# Delayed Feedback Control of Hidden Chaos in the Unified Chaotic System between the Sprott C System and Yang System

Authors: Huijian Zhu; Lijie Li
Journal: Complexity
 (2021)

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## Abstract

In this paper, the influence of delayed feedback on the unified chaotic system from the Sprott C system and Yang system is studied. The Hopf bifurcation and dynamic behavior of the system are fully studied by using the central manifold theorem and bifurcation theory. The explicit formula, bifurcation direction, and stability of the periodic solution of bifurcation are given correspondingly. The Hopf bifurcation diagram and chaotic phenomenon are also analyzed by numerical simulation to prove the correctness of the theory. It shows that this delay control can only be applied to the hidden chaos with two stable equilibria.


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## Body

## 1. Introduction

Since Lorentz inadvertently discovered chaos in a three-dimensional autonomous system [1] in 1963, more and more scholars began to study the chaos of various systems. Sprott [2–4] found nineteen simple chaotic systems, which may have no equilibra, one equilibrium, or two equilibra. Some classical three-dimensional autonomous chaotic systems, such as Lorentz system [1] and Chen system [5], have a saddle point and two unstable saddle foci. Other three-dimensional chaotic systems [6] have two unstable saddle foci. A chaotic system with one saddle and two stable node foci was discovered by Yang and Chen [7]. Some current studies on these systems including theoretical proof and numerical simulation can be found in the literature [8–19].In 2000, Yang, Wei, and Chen [20] introduced a new three-dimensional chaotic system that is very similar to Lorentz system and Chen system, but it has only two stable node foci. The related types of chaotic systems have been analyzed and numerically studied in detail [21–24]. It has become very important to study the local and global properties of systems with chaotic phenomena. As it is known, hidden attractors are known as the kind of attractors whose domain of attraction is outside the equilibrium points; meanwhile, the domain of attraction in self-excited attractors is related to the unstable equilibrium points [25–27]. Finding hidden attractors in complex variable chaotic systems is even more difficult than finding their real variable counterparts. In recent years, an increasing number of scholars have paid attention to the control and utilization of chaos [28–31] for stabilizing the system with chaotic behaviors. It is also worth noting that theories and methods of controlling hidden chaos in continuous dynamical systems have been developed. This paper mainly studies the following unified chaotic system from [2, 20, 24]:(1)x˙=ay−xy˙=−cy−xz+kyt−τ−yz˙=−b+k1xy+k2y2,,where a,b,andc are the positive real parameters, k,k1,andk2 are the control parameters, and τ is the delay parameter. For parameter values a,b,c,k,k1,k2=1,1,0,0,0,1, system (1) becomes the Sprott C system. For parameter values k,k1,k2=0,1,0, system (1) is a general expression for the Yang system. Moreover, for parameter values a,b,c,k,k1,k2=10,100,9.8,0,0.08,0.01, system (1) without delay has two stable equilibria, whose three characteristic values are λ1=−19.3894 and λ2,3=−0.2053±10.1542i. Figures 1(a) and 1(b), respectively, show time series and the projection of chaotic attractors on the y−z plane. Therefore, system (1) has a hidden attractor coexisting with two stable node foci for initial values 0.98,−1.82,0.49. Therefore, we want to consider the stability of system (1) with direct delay feedback from theoretical analysis and numerical study. At the same time, when all equilibria are stable, we want to show the results that system (1) will turn to chaotic attractor from Hopf bifurcation [32].Figure 1
For parameter valuesa,b,c,k,k1,k2=10,100,9.8,0,0.08,0.01, system (1) without delay has two stable equilibria: (a) system (1) without delay tends to stable equilibrium when IC=30,30,−9; (b) system (1) without delay has a chaotic attractor when IC=0.98,−1.82,0.01.
(a)(b)The organization of this paper is as follows: In Section2, the bifurcation conditions of Hopf bifurcation in delayed system (1) are discussed. In Section 3, based on the central manifold theorem and bifurcation theory, the direction and stability of Hopf bifurcation are analyzed in detail. In Section 4, numerical simulations illustrate our theoretical results. Finally, the conclusion is given in Section 5.


