# Multifarious Chaotic Attractors and Its Control in Rigid Body Attitude Dynamical System

Authors: Yang Wang; Zhen Wang; Dezhi Kong; Lingyun Kong; Yukun Qiao
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

The Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors under the action of different external torques. Many special types of new chaotic attractors are presented, including hidden attractors, double-body-double-core chaotic attractors, and single-body-three-core-tree-wing chaotic attractors. The position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its influence on the formation of the chaotic attractor are analyzed. An improved nonlinear relay control law based on Euler angle feedback is developed to stabilize a new chaotic spacecraft attitude motion to an appointed equilibrium point or a periodic orbit.


---

## Body

## 1. Introduction

The detailed exploration of the chaotic attitude motion of spacecraft and satellites still remains one of the main problems of rigid body attitude dynamics. A number of investigators have demonstrated that spinning satellite [1], dual-spin spacecraft (DSSC) [2–4], multispin spacecraft (MSSC) [5], gyrostat satellite [6–12], tethered satellite, and other complicated satellites would exhibit chaotic attitude motion in gravitational fields [13], geomagnetic field [14, 15], and sunlight flux. Beletsky et al [14] discussed the chaotic motion of a magnetic spacecraft in circular polar orbit without damping and gravitational torque. Chen and Liu [15, 16] investigated chaotic attitude motion of a magnetic rigid spacecraft with internal damping in a circular orbit near the equatorial plane of the earth. These works mainly analyzed the dynamics of different mathematical models of rigid body attitude motion from a different point of view.The attitude motion of a rigid body could be described by the Eulerian dynamic equations. This nonlinear equation will present very complex dynamic behaviors, including various forms of chaotic motion. Leipnik and Newton [17] found two strange attractors from this system. In the previous paper [18, 19], we introduced several new chaotic attractors but did not analyze the influence of the properties of equilibrium point on the formation of chaotic attractors.Different control techniques have been employed to suppress or manipulate chaotic attitude motion of spacecraft and satellites, including sliding mode variable structure control [20, 21], time-delayed feedback control [22, 23], observer-based control [24, 25], impulsive control, adaptive control, and open-plus-closed-loop control. Alban and Antonia [26] made use of three methods to control a six-dimensional chaotic system that describes the attitude dynamics of a rigid body spacecraft subjected to deterministic external perturbations that induce chaotic motion when no control is acted. The three techniques are a simple delayed feedback control method, the Otani–Jones technique, and a higher dimensional variation of the OGY method. Other control methods were employed to suppress chaos by Meehan and Asokanthan [1] and Awad [20]. Generally, the angular velocity of the spacecraft ω was chosen as feedback control variable. However, ω can be measured directly in practice.Different spacecraft have different mission requirements and need different control methods. A spinning spacecraft usually is required to rotate along one axis with constant rotation rate and not to rotate along the other two axes. When a spinning spacecraft is disturbed and produces unexpected rotation, the nozzle thruster is usually used to control it. Due to the coupling effect of the three channels, the single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft.The innovation of this paper lies in the following:(1)
Many special types of new chaotic attractors are presented, including hidden attractors, double-body-double-core chaotic attractors, and single-body-three-core-tree-wing chaotic attractors.(2)
The position of equilibrium points in several typical cases of the Euler dynamic equation is solved, and the stability of linearized equation at each equilibrium point and its influence on the formation of chaotic attractor are analyzed.(3)
A more practicable and efficient control techniques are utilized to suppress chaos and control state of the system to an appointed fixed point or a periodic orbit.


## 2. The Attitude Motion Equations of a Rigid Body

The attitude motion of a rigid body could be described by the Eulerian equations, which consists of kinematic equations and dynamic equations.The attitude orientation of spacecraft at a given point can be locally described in terms of three Eulerian anglesφ, θ, and ψ which are successive clockwise rotations about inertial axes X, Y, and Z, respectively. These successive rotations transform the inertially fixed set of orthonormal axes X, Y, and Z (regarded as initially instantaneously coincident with the body axes) into the axes x, y, and z fixed in the body. The kinematic equation of the rigid body can be expressed as [21](1)ωx=φ.cosψcosθ+θ.sinψ,ωy=−φ.sinψcosθ+θ.cosψ,ωz=ψ˙+φ.sinθ,or the following form(2)φ.=ωxcosψ−ωysinψcosθ,θ.=ωxsinψ+ωycosψ,ψ˙=ωz−ωxcosψ−ωysinψtanθ.Here,ω=ωx,ωy,ωz are the angular velocities of a rigid body. The dynamical motion equations of a spacecraft with principle axes at the center of mass are [21](3)Ixω˙x=ωyωzIy−Iz+Mx+ux,Iyω˙y=ωxωzIz−Ix+My+uy,Izω˙z=ωyωxIx−Iy+Mz+uz.Here,Ix, Iy, and Iz are the principal moments of inertia with respect to body axes x, y, and z; ux, uy, and uz are the component of the control torques; and Mx, My, and Mz are the perturbing torques. The total disturbing torque M which is put on a spacecraft can be written as(4)M=Mg+Mm+Mc+Mf+Md+L.Here,Mg and Mm are the gravitational and magnetic torque; Md is the internal damping torque which is proportional to the angular velocity of the body with coefficient c; Mc and Mf are atmosphere resistance torque and solar pressure torque. They all have relation to the attitude angles φ,θ, and ψ, the attitude angular velocities ω, and the orbital angular velocities ω0. For example, the component of gravitational torque Mgz on Z-axis can be written as(5)Mgz=−ω02Iy−Izsinφcosφ.The magnetic torque about Z-axis can be obtained as(6)Mmz=ksinα3cosφ−β−γ−cosφ+β+γ.Here,k is a coefficient with respect to the magnetic moment constant of the earth and the magnetic moment of the spacecraft and the distance between spacecraft and the earth. α is the angle of inclination of orbital plane. β is the argument of perigee. γ is the true anomaly of spacecraft as the angular coordinate measured from perigee.Equation (3) can be rewritten as(7)ω⟶.=axωyωzayωxωzazωxωyT+A⟶ω⟶+ω⟶.u.Here,ax=Iy−Iz/Ix,ay=Iz−Ix/Iy,az=Ix−Iy/Iz, ω⟶=ωxωyωzT,ω⟶.u=ux/Ixuy/Iyuz/IzT, A⟶ω⟶=Mx/IxMy/IyMz/IzT, A⟶=aij∈R3×3i,j=1,2,3 is a perturbing frequency matrix.


