthermore, this method is simple in actual operation and will hopefully enlighten for revealing the amplitude modulation mechanism of chaotic system.This paper is organized as follows. Following the introduction, we propose a chaotic system with partial coefficient of nonlinearity employed to control amplitude. In Section3, we introduce another chaotic system with no coefficient of nonlinearity employed to control amplitude. The principle of amplitude modulation is addressed in Section 4. Finally, some concluding remarks are drawn in Section 5.


## 2. Chaotic System with Partial Coefficient of Nonlinearity for Amplitude Control

### 2.1. System Description

The reported system possesses four linear terms and three quadratic cross-product terms, which is given by the following ordinary differential equations:(1)x˙1=ax2−x1+dx2x3,x˙2=bx2−ex1x3,x˙3=−cx3+fx1x2.It is easy to know that the proposed system is symmetric with respect to thex3-axis, as shown by the coordinate transformation (x1, x2, x3) ⟶ (−x1, −x2, x3).By considering the equilibrium conditionax2−x1+dx2x3=0, bx2−ex1x3=0, −cx3+fx1x2=0, five equilibrium points of system (1) are determined as(2)P00,0,0,P1bcef,−ace+cea2+4abd/e2dbf,−a+a2+4abd/e2d,P2bcef,−ace−cea2+4abd/e2dbf,−a−a2+4abd/e2d,P3−bcef,ace+cea2+4abd/e2dbf,−a+a2+4abd/e2dbf,P4−bcef,ace−cea2+4abd/e2dbf,−a−a2+4abd/e2dbf.And the characteristic equation is deduced as(3)φλ=−λ3+−a+b−cλ2+ab+bc−ac+dfx22−aex3−dex32−efx12λ+abc−aefx1x2−bdfx22−acex3−cdex32−aefx12.When selecting the parameter seta = 28, b = 20, c = 3, d = 1, e = 5, and f = 1, the four nonzero equilibrium points are calculated as P1 (3.4641, 3.0743, 3.5499), P2 (3.4641, −27.3230, −31.5499), P3 (−3.4641, 27.3230, −31.5499), and P4 (−3.4641, −3.0743, 3.5499). And the corresponding characteristic roots are(4)P0:λ1=−28,λ2=20,λ3=−3,P1,4:λ1=−18.1945,λ2=3.5973+13.8748i,λ3=3.5973−13.8748i,P2,3:λ1=−43.6337,λ2=16.3169+22.2518i,λ3=16.3169−22.2518i.Obviously, equilibrium pointP0 is a saddle-node with two-dimensional stable manifold and one-dimensional unstable manifold. But for the remaining nonzero equilibrium points P1 to P4, λ1 is a negative real number and λ2 and λ3 become a pair of complex conjugate roots with positive real parts. Accordingly, the four equilibrium points are saddle-focus points with two-dimensional unstable manifold and one-dimensional stable manifold.When selectinga = 28, b = 20, c = 3, d = 1, e = 5, and f = 1 and computing time 5000 s, the three finite time Lyapunov exponents of system (1) are calculated by orthogonal method as 3.29613 > 0, 0.01355, −19.34679 < 0. And the Kaplan–Yorke dimension is obtained as DKY = 2 + (3.29613 +0.01355)/19.34679 = 2.1711, revealing a fractional feature. Therefore, system (1) is chaotic. The corresponding chaotic phase diagrams and Poincare mapping on plane x2 = 0 are depicted in Figure 1.Figure 1
(a)x1-x2-x3 phase portrait; (b) x1-x3 phase portrait; (c) Poincare mapping.
(a)
(b)
(c)