## 2. Existence of Hopf Bifurcation in System (1)

Ifk1+k2>0, system (1) possesses two equilibria E1,2±b/k1+k2,±b/k1+k2,−c. Because of the symmetry of E1 and E2, it is sufficient to analyze the properties of only one of them. So, the rest of the discussion is going to be about E1. By the following linear transformation to shift E1 to the origin,(2)x1=x−bk1+k2,y1=y−bk1+k2,z1=x+c,the controlled system (1) is(3)x˙1=ay1−x1,y˙1=cx1−y1−bk1+k2z1−x1z1+ky1t−τ−y1,z˙1=bk1k1+k2x1+bk1+2k2k1+k2y1+k1x1y1+k2y12..The characteristic equation corresponding to the linear matrix of equation (3) is(4)λ3+a+c+kλ2+k1+2k2k1+k2b+akλ+2ab−λ2+aλke−λτ=0.Whenτ=0, equation (4) becomes(5)λ3+a+cλ2+k1+2k2k1+k2bλ+2ab=0.According to the Routh–Hurwitz criterion, in equation (5), under the following conditions, there are three roots in the negative real part:(6)k2>max−k1,−12c−ak1.The classification conditions of reference equilibrium point,E1, and E2 are local stable nodes or focal points.For the sake of analysis, let us reduce equation (4) to(7)λ3+a2λ2+a1λ+a0+b2λ2+b1λe−λτ=0,where a2=a+c+k,a1=k1+2k2/k1+k2b+ak,a0=2ab,b2=−k,andb1=−ak.Since Hopf bifurcations must have a pair of pure imaginary roots at the system equilibrium point, we might as well establish system (1) having a pair of pure imaginary roots, that is, iρ, so we can substitute the roots into equation (4):(8)−ρ3i−a2ρ2+a1ρi+a0+−b2ρ2+b1ρicosρτ−isinρτ=0,where we separate the real part from the imaginary part:(9)ρ3−a1ρ=b2ρ2sinρτ+b1ρcosρτ,a2ρ2−a0=b1ρsinρτ−b2ρ2cosρτ,which leads to(10)ρ6+a22−b22−2a1ρ4+a12−2a0a2−b12ρ2+a02=0.Lets=ρ2 and let us denote p=a22−b22−2a1,q=a12−2a0a2−b12, and r=a02; then, equation (10) becomes(11)s3+ps2+qs+r=0.Let(12)hs=s3+ps2+qs+r.From equation (11), we have(13)h′s=3s2+2ps+q.DenoteΔ=p2−3q. When Δ>0 and r>0, we can solve for two real roots of equation h′s=0 as follows:(14)s1∗=−p+Δ3,s2∗=−p−Δ3.Noticing thatr=a02>0 and lims⟶+∞hs=+∞, we can get results similar to [9].Lemma 1.
The following results hold:(1)
Equation (11) does not have positive real roots if Δ=p2−3q≤0(2)
Otherwise, if and only ifs1∗=−p+Δ/3>0 and hs1∗≤0, equation (11) has positive roots
From the second point of Lemma1, we can make an assumption to obtain the two positive roots of equation (11), and when s1<s2, then h′s1<0 and h′s2>0:(15)Δ=p2−3q>0,s1∗=−p+Δ3>0,hs1∗<0.Substitutingρk=kk=1,2 into equation (9), we have(16)τkj=1ρkarccosP+2jπ,Q≥0,1ρk2π−arccosP+2jπ,Q<0,where(17)P=b1ρk2−a1b1+a0b2−a2b2ρk2b22ρk2+b12,Q=b2ρk4−a1b2ρk2+a2b1ρk2−a0b1ρkb22ρk2+b12,and j=0,1,…. The following lemma comes naturally.Lemma 2.
If Lemma2 holds, when τ=τkjk=1,2;j=0,1,2,…, the root of a system (7) consists of a number of pure imaginary roots and nonzero real parts.By substitutingλτ into equation (7) and taking the derivative of τ, we can obtain(18)dReλdττ=τkj−1=Re3λ2+2a2λ+a1λb2λ2+b1λe−λττ=τkj+Re2b2λ+b1λb2λ2+b1λτ=τkj=Re3λ2+2a2λ+a1−ρk2b1+ib2ρkcosρkτkj+isinρkτkj+Re2b2λ+b1−ρk2b1+ib2ρk=skΛ3ρk4+2a22−b22−2a1ρk2+a12−2a0a2−b12=skΛh′sk,where Λ=ρk4b12+b22ρk2. Since sk>0, we conclude that dReλ/dττ=τkj−1 and h′sk have the same sign. Note that h′s1<0 and h′s2>0.Thus, the following crucial lemma can be obtained.Lemma 3.
If (15) holds, then the transversality condition of Hopf bifurcation holds: dReλ/dττ=τ1j−1<0,dReλ/dττ=τ2j−1>0, where j=0,1,….The results discussed above and the basic conditions for Hopf bifurcation (transversality and nondegradation) are also applicable to differential equations with time delay, and the important theorem in this section holds [33].Theorem 1.
Suppose that (6) and (15) are satisfied, then system (1) undergoes a Hopf bifurcation at the equilibria E1,2 when τ=τkjk=1,2;j=0,1,2,…. Moreover, if τ10>τ20, then there exists m∈N such that τ20<τ10<τ21<τ11<⋅⋅⋅<τ2m<τ1m<τ2m+1<τ2m+2<τ1m+1 and equilibria E1,2 of system (1) are asymptotically stable for τ∈0,τ20∪τ10,τ21∪⋯∪τ1m−1,τ2m∪τ1m,τ2m+1 and unstable for τ∈τ20,τ10∪τ21,τ11∪⋯∪τ2m,τ1m∪τ2m+1,+∞. Furthermore, system (1) undergoes a Hopf bifurcation at the equilibria E1,2 when τ=τkjk=1,2;j=0,1,….Remark.
Theorem 3 shows that when the delay passes a certain critical value, the chaotic attractor generated by system (1) with only two stable node foci can be transformed into stable and unstable periodic orbits or another chaotic attractor. Thus, the chaos generated by system (1) is controllable.