## 3. Multifarious Chaotic Attractors

Equation (7) is a more generalized 3-dimensional nonlinear system which will exhibit complex (periodic, quasiperiodic, or chaotic) dynamic behaviors under the action of different external torques.


### 3.1. Well-Known Chaotic Attractors in This System

A spinning rigid spacecraft is symmetrical with respect to the body axesOx, i.e., Iy=Iz. In general, it is appropriate to choose Ix=2Iy=2Iz for stabilizing the spinning motion of the spacecraft, i.e., ax,ay,az=0,−1,1. When ux=uy=uz=0, A⟶=−aa0;c−10;00−b, a, b, and c are all perturbing frequency parameters, and equation (7) is equal to the normal Lorenz equations [22]:(8)ω˙x=−aωx+aωy,ω˙y=cωx−ωy−ωxωz,ω˙z=−bωz+ωxωy.It will exhibit chaotic motion whena,b,c=10,−8/3,28 as we all know. When ax,ay,az=10,−5,5, ux=uy=uz=0, A⟶=−0.410;−1−0.40;00α, (α is alterable parameter), system (7) is a chaotic system named the Leipnik–Newton system [22, 23]:(9)ω˙x=−aωx+ωy+10ωyωz,ω˙y=−ωx−0.4ωy+5ωxωz,ω˙z=bωz−5ωxωy.There are two strange attractors in this system whena=0.4andb=0.175 as we all know.Earlier papers [1] have taken Ix=3, Iy=2, Iz=1, ax,ay,az=1/3,−1,1,A⟶=−0.401/6;00.1750;−60−0.4 in (7). These torques are chosen to be sufficiently large to induce chaotic motion and are comparable in magnitude with the available thruster torques. The dynamics of the satellite will then exhibit chaotic motion. We constructed the chaotic attractor of this system shown in Figure 1.Figure 1
One chaotic attractor of system (7), starting point 0.3694,0.0473,0.1105.


### 3.2. New Chaotic Attractors

#### 3.2.1. A Type of Hidden Attractors

In the study of the chaotic motion of a nonlinear system, it is often concerned with the chaotic attractors near the equilibrium point. Some hidden attractors exist widely in the greater region or the smaller region of the phase space.Let us take Leipnik–Newton system (9) as an example. There are five equilibriums in equation (9). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(10)ωxe1,2,3,4=±4a−15b±b225−80a250a,ωye1,2,3,4=±15−8ab±b225−80a200,ωze1,2,3,4=±54a−15±225−80a250a⋅15−8a±225−80a400.Example 1.
Leta=0.4andb=0.3, and the four equilibriums are(11)S1,S2:±0.0413,∓0.1602,−0.1103,S3,S4:±0.3129,∓0.0403,0.2103.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:S1,S2,S3,S4:λ1=−0.8,λ2,3=0.15±1.1419i. In the phase space, there exist one chaotic attractor near S1 and S2, and one periodic attractor near S3 and S4, and one big hidden attractor far away from equilibriums. The phase trajectories of the three attractors are shown in Figure 2. The starting point of the small periodic attractor is selected at −0.0459,0.0802,0.2871. The starting point of the big hidden attractor is selected at −74.2294,52.3574,−29.0246.Figure 2
One chaotic attractor, one periodic attractor, and one big hidden attractor exist at the same phase space of system (9): (a) small periodic attractor; (b) big hidden attractor.
(a)(b)


#### 3.2.2. Another Type of Double-Body-Double-Core Chaotic Attractors

Investigating the reduced form of equation (7),(12)ω˙x=axωyωz+a11ωx,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equations (12). Obviously, the origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(13)ωxe=±a22a33ayaz=±μx,ωye=±a11a33axaz=±μy,ωze=±a22a11ayax=±μz.When the feedback coefficientsa11,a22, and a33 make 0≥μk∈R,k=x,y,z, equation (12) degenerates to the single equilibrium point system.Here, let us just discuss the case of0≤μk∈R,k=x,y,z.Let us assumeIx>Iy>Iz>0, ax>0,ay<0,az>0, and there are two cases of combination forms.Case A:ax>0,ay<0,az>0, a22>0,a11<0,a33<0. The other four equilibriums are expressed as(14)S1:μx,μy,μz,S2:μx,−μy,−μz,S3:−μx,−μy,μz,S4:−μx,μy,−μz.Case B:ax>0,ay<0,az>0,a22<0,a11>0,a33>0. The other four equilibriums are expressed as(15)S1′:μx,−μy,μz,S2′:μx,μy,−μz,S3′:−μx,μy,μz,S4′:−μx,−μy,−μz.Equation (12) is linearized at equilibrium point S1:μx,μy,μz, and the Jacobian matrix is(16)J⟶=a11axμzaxμyayμza22ayμxazμyazμxa33.The characteristic equation of the Jacobian linearization system is(17)λ3−a11+a22+a33λ2+4a11a22a33=0.The eigenvalues can be calculated by the following steps:(18)λ1=u+v+a11+a22+a333,λ2,3=−u+v2+a11+a22+a333±iu−v23.Here,(19)u=−q2+Q3,v=−q2−Q3,Q=p33+q22,q=2a11+a22+a3333+4a11a22a33,p=−a11+a22+a3323.Simulation studies indicate that system (12) will exhibit complex dynamic behaviors under the different feedback coefficients.Whena22=a11−a33, for example, a11=−1, a22=2, a33=−3, there exists a single cycle attractor.Whena22>a11−a33, for example, a11=−1, a22=2, a33=−2.8,−2.5,−2,−1.5, system (12) exhibits a divergent trajectory, and the pace of trajectory divergence speeds up with the increases of the difference between a11 and a33.Whena22<a11−a33, for example, a11=−1, a22=2, and a33=−3,−3.4,−3.42,−3.43,−3.44,−3.45L…, system (12) exists, respectively, 1–cycle, 2–cycle, 3–cycle, 4–cycle, 6–cycle, 8–cycle attractor ⋯ until the chaotic attractor.Example 2.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, and the other four equilibriums are(20)S1:24,32,12,S2:24,−32,−12,S3:−24,−32,12,S4:−24,32,−12,λ1=−4.5474,λ2,3=0.7734±2.5374i.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:λ1=−4.5474, λ2,3=0.7734±2.5374i. There exist double-body-double-core chaotic attractors in the phase space. The phase trajectories of the two attractors are shown in Figure 3.Figure 3
One double-body-double-core chaotic attractor of system (12): (a) chaotic attractor; (b) the projection on the Oωxωy plane.
(a)(b)