### 2.2. Analysis of Amplitude Modulation

Theorem 1.
The parameterf in cross-product term x1x2 is a local parameter of nonlinear amplitude modulation, which can control the signal amplitude of x1,x2 by the power function of index −1/2 respectively, but the amplitude of variable x3 remains in the same range; besides, the Lyapunov exponent spectrum remains unchanged with the variation of parameter f.Proof.
Considering the variable substitutionx1=u1/h,x2=u2/h,x3=u3 (h>0), system (1) is turned to(5)u˙1=au2−u1+du2u3,u˙2=bu2−eu1u3,u˙3=−cu3+f/hu1u2.
Therefore, when parameterf increases linearly, the signal amplitude of system variables x1,x2 change according to the power function of index −1/2, respectively, but the amplitude of variable x3 is in the same range.
When substituting the equilibrium pointP0 into characteristic equation (3), it holds(6)φλ=−λ3+−a+b−cλ2+ab+bc−acλ+abc.
In equation (6), the influence of parameter f is eliminated. We can draw a similar conclusion for the other equilibrium points. As an illustration, we insert the equilibrium P1 into expression (3) obtaining(7)φλ=−λ3+−a+b−cλ2+ab+bc−ac+ce2a2+4abd/e−2aa2+4abd/e4bd−ae−a+a2+4abd/e2d−2a2e+4abd−2aea2+4abd/e4d−bcλ+abc−aecbe−a+a2+4abd/e2dbe−ce2a2+4abd/e−2aa2+4abd/e4d−ace−a+a2+4abd/e2d−cde2a2+4abd/e−2aa2+4abd/e4d2−abc.
That is, parameterf does not produce effect on the characteristic root of equation (7). Therefore, when parameter f varies in field of real number, the Lyapunov exponent spectrum remains constant. This completes the proof.
The corresponding bifurcation diagram and Lyapunov exponent spectrum versusf are shown in Figure 2, which authenticates the theoretical results.
It is generally accepted that for the quadratic chaotic systems, the coefficients of nonlinear terms can modulate the signal amplitude of partial or total state variables [11–14]. The bifurcation diagram for e∈0,20 is shown in Figure 3(a). Superficially, the coefficient e can modulate the signal amplitude nonlinearly. But from the enlarged view, one can see that there emerges a visible periodic window, and the Lyapunov exponent spectrum further verifies the observation, as depicted in Figures 3(b) and 3(c). The concrete bifurcation diagram and Lyapunov exponent spectrum versus d also show that not all coefficients of quadratic terms can modulate the signal amplitude, which is illustrated in Figure 4. In spite of this, the chaos of the reported system is still robust in a large range of parameters d and e. Consequently, the system can be recommended as an important candidate in secure communication.Figure 2
(a, b, c) Bifurcation diagram versusf; (d) finite time Lyapunov exponent spectrum versus f.
(a)
(b)
(c)
(d)Figure 3
(a, b) Bifurcation diagram versuse; (c) finite time Lyapunov exponent spectrum versus e.
(a)
(b)
(c)Figure 4
(a, b) Bifurcation diagram versusd; (c) finite time Lyapunov exponent spectrum versus d.
(a)
(b)
(c)


## 2.1. System Description

The reported system possesses four linear terms and three quadratic cross-product terms, which is given by the following ordinary differential equations:(1)x˙1=ax2−x1+dx2x3,x˙2=bx2−ex1x3,x˙3=−cx3+fx1x2.It is easy to know that the proposed system is symmetric with respect to thex3-axis, as shown by the coordinate transformation (x1, x2, x3) ⟶ (−x1, −x2, x3).By considering the equilibrium conditionax2−x1+dx2x3=0, bx2−ex1x3=0, −cx3+fx1x2=0, five equilibrium points of system (1) are determined as(2)P00,0,0,P1bcef,−ace+cea2+4abd/e2dbf,−a+a2+4abd/e2d,P2bcef,−ace−cea2+4abd/e2dbf,−a−a2+4abd/e2d,P3−bcef,ace+cea2+4abd/e2dbf,−a+a2+4abd/e2dbf,P4−bcef,ace−cea2+4abd/e2dbf,−a−a2+4abd/e2dbf.And the characteristic equation is deduced as(3)φλ=−λ3+−a+b−cλ2+ab+bc−ac+dfx22−aex3−dex32−efx12λ+abc−aefx1x2−bdfx22−acex3−cdex32−aefx12.When selecting the parameter seta = 28, b = 20, c = 3, d = 1, e = 5, and f = 1, the four nonzero equilibrium points are calculated as P1 (3.4641, 3.0743, 3.5499), P2 (3.4641, −27.3230, −31.5499), P3 (−3.4641, 27.3230, −31.5499), and P4 (−3.4641, −3.0743, 3.5499). And the corresponding characteristic roots are(4)P0:λ1=−28,λ2=20,λ3=−3,P1,4:λ1=−18.1945,λ2=3.5973+13.8748i,λ3=3.5973−13.8748i,P2,3:λ1=−43.6337,λ2=16.3169+22.2518i,λ3=16.3169−22.2518i.Obviously, equilibrium pointP0 is a saddle-node with two-dimensional stable manifold and one-dimensional unstable manifold. But for the remaining nonzero equilibrium points P1 to P4, λ1 is a negative real number and λ2 and λ3 become a pair of complex conjugate roots with positive real parts. Accordingly, the four equilibrium points are saddle-focus points with two-dimensional unstable manifold and one-dimensional stable manifold.When selectinga = 28, b = 20, c = 3, d = 1, e = 5, and f = 1 and computing time 5000 s, the three finite time Lyapunov exponents of system (1) are calculated by orthogonal method as 3.29613 > 0, 0.01355, −19.34679 < 0. And the Kaplan–Yorke dimension is obtained as DKY = 2 + (3.29613 +0.01355)/19.34679 = 2.1711, revealing a fractional feature. Therefore, system (1) is chaotic. The corresponding chaotic phase diagrams and Poincare mapping on plane x2 = 0 are depicted in Figure 1.Figure 1
(a)x1-x2-x3 phase portrait; (b) x1-x3 phase portrait; (c) Poincare mapping.
(a)
(b)
(c)