## 3. Direction and Stability of Hopf Bifurcation

The Hopf bifurcation theory of the smooth autonomous system has been very advanced [33–35]. In this section, we use bifurcation theory to study Hopf bifurcation of system (1), determine the bifurcation direction and stability, and obtain the corresponding parameter conditions through detailed calculation. Due to the symmetry of the equilibrium point, we only study the Hopf bifurcation of E1 at τ=τk.Letm1=x−x0,m2=y−y0,m3=z−z0,mi¯t=miτt,andτ=ω+τk; for convenience, we are dropping the bars. Nonlinear system (1) can be transformed into an FDE in C∈C−1,0,R3 as(19)m˙t=Lωmt+gω,mt,where mt=m1t,m2t,m3tT∈R3 and Lω:C⟶R3 and g:R×C⟶R are given, respectively, by(20)Lωϕ=ω+τk−aa0c−k−c−bk1+k2bk1k1+k2bk1+2k2k1+k20ϕ10ϕ20ϕ30+ω+τk0000k0000ϕ1−1ϕ2−1ϕ3−1,gω,mt=ω+τk0−ϕ10ϕ30k1ϕ10ϕ20+k2ϕ220.Based on the Reese representation theorem in functional analysis,(21)Lωϕ=∫−10dμε,ωϕε,forϕ∈C.where με,ω is a bounded variation function in ε∈−1,0 and can be selected as(22)με,ω=ω+τk−aa0c−k−c−bk1+k2bk1k1+k2bk1+2k2k1+k20δε−ω+τk0000k0000δε+1,where δ⋅ is a Dirac function.Let us defineAωϕ and Rωϕ on ϕ∈C−1,0,R3, and let mtε=mt+ε,ε∈−1,0. Rewrite the system as equation (31):(23)Aωϕ=dϕεdε,ε∈−1,0,∫−10dμε,sϕs,ε=0,and(24)Rωϕ=0,ε∈−1,0,gω,ϕ,ε=0,m˙t=Aωmt+Rωmt.Forψ∈C10,1,R3∗, we can define A∗ψθ; when θ=0, A∗ψθ=∫−10dμTt,0ψ−t; Otherwise, when θ∈0,1,A∗ψθ=−dψθ/dθ and we get a bilinear inner product(25)ψ,ϕ=ψ¯0ϕ0−∫−10∫ξ=0εψ¯ξ−εdμεϕξdξ,where με=με,0.According to the properties of matrix eigenvalues, we can get that the eigenvalues ofA∗ are the same as those of A0. Therefore, the eigenvalue of A0 is ±iρkτk. We need to calculate the iρkτk and −iρkτk corresponding to the eigenvectors of A0 and A∗. Let A0qε=iρkτkqε, i.e., qε=eiερkτk1,α,βT, be the eigenvectors of A0; then, we have(26)iρk+a−a0−ciρk+k+c−ke−iρkτkbk1+k2−bk1k1+k2−bk1+2k2k1+k2iρkq0=000.Through calculation, there are(27)q0=1,α,βT=1,a+iρka,b−2iak1+k2+k1+2k2wkak1+k2wkT.Similarly, we can assume thatq∗θ=eiθρkτkD1,α∗,β∗ is the eigenvector of A∗ corresponding to −iρkτk, that is, A∗q∗θ=−iρkτkq∗θ, and we have(28)q∗θ=D1,α∗,β∗eiθρkτk=D1,k1+k2a−iρkρk−ibk1+ck1+k2ρk,bk1+k2a−iρkbk1+ick1+k2ρkeiθρkτk.By (25), we can replace ψ and ϕ with q0 and q∗θ. At this time, <q∗θ,qε>=1, so we can calculate D, shown as