#### 3.2.3. A Type of Single-Body-Three-Core Chaotic Attractors

Investigating another reduced form of equation (7),(21)ω˙x=axωyωz+a11ωx+a12ωy+a13ωz,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equation (13). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following values:(22)ωxe1,2,3,4=±a22a33ayaz,ωye1,2=a12a13−a13aza22a33/ayaz2axaza22a33/ayaz±a12a13−a13aza22a33/ayaz2+4axa11a22a332/ay2axaza22a33/ayaz,ωye3,4=a12a13+a13aza22a33/ayaz−2axaza22a33/ayaz±a12a13+a13aza22a33/ayaz2+4axa11a22a332/ay−2axaza22a33/ayaz,ωze1,2=a22a13−a12aya22a33/ayaz2axaya22a33/ayaz±a22a13−a12aya22a33/ayaz2+4axa11a33a222/az2axaya22a33/ayaz,ωze3,4=a22a13+a12aya22a33/ayaz−2axaya22a33/ayaz±a22a13+a12aya22a33/ayaz2+4axa11a33a222/az−2axaya22a33/ayaz.Ifay and a22 are the opposite sign, then az and a33 must be the opposite sign. When ωxe>0, ωye and ωze must be the same sign. When ωxe<0, ωye and ωze must be the opposite sign.Compared with equation (12), feedback control parameters a12ωy and a13ωy are applied to the system, the four equilibriums are no longer symmetric, and the dynamic properties of equation (13) near the four equilibriums present a complicated situation. When the real part of the eigenvalues of Jacobian linearization matrix at four equilibriums are greater than zero: S1,S2,S3,S4:Reλ2,3>0, the four equilibrium points are unstable saddle odd points, and the system will produce a variety of single-body-four-core-four-wing chaotic attractors. When one of the real part is less than zero: S4:Reλ2,3<0, this equilibrium point is stable, and the system will produce a type of single-body-three-core-tree-wing chaotic attractor.Example 3.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, a12=2, a13=2, and the other four equilibriums are S1:4.899,1.43,0.871,S2:4.899,−22.49,−13.77, S3:−4.899,8.729,−5.346,andS4:−4.899,−3.666,2.245.
The eigenvalues of the Jacobian linearization matrix at four equilibriums areS1:λ1=−3.7695,λ2,3=0.3848±2.0893i, S2:λ1=−9.0734,λ2,3=3.0367±4.5174i, S3:λ1=−6.8623,λ2,3=1.9312±2.0383i, S4:λ1=−2.9554,λ2,3=−0.0223±2.7725i.
Obviously,S1,S2,S3:Reλ2,3>0 but S4:Reλ2,3<0, S4 is stable equilibrium point, and S1,S2,S3 are unstable. There is a small attraction basin around S4 (shown in Figure 4(a)) and a single-body-three-core-tree-wing chaotic attractor which are formed by S1,S2, and S3 in the larger domain. The phase trajectories of the attractors and the attraction basin (red point) are shown in Figure 4.Figure 4
One single-body-three-core-tree-wing chaotic attractor of system (13): (a) attraction plane around S4; (b) chaotic attractor; (c) the projection on the Oωxωy plane; (d) the projection on the Oωxωy plane.
(a)(b)(c)(d)


#### 3.2.4. Multifarious Chaotic Attractors

Keepingωu=0, changing parameter value of ax, ay, and az and matrix A, system (7) will exhibit chaotic attitude motion, and a lot of new chaotic attractors are found in state space, as follows:Example 4.
ax,ay,az=10,−10,10 and A⟶=−36360;0200;00−3, as shown in Figure 5(a).Figure 5
Four new chaotic attractor of system (7): (a) example 4; (b) example 5; (c) example 6; (d) example 7.
(a)(b)(c)(d)Example 5.
ax,ay,az=4,−1,2 and A⟶=−88−2;10−1−1;0.83−3, as shown in Figure 5(b).Example 6.
ax,ay,az=0,−1,1 and A⟶=−77−6;40−1−33;550−3, as shown in Figure 5(c).Example 7.
ax,ay,az=1,−1,1 and A=−660;05−0.8;1010−1, as shown in Figure 5(d).