## 2.2. Analysis of Amplitude Modulation

Theorem 1.
The parameterf in cross-product term x1x2 is a local parameter of nonlinear amplitude modulation, which can control the signal amplitude of x1,x2 by the power function of index −1/2 respectively, but the amplitude of variable x3 remains in the same range; besides, the Lyapunov exponent spectrum remains unchanged with the variation of parameter f.Proof.
Considering the variable substitutionx1=u1/h,x2=u2/h,x3=u3 (h>0), system (1) is turned to(5)u˙1=au2−u1+du2u3,u˙2=bu2−eu1u3,u˙3=−cu3+f/hu1u2.
Therefore, when parameterf increases linearly, the signal amplitude of system variables x1,x2 change according to the power function of index −1/2, respectively, but the amplitude of variable x3 is in the same range.
When substituting the equilibrium pointP0 into characteristic equation (3), it holds(6)φλ=−λ3+−a+b−cλ2+ab+bc−acλ+abc.
In equation (6), the influence of parameter f is eliminated. We can draw a similar conclusion for the other equilibrium points. As an illustration, we insert the equilibrium P1 into expression (3) obtaining(7)φλ=−λ3+−a+b−cλ2+ab+bc−ac+ce2a2+4abd/e−2aa2+4abd/e4bd−ae−a+a2+4abd/e2d−2a2e+4abd−2aea2+4abd/e4d−bcλ+abc−aecbe−a+a2+4abd/e2dbe−ce2a2+4abd/e−2aa2+4abd/e4d−ace−a+a2+4abd/e2d−cde2a2+4abd/e−2aa2+4abd/e4d2−abc.
That is, parameterf does not produce effect on the characteristic root of equation (7). Therefore, when parameter f varies in field of real number, the Lyapunov exponent spectrum remains constant. This completes the proof.
The corresponding bifurcation diagram and Lyapunov exponent spectrum versusf are shown in Figure 2, which authenticates the theoretical results.
It is generally accepted that for the quadratic chaotic systems, the coefficients of nonlinear terms can modulate the signal amplitude of partial or total state variables [11–14]. The bifurcation diagram for e∈0,20 is shown in Figure 3(a). Superficially, the coefficient e can modulate the signal amplitude nonlinearly. But from the enlarged view, one can see that there emerges a visible periodic window, and the Lyapunov exponent spectrum further verifies the observation, as depicted in Figures 3(b) and 3(c). The concrete bifurcation diagram and Lyapunov exponent spectrum versus d also show that not all coefficients of quadratic terms can modulate the signal amplitude, which is illustrated in Figure 4. In spite of this, the chaos of the reported system is still robust in a large range of parameters d and e. Consequently, the system can be recommended as an important candidate in secure communication.Figure 2
(a, b, c) Bifurcation diagram versusf; (d) finite time Lyapunov exponent spectrum versus f.
(a)
(b)
(c)
(d)Figure 3
(a, b) Bifurcation diagram versuse; (c) finite time Lyapunov exponent spectrum versus e.
(a)
(b)
(c)Figure 4
(a, b) Bifurcation diagram versusd; (c) finite time Lyapunov exponent spectrum versus d.
(a)
(b)
(c)