follows:(29)q∗θ,qε=D¯1+αα∗¯+ββ∗¯+kτkαα∗¯e−iρkτk=1.Therefore, we can obtain(30)D=11+αα∗¯+ββ∗¯+kτkαα∗¯e−iρkτk.The central manifoldC0 must be calculated at ω=0. We can make mt the solution of (29) when ω=0. Define(31)zt=q∗,mt,Gt,ε=mtε−2Reztqε.Then,(32)Gt,ε=Gzt,z¯t,ε=G20εz22+G11εzz¯+G02εz¯22+G30εz36+⋯,where z and z¯ are the local coordinates for the center manifold C0 in the directions of q∗ and q¯∗. Note that since mt is real, then G is also real, so we only deal with real solutions. For solution mt∈C0, since ω=0, we have(33)z˙t=iρkτkz+q∗ε,f0,Gzt,z¯t,ε+2Reztqε=iρkτkz+q∗0f0,Gzt,z¯t,0+2Reztq0.Letg0,Gzt,z¯t,0+2Reztq0=g0z,z¯; then,(34)z˙t=iρkτkz+q∗0g0z,z¯.Now, we consider(35)z˙t=iρkτkz+fz,z¯,where(36)fz,z¯=f20z22+f11zz¯+f02z¯22+f21z2z¯2+⋯.Sinceqε=1,α,βTeiερkτk and mtε=m1tε,m2tε,m3tε=Gt,ε+ztqε+z¯tq¯ε, we have(37)m1t0=z+z¯+G201z22+G111zz¯+G021z¯22+⋯,m2t0=αz+α¯z¯+G202z22+G112zz¯+G022z¯22+⋯,m3t0=βz+β¯z¯+G203z22+G113zz¯+G023z¯22+⋯.From (36), we have(38)fz,z¯=q∗0g0z,z¯=D¯τk1,α∗,β∗0−m1t0m3t0k1m1t0m2t0+k2m2t20=D¯τk−α∗z+z¯+G201z22+G111zz¯+G021z¯22+⋯βz+β¯z¯+G203z22+G113zz¯+G023z¯22+⋯+k1β∗z+z¯+G201z22+G111zz¯+G021z¯22+⋯αz+α¯z¯+G202z22+G112zz¯+G022z¯22+⋯+k2β∗αz+α¯z¯+G202z22+G112zz¯+G022z¯22+⋯2.Comparing the coefficients with (36), we have(39)f20=2D¯τkk1αβ∗¯−βα∗¯+k2α2β∗,f11=D¯τk2k1β∗Reα−2α∗¯Reβ+2k2β∗αα¯,f02=2D¯τkk1β∗¯α¯−α∗¯β¯+k2α¯2,f21=−D¯τkα∗2G1130+G2030+2βG1110+β¯G2010+k1D¯τkβ∗2G1120+G2020+2αG1110+α¯G2010+2k2D¯τkβ∗α¯G2020+2αG1120.Then, we need to computeG20ε and G11ε. From (24) and (31), we have(40)G˙=m˙t−z˙q−z¯˙q¯=A0G−2Req∗¯0g0qε,ε∈−1,0A0G−2Req∗¯0g0qε+g0,ε=0.Let(41)Mz,z¯,ε=2Req∗¯0g0qε,ε∈−1,02Req∗¯0g0qε+g0,ε=0..We can rewrite (40) as(42)G˙=A0G+Mz,z¯,ε,where(43)Mz,z¯,ε=M20εz22+M11εzz¯+M02εz¯22+⋯.According to the definition of (40) and (43) and G, using the series expansion and comparison of coefficients, we have(44)A0−2iρkτkG20ε=−M20ε,A0G11ε=−M11ε,….From (40), we know that for ε∈−1,0,(45)Mz,z¯,ε=−q∗¯0g0qε−q∗0g0¯q¯ε=−fz,z¯qε−f¯z,z¯q¯ε.Comparing the coefficients of equation (45) with those of equation (43), we have(46)M20ε=−f20qε−f¯02q¯ε,M11ε=−f11qε−f¯11q¯ε.Therefore, the following equation can be obtained:(47)G20.