## 3.1. Well-Known Chaotic Attractors in This System

A spinning rigid spacecraft is symmetrical with respect to the body axesOx, i.e., Iy=Iz. In general, it is appropriate to choose Ix=2Iy=2Iz for stabilizing the spinning motion of the spacecraft, i.e., ax,ay,az=0,−1,1. When ux=uy=uz=0, A⟶=−aa0;c−10;00−b, a, b, and c are all perturbing frequency parameters, and equation (7) is equal to the normal Lorenz equations [22]:(8)ω˙x=−aωx+aωy,ω˙y=cωx−ωy−ωxωz,ω˙z=−bωz+ωxωy.It will exhibit chaotic motion whena,b,c=10,−8/3,28 as we all know. When ax,ay,az=10,−5,5, ux=uy=uz=0, A⟶=−0.410;−1−0.40;00α, (α is alterable parameter), system (7) is a chaotic system named the Leipnik–Newton system [22, 23]:(9)ω˙x=−aωx+ωy+10ωyωz,ω˙y=−ωx−0.4ωy+5ωxωz,ω˙z=bωz−5ωxωy.There are two strange attractors in this system whena=0.4andb=0.175 as we all know.Earlier papers [1] have taken Ix=3, Iy=2, Iz=1, ax,ay,az=1/3,−1,1,A⟶=−0.401/6;00.1750;−60−0.4 in (7). These torques are chosen to be sufficiently large to induce chaotic motion and are comparable in magnitude with the available thruster torques. The dynamics of the satellite will then exhibit chaotic motion. We constructed the chaotic attractor of this system shown in Figure 1.Figure 1
One chaotic attractor of system (7), starting point 0.3694,0.0473,0.1105.


## 3.2. New Chaotic Attractors

### 3.2.1. A Type of Hidden Attractors

In the study of the chaotic motion of a nonlinear system, it is often concerned with the chaotic attractors near the equilibrium point. Some hidden attractors exist widely in the greater region or the smaller region of the phase space.Let us take Leipnik–Newton system (9) as an example. There are five equilibriums in equation (9). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(10)ωxe1,2,3,4=±4a−15b±b225−80a250a,ωye1,2,3,4=±15−8ab±b225−80a200,ωze1,2,3,4=±54a−15±225−80a250a⋅15−8a±225−80a400.Example 1.
Leta=0.4andb=0.3, and the four equilibriums are(11)S1,S2:±0.0413,∓0.1602,−0.1103,S3,S4:±0.3129,∓0.0403,0.2103.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:S1,S2,S3,S4:λ1=−0.8,λ2,3=0.15±1.1419i. In the phase space, there exist one chaotic attractor near S1 and S2, and one periodic attractor near S3 and S4, and one big hidden attractor far away from equilibriums. The phase trajectories of the three attractors are shown in Figure 2. The starting point of the small periodic attractor is selected at −0.0459,0.0802,0.2871. The starting point of the big hidden attractor is selected at −74.2294,52.3574,−29.0246.Figure 2
One chaotic attractor, one periodic attractor, and one big hidden attractor exist at the same phase space of system (9): (a) small periodic attractor; (b) big hidden attractor.
(a)(b)


### 3.2.2. Another Type of Double-Body-Double-Core Chaotic Attractors

Investigating the reduced form of equation (7),(12)ω˙x=axωyωz+a11ωx,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equations (12). Obviously, the origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(13)ωxe=±a22a33ayaz=±μx,ωye=±a11a33axaz=±μy,ωze=±a22a11ayax=±μz.When the feedback coefficientsa11,a22, and a33 make 0≥μk∈R,k=x,y,z, equation (12) degenerates to the single equilibrium point system.Here, let us just discuss the case of0≤μk∈R,k=x,y,z.Let us assumeIx>Iy>Iz>0, ax>0,ay<0,az>0, and there are two cases of combination forms.Case A:ax>0,ay<0,az>0, a22>0,a11<0,a33<0. The other four equilibriums are expressed as(14)S1:μx,μy,μz,S2:μx,−μy,−μz,S3:−μx,−μy,μz,S4:−μx,μy,−μz.Case B:ax>0,ay<0,az>0,a22<0,a11>0,a33>0. The other four equilibriums are expressed as(15)S1′:μx,−μy,μz,S2′:μx,μy,−μz,S3′:−μx,μy,μz,S4′:−μx,−μy,−μz.Equation (12) is linearized at equilibrium point S1:μx,μy,μz, and the Jacobian matrix is(16)J⟶=a11axμzaxμyayμza22ayμxazμyazμxa33.The characteristic equation of the Jacobian linearization system is(17)λ3−a11+a22+a33λ2+4a11a22a33=0.The eigenvalues can be calculated by the following steps:(18)λ1=u+v+a11+a22+a333,λ2,3=−u+v2+a11+a22+a333±iu−v23.Here,(19)u=−q2+Q3,v=−q2−Q3,Q=p33+q22,q=2a11+a22+a3333+4a11a22a33,p=−a11+a22+a3323.Simulation studies indicate that system (12) will exhibit complex dynamic behaviors under the different feedback coefficients.Whena22=a11−a33, for example, a11=−1, a22=2, a33=−3, there exists a single cycle attractor.Whena22>a11−a33, for example, a11=−1, a22=2, a33=−2.8,−2.5,−2,−1.5, system (12) exhibits a divergent trajectory, and the pace of trajectory divergence speeds up with the increases of the difference between a11 and a33.Whena22<a11−a33, for example, a11=−1, a22=2, and a33=−3,−3.4,−3.42,−3.43,−3.44,−3.45L…, system (12) exists, respectively, 1–cycle, 2–cycle, 3–cycle, 4–cycle, 6–cycle, 8–cycle attractor ⋯ until the chaotic attractor.Example 2.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, and the other four equilibriums are(20)S1:24,32,12,S2:24,−32,−12,S3:−24,−32,12,S4:−24,32,−12,λ1=−4.5474,λ2,3=0.7734±2.5374i.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:λ1=−4.5474, λ2,3=0.7734±2.5374i. There exist double-body-double-core chaotic attractors in the phase space. The phase trajectories of the two attractors are shown in Figure 3.Figure 3
One double-body-double-core chaotic attractor of system (12): (a) chaotic attractor; (b) the projection on the Oωxωy plane.
(a)(b)