## 3. Chaotic System with Noncoefficient of Nonlinearity for Amplitude Control

### 3.1. System Description

We consider another three-dimensional autonomous system with five linear terms and three quadratic cross-product terms, as follows:(8)x˙1=ax2−gx1+dx2x3,x˙2=bx1−ex1x3,x˙3=kx1−cx3+fx1x2.System (8) possesses three equilibrium points, which are, respectively, described by(9)P00,0,0,P1ae+bd−k+k2+4bcgf/ae+bd2gef,−k+k2+4bcgf/ae+bd2f,be,P2ae+bd−k−k2+4bcgf/ae+bd2gef,−k−k2+4bcgf/ae+bd2f,be.When selecting the parameter seta = 24, b = 12, c = 1, d = 1, e = 1, f = 1, k = 1, and g = 6, the two nonzero equilibrium points are obtained as P1 (6, 1, 12) and P2 (−12, −2, 12). The corresponding characteristic roots are(10)P0:λ1=−1,λ2=14.2337,λ3=−20.2337,P1:λ1=−9.7318,λ2=1.3659+8.0449i,λ3=1.3659−8.0449i,P2:λ1=−8.2073,λ2=0.6036+12.5517i,λ3=0.6036−12.5517i.Obviously, equilibrium pointP0 is a saddle-node with two-dimensional stable manifold and one-dimensional unstable manifold. And the equilibrium points P1 and P2 are saddle-focus points with two-dimensional unstable manifold and one-dimensional stable manifold.The corresponding finite time Lyapunov exponents by orthogonal method are calculated as 0.926059, 0.051479, and −10.972296. And the Kaplan–Yorke dimension is obtained asDKY = 2 + (0.926059 + 0.051479)/10.972296 = 2.0891, revealing a fractional feature. Therefore, system (8) is chaotic. The typical chaotic phase diagrams and Poincare mapping on plane x2 = 0 are depicted in Figure 5.Figure 5
(a)x1-x2-x3 phase portrait; (b) x1-x3 phase portrait; (c) Poincare mapping.
(a)
(b)
(c)


### 3.2. Analysis of Phase Modulation

In system (8), coefficient k is a phase parameter, which can control the signal phase of x1,x2 simultaneously. The phase reversal can be verified by the invariance of the transformation x1,x2,x3,a,b,c,d,e,f,g,k ⟶ −x1,−x2,x3,a,b,c,d,e,f,g,−k. The bifurcation diagrams for x1 and x2 are reverse symmetrical, and the Lyapunov exponent spectrum is symmetrical about k = 0, as depicted in Figure 6. This further demonstrates that the sign of k can control the polarity of x1 and x2, independent of the dynamics behavior.Figure 6
(a, b) Bifurcation diagram versusk; (c) finite time Lyapunov exponent spectrum versus k.
(a)
(b)
(c)


### 3.3. Analysis of Amplitude Modulation

In the search for the property of amplitude modulation, it is surprising to find that for the presented system (8), there is no coefficient of nonlinearity employed to control the signal amplitude. The obtained result is different from the existing quadratic system [11–14] and the reported system (1). The most intuitive interpretation is the numerical simulations of bifurcation diagram and Lyapunov exponent spectrum versus the nonlinear parameter, as shown in Figures 7–9. Theoretically, according to the proposed method in [19], we cannot access appropriate variable substitution to realize the normalization of the system equation and characteristic equation [19].Figure 7
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versusd.
(a)
(b)Figure 8
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versuse.
(a)
(b)Figure 9
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versusf.
(a)
(b)


## 3.1. System Description

We consider another three-dimensional autonomous system with five linear terms and three quadratic cross-product terms, as follows:(8)x˙1=ax2−gx1+dx2x3,x˙2=bx1−ex1x3,x˙3=kx1−cx3+fx1x2.System (8) possesses three equilibrium points, which are, respectively, described by(9)P00,0,0,P1ae+bd−k+k2+4bcgf/ae+bd2gef,−k+k2+4bcgf/ae+bd2f,be,P2ae+bd−k−k2+4bcgf/ae+bd2gef,−k−k2+4bcgf/ae+bd2f,be.When selecting the parameter seta = 24, b = 12, c = 1, d = 1, e = 1, f = 1, k = 1, and g = 6, the two nonzero equilibrium points are obtained as P1 (6, 1, 12) and P2 (−12, −2, 12). The corresponding characteristic roots are(10)P0:λ1=−1,λ2=14.2337,λ3=−20.2337,P1:λ1=−9.7318,λ2=1.3659+8.0449i,λ3=1.3659−8.0449i,P2:λ1=−8.2073,λ2=0.6036+12.5517i,λ3=0.6036−12.5517i.Obviously, equilibrium pointP0 is a saddle-node with two-dimensional stable manifold and one-dimensional unstable manifold. And the equilibrium points P1 and P2 are saddle-focus points with two-dimensional unstable manifold and one-dimensional stable manifold.The corresponding finite time Lyapunov exponents by orthogonal method are calculated as 0.926059, 0.051479, and −10.972296. And the Kaplan–Yorke dimension is obtained asDKY = 2 + (0.926059 + 0.051479)/10.972296 = 2.0891, revealing a fractional feature. Therefore, system (8) is chaotic. The typical chaotic phase diagrams and Poincare mapping on plane x2 = 0 are depicted in Figure 5.Figure 5
(a)x1-x2-x3 phase portrait; (b) x1-x3 phase portrait; (c) Poincare mapping.
(a)
(b)
(c)