=2iρkτkG20ε+f20qε+f¯02q¯ε.Fromqε=1,α,βTeiεωkτk, we can calculate the solution of the previous equation:(48)G20ε=if20ρkτkq0eiερkτk+if¯023ρkτkq¯0e−iερkτk+W1e2iερkτkand, similarly,(49)G11ε=if11ρkτkq0eiερkτk+if¯11ρkτkq¯0e−iερkτk+W2,where W1=W11,W12,W13T∈R3 and W2=W21,W22,W23T∈R3 are the constant vectors corresponding to the initial conditions.We find the values ofW1 and W2 now. For (44), we have(50)G˙20ε=∫−10dμεG20ε=2iερkτkG200−M200and(51)G˙11ε=∫−10dμεG11ε=−M110,where με=με,0. From equation (40), we have(52)M200=−f20q0−f¯02q¯0+2τk0,−β,k1α+k2α2Tand(53)M110=−f11q0−f¯11q¯0+2τk0,−Reβ,k1Reα+k2αα¯T.The eigenvector corresponding to eigenvalueiρkτk by A0 is q0. We obtain(54)iρkτk−∫−10eiερkτkdμfq0=0,−iρkτk−∫−10e−iερkτkdμfq¯0=0.Substituting equations (48) and (52) into equation (50), we obtain(55)2iρkτkI−∫−10e2iερkτkdμfE1=2τk0,−β,k1α+k2α2T.That is,(56)2iρk+a−a0−c2iρk+k+c−ke−iρkτkbk1+k2−bk1k1+k2−bk1+2k2k1+k22iρkE1=20−βk1α+k2α2.It follows that(57)W11=Δ11Δ1,W12=Δ12Δ1,W13=Δ13Δ1,where(58)Δ11=20−a0−β2iρk+k+c−ke−iρkτkbk1+k2k1α+k2α2−bk1+2k2k1+k22iρk,Δ12=22iρk+a00−c−βbk1+k2−bk1k1+k2k1α+k2α22iρk,Δ13=22iρk+a−a0−c2iρk+k+c−ke−iρkτk−β−bk1k1+k2−bk1+2k2k1+k2k1α+k2α2,Δ1=2iρk+a−a0−c2iρk+k+c−ke−iρkτkbk1+k2−bk1k1+k2−bk1+2k2k1+k22iρk.Similarly, substituting equations (49) and (53) into equation (51), we have(59)a−a0−ccbk1+k2−bk1k1+k2−bk1+2k2k1+k20E2=20−Reβk1Reα+k2αα¯.It follows that(60)W21=Δ21Δ2,W22=Δ22Δ2,W23=Δ23Δ2,where(61)Δ21=20−a0−Reβcbk1+k2k1Reα+k2αα¯−bk1+2k2k1+k20,Δ22=2a00−c−Reβbk1+k2−bk1k1+k2k1Reα+k2αα¯0,Δ23=2a−a0−cc−Reβ−bk1k1+k2−bk1+2k2k1+k2k1Reα+k2αα¯,Δ2=a−a0−ccbk1+k2−bk1k1+k2−bk1+2k2k1+k20.We can determineG200 and G110, and therefore, all fij can be determined by (39).In summary, the properties of Hopf bifurcation are determined by the following parameters:ω2 determines the direction of Hopf bifurcation, β2 determines the stability of bifurcation periodic solutions, and T2 determines the period of bifurcation periodic solutions, and the specific values are shown as follows. The main theories and methods are from [34,35]:(62)C10=i2ρkτkf20f11−2f112−13f022+f212,ω2=−ReC10Redλτk/dτ,T2=−ImC10+ω2Imdλτk/dτρkτk,β2=2ReC10.Therefore, the following main results are obtained in this section.Theorem 2.
In equation (62), when τ>τkorτ<τk, system (1) has Hopf bifurcations with the following properties: if ω2>0ω2<0, the Hopf bifurcation is supercritical (subcritical); if β2<0β2>0, the orbit is stable (unstable); if T2>0T2<0, then the period increases (decreases).