### 3.2.3. A Type of Single-Body-Three-Core Chaotic Attractors

Investigating another reduced form of equation (7),(21)ω˙x=axωyωz+a11ωx+a12ωy+a13ωz,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equation (13). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following values:(22)ωxe1,2,3,4=±a22a33ayaz,ωye1,2=a12a13−a13aza22a33/ayaz2axaza22a33/ayaz±a12a13−a13aza22a33/ayaz2+4axa11a22a332/ay2axaza22a33/ayaz,ωye3,4=a12a13+a13aza22a33/ayaz−2axaza22a33/ayaz±a12a13+a13aza22a33/ayaz2+4axa11a22a332/ay−2axaza22a33/ayaz,ωze1,2=a22a13−a12aya22a33/ayaz2axaya22a33/ayaz±a22a13−a12aya22a33/ayaz2+4axa11a33a222/az2axaya22a33/ayaz,ωze3,4=a22a13+a12aya22a33/ayaz−2axaya22a33/ayaz±a22a13+a12aya22a33/ayaz2+4axa11a33a222/az−2axaya22a33/ayaz.Ifay and a22 are the opposite sign, then az and a33 must be the opposite sign. When ωxe>0, ωye and ωze must be the same sign. When ωxe<0, ωye and ωze must be the opposite sign.Compared with equation (12), feedback control parameters a12ωy and a13ωy are applied to the system, the four equilibriums are no longer symmetric, and the dynamic properties of equation (13) near the four equilibriums present a complicated situation. When the real part of the eigenvalues of Jacobian linearization matrix at four equilibriums are greater than zero: S1,S2,S3,S4:Reλ2,3>0, the four equilibrium points are unstable saddle odd points, and the system will produce a variety of single-body-four-core-four-wing chaotic attractors. When one of the real part is less than zero: S4:Reλ2,3<0, this equilibrium point is stable, and the system will produce a type of single-body-three-core-tree-wing chaotic attractor.Example 3.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, a12=2, a13=2, and the other four equilibriums are S1:4.899,1.43,0.871,S2:4.899,−22.49,−13.77, S3:−4.899,8.729,−5.346,andS4:−4.899,−3.666,2.245.
The eigenvalues of the Jacobian linearization matrix at four equilibriums areS1:λ1=−3.7695,λ2,3=0.3848±2.0893i, S2:λ1=−9.0734,λ2,3=3.0367±4.5174i, S3:λ1=−6.8623,λ2,3=1.9312±2.0383i, S4:λ1=−2.9554,λ2,3=−0.0223±2.7725i.
Obviously,S1,S2,S3:Reλ2,3>0 but S4:Reλ2,3<0, S4 is stable equilibrium point, and S1,S2,S3 are unstable. There is a small attraction basin around S4 (shown in Figure 4(a)) and a single-body-three-core-tree-wing chaotic attractor which are formed by S1,S2, and S3 in the larger domain. The phase trajectories of the attractors and the attraction basin (red point) are shown in Figure 4.Figure 4
One single-body-three-core-tree-wing chaotic attractor of system (13): (a) attraction plane around S4; (b) chaotic attractor; (c) the projection on the Oωxωy plane; (d) the projection on the Oωxωy plane.
(a)(b)(c)(d)


### 3.2.4. Multifarious Chaotic Attractors

Keepingωu=0, changing parameter value of ax, ay, and az and matrix A, system (7) will exhibit chaotic attitude motion, and a lot of new chaotic attractors are found in state space, as follows:Example 4.
ax,ay,az=10,−10,10 and A⟶=−36360;0200;00−3, as shown in Figure 5(a).Figure 5
Four new chaotic attractor of system (7): (a) example 4; (b) example 5; (c) example 6; (d) example 7.
(a)(b)(c)(d)Example 5.
ax,ay,az=4,−1,2 and A⟶=−88−2;10−1−1;0.83−3, as shown in Figure 5(b).Example 6.
ax,ay,az=0,−1,1 and A⟶=−77−6;40−1−33;550−3, as shown in Figure 5(c).Example 7.
ax,ay,az=1,−1,1 and A=−660;05−0.8;1010−1, as shown in Figure 5(d).


## 3.2.1. A Type of Hidden Attractors

In the study of the chaotic motion of a nonlinear system, it is often concerned with the chaotic attractors near the equilibrium point. Some hidden attractors exist widely in the greater region or the smaller region of the phase space.Let us take Leipnik–Newton system (9) as an example. There are five equilibriums in equation (9). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(10)ωxe1,2,3,4=±4a−15b±b225−80a250a,ωye1,2,3,4=±15−8ab±b225−80a200,ωze1,2,3,4=±54a−15±225−80a250a⋅15−8a±225−80a400.Example 1.
Leta=0.4andb=0.3, and the four equilibriums are(11)S1,S2:±0.0413,∓0.1602,−0.1103,S3,S4:±0.3129,∓0.0403,0.2103.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:S1,S2,S3,S4:λ1=−0.8,λ2,3=0.15±1.1419i. In the phase space, there exist one chaotic attractor near S1 and S2, and one periodic attractor near S3 and S4, and one big hidden attractor far away from equilibriums. The phase trajectories of the three attractors are shown in Figure 2. The starting point of the small periodic attractor is selected at −0.0459,0.0802,0.2871. The starting point of the big hidden attractor is selected at −74.2294,52.3574,−29.0246.Figure 2
One chaotic attractor, one periodic attractor, and one big hidden attractor exist at the same phase space of system (9): (a) small periodic attractor; (b) big hidden attractor.
(a)(b)