## 3.2. Analysis of Phase Modulation

In system (8), coefficient k is a phase parameter, which can control the signal phase of x1,x2 simultaneously. The phase reversal can be verified by the invariance of the transformation x1,x2,x3,a,b,c,d,e,f,g,k ⟶ −x1,−x2,x3,a,b,c,d,e,f,g,−k. The bifurcation diagrams for x1 and x2 are reverse symmetrical, and the Lyapunov exponent spectrum is symmetrical about k = 0, as depicted in Figure 6. This further demonstrates that the sign of k can control the polarity of x1 and x2, independent of the dynamics behavior.Figure 6
(a, b) Bifurcation diagram versusk; (c) finite time Lyapunov exponent spectrum versus k.
(a)
(b)
(c)


## 3.3. Analysis of Amplitude Modulation

In the search for the property of amplitude modulation, it is surprising to find that for the presented system (8), there is no coefficient of nonlinearity employed to control the signal amplitude. The obtained result is different from the existing quadratic system [11–14] and the reported system (1). The most intuitive interpretation is the numerical simulations of bifurcation diagram and Lyapunov exponent spectrum versus the nonlinear parameter, as shown in Figures 7–9. Theoretically, according to the proposed method in [19], we cannot access appropriate variable substitution to realize the normalization of the system equation and characteristic equation [19].Figure 7
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versusd.
(a)
(b)Figure 8
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versuse.
(a)
(b)Figure 9
(a) Bifurcation diagram and (b) finite time Lyapunov exponent spectrum versusf.
(a)
(b)