## 4. Numerical Results

In the previous two sections, we have proved the parameter conditions for Hopf bifurcation in system (1) and analyzed the bifurcation direction and bifurcation stability. In this section, we select appropriate parameters and use the MATLAB toolkit for numerical simulation to verify our theoretical analysis. When the parameter values are a=10,c=9.8,andb=100, the two equilibria are E1100/3,100/3,−9.8 and E2−100/3,−100/3,−9.8.(63)x˙=10y−x,y˙=−9.8y−xz+kyt−τ−yz˙=−100+xy.,Through the previous analysis and numerical simulation from the finite difference method (FDM), as shown in Figure1, the equilibrium points E1,2 of system (63) are asymptotically stable when τ=0 and a chaotic attractor appears. If we choose the parameter k=−0.5, we can get that equation (10) has two positive roots ρ1=10.0521 and ρ2=10.3648. Therefore, there are, respectively,(64)τ1j=0.467983+0.625063j,τ2j=0.211299+0.606205j,where j=0,1,2,…. From formula (62), it follows that τ20=0.211299, C10=0.000588642+0.000658409i,ω2=−0.000362443,T2=−0.000204142,andβ2=0.00117728. Therefore, as shown by computer simulation, when 0<τ<τ20, the equilibria E1,2 are stable (see Figure 2). E1 loses its stability and Hopf bifurcation occurs as τ crosses the critical value τ20. According to the properties of Hopf bifurcation, Hopf bifurcation is subcritical and the bifurcation direction is τ<τ20, when ω2<0 and β2>0. At this time, an unstable bifurcation periodic solution appears, as shown in Figures 3(a) and 3(b).Figure 2
For system (1) with parameter values a,b,c,k,k1,k2=10,100,9.8,−0.5,0.08,0.01 and initial values 30,30,−9 when τ=0.16, the equilibrium E1 is asymptotically stable.Figure 3
A bifurcation diagram for system (1) with parameter values a,b,c,k,k1,k2=10,100,9.8,−0.5,0.08,0.01 and initial values 30,30,−9 when τ=0.2095<τ20 is close to τ20: (a) phase portrait; (b) time series of zt.
(a)(b)Otherwise, as shown in Figures4(a) and 4(b), numerical simulation shows that when τ reaches the region τ>τ20, Hopf bifurcation periodic solution disappears and chaos occurs.Figure 4
Whenτ is closer to τ20, τ=0.23>τ20 is taken and system (1) is chaotic at the initial values 30,30,−9: (a) the chaotic attractor of system (1); (b) the time series of zt.
(a)(b)


## 5. Conclusion

In previous studies, few scholars have analyzed the time-delay feedback chaotic system with two stable node foci coexisting. In this paper, a unified chaotic system control model is established by using the delay feedback control law. The corresponding parameter range is obtained according to the conditions of Hopf bifurcation. Central manifold theory and normal form method are the most classical methods to study the properties of Hopf bifurcation. This paper also uses this method to study the direction of Hopf bifurcation and the stability of bifurcation periodic solution of system (1). Theoretical results and numerical simulation show that chaos can be controlled using a delay system (1). Numerical simulation shows that the periodic solution is transformed into a chaotic attractor with further increase in delay. It is worth noting that the results obtained in this paper are of great significance for controlling chaos in systems with only two stable node foci. The dynamic behavior of the new system is still rich and complex, and its topology needs to be thoroughly studied and developed. In future studies, we will provide more credible theoretical analysis and data results.



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