## 3.2.2. Another Type of Double-Body-Double-Core Chaotic Attractors

Investigating the reduced form of equation (7),(12)ω˙x=axωyωz+a11ωx,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equations (12). Obviously, the origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following three values:(13)ωxe=±a22a33ayaz=±μx,ωye=±a11a33axaz=±μy,ωze=±a22a11ayax=±μz.When the feedback coefficientsa11,a22, and a33 make 0≥μk∈R,k=x,y,z, equation (12) degenerates to the single equilibrium point system.Here, let us just discuss the case of0≤μk∈R,k=x,y,z.Let us assumeIx>Iy>Iz>0, ax>0,ay<0,az>0, and there are two cases of combination forms.Case A:ax>0,ay<0,az>0, a22>0,a11<0,a33<0. The other four equilibriums are expressed as(14)S1:μx,μy,μz,S2:μx,−μy,−μz,S3:−μx,−μy,μz,S4:−μx,μy,−μz.Case B:ax>0,ay<0,az>0,a22<0,a11>0,a33>0. The other four equilibriums are expressed as(15)S1′:μx,−μy,μz,S2′:μx,μy,−μz,S3′:−μx,μy,μz,S4′:−μx,−μy,−μz.Equation (12) is linearized at equilibrium point S1:μx,μy,μz, and the Jacobian matrix is(16)J⟶=a11axμzaxμyayμza22ayμxazμyazμxa33.The characteristic equation of the Jacobian linearization system is(17)λ3−a11+a22+a33λ2+4a11a22a33=0.The eigenvalues can be calculated by the following steps:(18)λ1=u+v+a11+a22+a333,λ2,3=−u+v2+a11+a22+a333±iu−v23.Here,(19)u=−q2+Q3,v=−q2−Q3,Q=p33+q22,q=2a11+a22+a3333+4a11a22a33,p=−a11+a22+a3323.Simulation studies indicate that system (12) will exhibit complex dynamic behaviors under the different feedback coefficients.Whena22=a11−a33, for example, a11=−1, a22=2, a33=−3, there exists a single cycle attractor.Whena22>a11−a33, for example, a11=−1, a22=2, a33=−2.8,−2.5,−2,−1.5, system (12) exhibits a divergent trajectory, and the pace of trajectory divergence speeds up with the increases of the difference between a11 and a33.Whena22<a11−a33, for example, a11=−1, a22=2, and a33=−3,−3.4,−3.42,−3.43,−3.44,−3.45L…, system (12) exists, respectively, 1–cycle, 2–cycle, 3–cycle, 4–cycle, 6–cycle, 8–cycle attractor ⋯ until the chaotic attractor.Example 2.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, and the other four equilibriums are(20)S1:24,32,12,S2:24,−32,−12,S3:−24,−32,12,S4:−24,32,−12,λ1=−4.5474,λ2,3=0.7734±2.5374i.
The eigenvalues of the Jacobian linearization matrix at four equilibriums are the same:λ1=−4.5474, λ2,3=0.7734±2.5374i. There exist double-body-double-core chaotic attractors in the phase space. The phase trajectories of the two attractors are shown in Figure 3.Figure 3
One double-body-double-core chaotic attractor of system (12): (a) chaotic attractor; (b) the projection on the Oωxωy plane.
(a)(b)


## 3.2.3. A Type of Single-Body-Three-Core Chaotic Attractors

Investigating another reduced form of equation (7),(21)ω˙x=axωyωz+a11ωx+a12ωy+a13ωz,ω˙y=ayωxωz+a22ωy,ω˙z=azωyωx+a33ωz.There are five equilibriums in equation (13). The origin of coordinates S0:0,0,0 is one of five equilibriums. The other four equilibriums are composed of the proper combination of the following values:(22)ωxe1,2,3,4=±a22a33ayaz,ωye1,2=a12a13−a13aza22a33/ayaz2axaza22a33/ayaz±a12a13−a13aza22a33/ayaz2+4axa11a22a332/ay2axaza22a33/ayaz,ωye3,4=a12a13+a13aza22a33/ayaz−2axaza22a33/ayaz±a12a13+a13aza22a33/ayaz2+4axa11a22a332/ay−2axaza22a33/ayaz,ωze1,2=a22a13−a12aya22a33/ayaz2axaya22a33/ayaz±a22a13−a12aya22a33/ayaz2+4axa11a33a222/az2axaya22a33/ayaz,ωze3,4=a22a13+a12aya22a33/ayaz−2axaya22a33/ayaz±a22a13+a12aya22a33/ayaz2+4axa11a33a222/az−2axaya22a33/ayaz.Ifay and a22 are the opposite sign, then az and a33 must be the opposite sign. When ωxe>0, ωye and ωze must be the same sign. When ωxe<0, ωye and ωze must be the opposite sign.Compared with equation (12), feedback control parameters a12ωy and a13ωy are applied to the system, the four equilibriums are no longer symmetric, and the dynamic properties of equation (13) near the four equilibriums present a complicated situation. When the real part of the eigenvalues of Jacobian linearization matrix at four equilibriums are greater than zero: S1,S2,S3,S4:Reλ2,3>0, the four equilibrium points are unstable saddle odd points, and the system will produce a variety of single-body-four-core-four-wing chaotic attractors. When one of the real part is less than zero: S4:Reλ2,3<0, this equilibrium point is stable, and the system will produce a type of single-body-three-core-tree-wing chaotic attractor.Example 3.
Letax=0.25,ay=−2/3,az=0.5, a11=−1, a22=2, a33=−4, a12=2, a13=2, and the other four equilibriums are S1:4.899,1.43,0.871,S2:4.899,−22.49,−13.77, S3:−4.899,8.729,−5.346,andS4:−4.899,−3.666,2.245.
The eigenvalues of the Jacobian linearization matrix at four equilibriums areS1:λ1=−3.7695,λ2,3=0.3848±2.0893i, S2:λ1=−9.0734,λ2,3=3.0367±4.5174i, S3:λ1=−6.8623,λ2,3=1.9312±2.0383i, S4:λ1=−2.9554,λ2,3=−0.0223±2.7725i.
Obviously,S1,S2,S3:Reλ2,3>0 but S4:Reλ2,3<0, S4 is stable equilibrium point, and S1,S2,S3 are unstable. There is a small attraction basin around S4 (shown in Figure 4(a)) and a single-body-three-core-tree-wing chaotic attractor which are formed by S1,S2, and S3 in the larger domain. The phase trajectories of the attractors and the attraction basin (red point) are shown in Figure 4.Figure 4
One single-body-three-core-tree-wing chaotic attractor of system (13): (a) attraction plane around S4; (b) chaotic attractor; (c) the projection on the Oωxωy plane; (d) the projection on the Oωxωy plane.
(a)(b)(c)(d)