## 4. Amplitude Modulation Principle of Chaotic System

As shown in the previous literatures [11–15], the coefficients of quadratic terms in smooth chaotic systems can modulate the signal amplitude of partial or total state variables. However, for systems (1) and (8) reported in this study, it is found that not all quadratic nonlinearity coefficients can be used to modulate the amplitude of signals. Therefore, when investigating dynamic properties of a chaotic system, we propose a naturally confusing but worthwhile question: “what is the possible principle for modulating the amplitude of chaotic signals?”The physical significance of equilibrium point of dynamic system can be explained as zero velocity point. When the trajectory of a chaotic attractor is rescaled, the nonzero equilibrium point will deviate from the initial position. On the contrary, when the nonzero equilibrium point deviates from the initial position, the signal amplitude of the phase space trajectory can be rescaled. Therefore, the amplitude modulation principle of chaotic system with multiple equilibrium points can be described as follows: (1) in the mathematical representation of nonzero equilibrium point, the amplitude parameter is axisymmetric and (2) the location of the nonzero equilibrium point in phase space can be controlled by the amplitude parameter.From the expression of nonzero equilibrium points of system (1), it can be seen that the parameters d and e of nonlinear term cannot modulate the location of equilibrium point P1, P2, P3, or P4. Accordingly, parameter d or e cannot modulate the signal amplitude. When we introduce a unified parameter d in x2x3 and x1x3, system (1) is deduced to(11)x˙1=ax2−x1+dx2x3,x˙2=bx2−dx1x3,x˙3=−cx3+fx1x2,and the four nonzero equilibrium points are(12)P1bcdf,−ac+ca2+4ab2bdf,−a+a2+4ab2d,P2bcdf,−ac−ca2+4ab2bdf,−a−a2+4ab2d,P3−bcdf,ac+ca2+4ab2bdf,−a+a2+4ab2dbf,P4−bcdf,ac−ca2+4ab2bdf,−a−a2+4ab2dbf.It is known that for equilibrium pointsP1 and P2, the set of symmetrical axis is 0,−ac/2bdf,−a/2d, and the set of symmetrical axis for equilibrium points P3 and P4 is 0,ac/2bdf,−a/2dbf. In addition, the parameter d can modulate the location of the nonzero equilibrium points according to 1/d, 1/d, and 1/d, respectively. As a result, the parameter d can modulate the amplitude of signal x1, x2, and x3 according to 1/d, 1/d, and 1/d, respectively. The signal amplitude and Lyapunov exponent spectrum versus d are depicted in Figure 10.Figure 10
(a) Signal amplitude and (b) finite time Lyapunov exponent spectrum versusd.
(a)
(b)For the expression of nonzero equilibrium points of system (8), parameters d, e, and f of quadratic terms cannot modulate the location of equilibrium point P1 or P2. Accordingly, parameter d, e, or f cannot modulate the signal amplitude. To realize amplitude modulation in system (8), we introduce a unified parameter d in each nonlinear term yielding(13)x˙1=ax2−gx1+dx2x3,x˙2=bx1−dx1x3,x˙3=kx1−cx3+dx1x2,and the two nonzero equilibrium points are deduced as(14)P1a+b−k+k2+4bcg/a+b2gd,−k+k2+4bcg/a+b2d,bd,P2a+b−k−k2+4bcg/a+b2gd,−k−k2+4bcg/a+b2d,bd.We know that for equilibrium pointsP1 and P2, the set of symmetrical axis is −ka+b/2gd,−k/2d,0. In addition, the parameter d can modulate the location of the nonzero equilibrium points according to 1/d, respectively. Similarly, parameter d can modulate the amplitude of x1, x2, and x3 according to 1/d, respectively. The signal amplitude and Lyapunov exponent spectrum versus d are shown in Figure 11.Figure 11
(a) Signal amplitude and (b) finite time Lyapunov exponent spectrum versusd.
(a)
(b)


## 5. Discussion and Conclusion

Exploring the amplitude modulation phenomenon of chaotic signal is attractive yet recent topic of interest. This paper reported two chaotic systems with three quadratic cross-product terms and analyzed the property of amplitude modulation. By making an exhaustive study on the characteristics of nonzero equilibrium points, we attempt to address the possible principle for amplitude modulation. That is to say, the amplitude parameter in the expression of nonzero equilibrium point is symmetrical about some axis and can modulate the location of the nonzero equilibrium point in phase space.The addressed principle can be popularized to high-dimensional chaotic system and other chaotic systems except with quadratic nonlinearity [19–22], not relying on the type of nonzero equilibrium point. However, for the dynamical system with none, single, or an infinite number of equilibrium points [23–26], the principle is out of our consideration and deserves an in-depth study.It must be reiterated that the proposed principle is just a prerequisite, but not a sufficient and necessary condition for amplitude modulation. As an interpretation, we consider the following system with cubic nonlinearity:(15)x˙1=−ax1+bx2x3,x˙2=−cx23+dx1x3,x˙3=ex3−fx1x2.System (15) holds five equilibrium points, and the nonzero equilibrium points are described by(16)P1efacbd,aebf,abcedf,P2−efacbd,aebf,−abcedf,P3−efacbd,−aebf,abcedf,P4efacbd,−aebf,−abcedf.Clearly, the set of symmetrical axis for equilibrium pointsP1, P2, P3, and P4 is (0, 0, 0). In addition, the parameter b can modulate the location of the nonzero equilibrium points according to 1/b, 1/b, and 1/b; the parameter c can modulate the location of the nonzero equilibrium points according to c, 1, and c; the parameter d can modulate the location of the nonzero equilibrium points according to 1/d, 1, and 1/d; and the parameter f can modulate the location of the nonzero equilibrium points according to 1/f, 1/f, and 1/f. However, it is found from the numerical experiment that only the nonlinear parameter d can modulate the amplitude of x1, x2, and x3 according to 1/d, 1, and 1/d, respectively, as plotted in Figure 12.Figure 12
Modulation property of signal amplitude for system (15).
(a)
(b)
(c)
(d)In spite of this, the principle provides a feasible mentality for analyzing amplitude modulation of chaotic signal, which can be briefly summarized from theoretical analysis to numerical confirmation. We hope that our work can constitute a stimulus and afford a subservient reference for further exploring the intrinsic amplitude modulation mechanism of chaotic systems.



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