## 3.2.4. Multifarious Chaotic Attractors

Keepingωu=0, changing parameter value of ax, ay, and az and matrix A, system (7) will exhibit chaotic attitude motion, and a lot of new chaotic attractors are found in state space, as follows:Example 4.
ax,ay,az=10,−10,10 and A⟶=−36360;0200;00−3, as shown in Figure 5(a).Figure 5
Four new chaotic attractor of system (7): (a) example 4; (b) example 5; (c) example 6; (d) example 7.
(a)(b)(c)(d)Example 5.
ax,ay,az=4,−1,2 and A⟶=−88−2;10−1−1;0.83−3, as shown in Figure 5(b).Example 6.
ax,ay,az=0,−1,1 and A⟶=−77−6;40−1−33;550−3, as shown in Figure 5(c).Example 7.
ax,ay,az=1,−1,1 and A=−660;05−0.8;1010−1, as shown in Figure 5(d).


## 4. Analysis of the Properties of New Chaotic Attractors

In order to analyze the structure of new chaotic attractors, the concept of attractive plane is introduced from the linear system:(23)dx⟶dt=B⟶x⟶,where B⟶∈R3×3 is nonsingular coefficient matrix. x⟶=x1,x2,x3T is state vector. If the matrix B⟶ has one real eigenvalue λ1 and a pair of complex conjugate eigenvalues λ2,3=α±βi, the solution of equation (16) can be written as(24)xit,xi0=Ci1eλ1t+Ci2eαtcosβt+Ci3eαtsinβt=Ci1eλ1t+ki″t,xi0,where Ciji,j=1,2,3 are coefficients in relation to structure parameters of system (23) and initial values. ki″t,xi0 is the partial solution of (23).Theorem 1.
If the matrixB⟶ has complex conjugate eigenvalues λ2,3=α±βi, there must be a fixed plane across coordinate origin (0, 0, 0) in the phase space, and the motion trajectory of the partial solution must be inside the fixed plane:(25)ki″t,x0=Ci2eαtcosβt+Ci3eαtsinβt,i=1,2,3.Proof.
it is assumed that the motion trajectory of the partial solutionki″t,x⟶0 is inside a plane across coordinate origin (0, 0, 0) in the phase space, and the normal vector of this plan is P⟶=p1,p2,p3. The point-norm form equation of plane can be expressed as(26)p1k1″+p2k2″+p3k3″=0,
Now, let us prove this plane exists, and its normal vectorP⟶ is constant and has nothing to do with the initial conditions. Put formulas (25) into (26):(27)p1C12+p2C22+p3C32eαtcosβt+p1C13+p2C23+p3C33eαtsinβt=0.
Whent=0, C12=x10,C22=x20,C32=x30, and x˙i0=αCi2+βCi3=αxi0+βCi3=ai1x10+ai2x20+ai3x30, then we have(28)p1x10+p2x20+p3x30eαtcosβt+p1a11x10−αx10+a12x20+a13x30+p2a21x10+a22x20−αx20+a23x30+p3a31x10+a32x20+a33x30−αx30eαtsinβt=0.
Asp1x10+p2x20+p3x30=0, we have(29)p1a11+p2a21+p3a31x10+p1a12+p2a22+p3a32x20+p1a13+p2a23+p3a33x30=0.
As long as letp1a11+p2a21+p3a31/p1=p1a12+p2a22+p3a32/p2=p1a13+p2a23+p3a33/p3=k, it can be derived as follows:(30)a11−ka21a31a12a22−ka32a13a23a33−kp1p2p3=0,where k is the ratio of vectors module. Arbitrarily choose one vector coordinate, for example, p1=1, and other coordinates p1,p2,p3 could be solved. Obviously, it has nothing to do with the initial conditions and the eigenvalue λ. Proof ends.Definition 1.
Assume the matrixB⟶ has one real eigenvalue λ1 and a pair of complex conjugate eigenvalues λ2,3=α±βi. When λ1<0 (or λ1>0), the fixed plane is named attractive plane (or repulsive plane) in which the partial solution ki″t,x⟶0 is inside.


### 4.1. Structural Properties of Chaotic Attractors

Properties 1.
The size of various attractors in equation (7) is determined by the relative positions of all equilibrium points. In other words, the relative distances of all equilibrium points are magnified k times, and the sizes of various attractors are magnified 1/k times synchronously, and the structure and shape remain unchanged.
The parametersax,ay, and az in equation (7) are simultaneously magnified k times, and the sizes of various attractors are magnified 1/k times, and the structure and shape remain unchanged. This conclusion can be easily drawn from equilibrium calculation formulas (13) and (22) of equations (12) and (21), respectively.Properties 2.
There is an attractive plane in the neighborhood of each equilibrium point in the chaotic system. The nonlinear motion mode near an equilibrium point of a complicated chaotic system can be approximately described with its linearized equation. Each chaotic motion comprises at least two nonlinear motion modes. The essential reason to form chaotic attractor is nonperiodic and asynchronous switch between two modes.


## 4.1. Structural Properties of Chaotic Attractors

Properties 1.
The size of various attractors in equation (7) is determined by the relative positions of all equilibrium points. In other words, the relative distances of all equilibrium points are magnified k times, and the sizes of various attractors are magnified 1/k times synchronously, and the structure and shape remain unchanged.
The parametersax,ay, and az in equation (7) are simultaneously magnified k times, and the sizes of various attractors are magnified 1/k times, and the structure and shape remain unchanged. This conclusion can be easily drawn from equilibrium calculation formulas (13) and (22) of equations (12) and (21), respectively.Properties 2.
There is an attractive plane in the neighborhood of each equilibrium point in the chaotic system. The nonlinear motion mode near an equilibrium point of a complicated chaotic system can be approximately described with its linearized equation. Each chaotic motion comprises at least two nonlinear motion modes. The essential reason to form chaotic attractor is nonperiodic and asynchronous switch between two modes.


## 5. Control of Chaotic Attitude Motion in a Spinning Spacecraft

A spinning spacecraft usually is required to rotate along one axis with constant rotation rate and not rotate along the other two axes. The goal of control is to stabilize the state of the system at the desired equilibrium pointωx,ωy,ωz=c,0,0. The angular velocity ω in system (7) cannot be measured directly in practice and ωy≠θ., ωz≠ψ˙ generally. The attitude stabilization system measures the attitude angle ϕ, θ, and ψ and the attitude angular velocities ϕ˙, θ., and ψ˙ of the spacecraft using attitude sensors such as earth sensors, solar sensors, and gyroscopes. Compared with the desired attitude angle and angular velocity, if there is a deviation, the switch control instruction is formed by the controller to suppress the unwanted rotation of the spacecraft.As all know, the nonlinear relay control law which is based on position and velocity feedback with dead band peculiarity is suitable for the nonchaotic system whenA⟶=0 in system (7). However, the situation is completely different for chaotic systems, because there are multiple equilibrium points in the system, and the control torques applied to the three axes are interacting. So, the attitude angle change of other direction must be considered into the design when the control law for one direction was designed.Let us control six-dimensional chaotic system (21) (its chaotic attractor is shown in Figure 5(a)):(31)ω˙x=10ωyωz−36ωx+36ωy+uxIx,ω˙y=−10ωxωz+20ωy+uyIy,ω˙z=10ωyωz−3ωz+uzIz,φ˙=ωxcosψ−ωysinψcosθ,θ.=ωxsinψ+ωycosψ,ψ˙=ωz−ωxcosψ−ωysinψtanθ.Adopting single-channel dead zone relay control law based on position and speed feedback,(32)uxφ,φ˙=−IxM,whenφ>φ1,φ˙>−φ˙1,0,whenφ<φ1,φ˙>φ˙1,IxM,whenφ<φ1,φ˙<φ˙1,uyθ,θ.=−IyM,whenθ>θ1,θ.>−θ.1,0,whenθ<θ1,θ.>θ.1,IyM,whenθ<θ1,θ.<θ.1,uzψ,ψ˙=−IzM,whenψ>ψ1,ψ˙>−ψ˙1,0,whenψ<ψ1,ψ˙>ψ˙1,IzM,whenψ<ψ1,ψ˙<ψ˙1,where M is control torque to be generated by nozzle thrusters. φ1,θ1,ψ1 is angle position error of dead band corresponding to the driving signal when the nozzle baffle is opened. There are five equilibriums in equations (21), and the point S1:−0.34,0.44,0.34 is one of five equilibriums. The control action may drive any state to approach a nearest equilibrium point from an initial state. An equilibrium state is not the desired state for a spinning spacecraft though the chaotic motion is suppressed. Figure 6 shows the time history of ωy and θ under the single-channel dead zone relay control action, starting from initial state (0.2, 0.1, 0.2), beginning to control on 1500 s, and ending in the equilibrium point S1:−0.34,0.44,0.34.Figure 6
The time history ofωy and θ under single-channel relay control action: (a) the time history of ωy; (b) the time history of θ.
(a)(b)Now, considering the mutual effect amongωx, ωy, and ωz, changing the dual-channel control law to be(33)uy=−IyM,whenθ>θ1,θ.>−θ.1,andφ<φ1,φ˙<φ˙1,IyM,whenθ>θ1,θ.>−θ.1,andφ>φ1,φ˙>−φ˙1,0,whenθ<θ1,θ.>θ.1,andφ<φ1,φ˙<φ˙1,−IyM,whenθ<θ1,θ.<θ.1,andφ<φ1,φ˙<φ˙1,IyM,whanθ<θ1,θ.<θ.1,andφ>φ1,φ˙>−φ˙1,ux=+IxM,whenθ>θ1,θ.>−θ.1,andφ<φ1,φ˙<−φ˙1,−IxM,whenθ>θ1,θ.>−θ.1,andφ>φ1,φ˙>−φ˙1,0,whenθ<θ1,θ.>θ.1,andφ<φ1,φ˙>φ˙1,+IxM,whenθ<θ1,θ.<θ.1,andθ>θ1,θ.>θ.1,−IxM,whenθ<θ1,θ.<θ.1,andθ<θ1,θ.<θ.1.Figure7 shows the time history of ωy and θ under the two-channel control action: initial state (0.2, 0.1, 0.2); end point (3, 0, 0).The control effect meets the desired operating requirements of a spinning spacecraft with constant rotation rate.Figure 7
The time history ofωy and θ under two-channel relay control action: (a) the time history of ωy; (b) the time history of θ.
(a)(b)


## 6. Conclusion

The Euler dynamical equation which describes the attitude motion of a rigid body will exhibit very complex dynamic behaviors and include many special types of chaotic attractors. In the same phase space, beyond chaotic attractors, there may be a very large hidden attractor or very small hidden attractors. Even if the system is stable near a certain equilibrium point, chaotic attractors may still be generated in a larger space. From the simulation analysis, we can see that the stability of the linearized equation near the equilibrium point determines the shape and size of the chaotic attractor. The single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft. The simulation results confirm that the single-channel dead zone relay control method cannot effectively suppress a chaotic motion of spinning spacecraft. The dual-channel control method proposed in this paper can control a chaotic system to an appointed equilibrium point or a periodic orbit.



---

*Source: 5958768-2020-11-21.xml*