Research article

A hyperchaos generated from Rabinovich system

  • Received: 08 July 2022 Revised: 29 September 2022 Accepted: 13 October 2022 Published: 20 October 2022
  • MSC : 34K18, 65P20

  • In this paper, we present a 4D hyperchaotic Rabinovich system which obtained by adding a linear controller to 3D Rabinovich system. Based on theoretical analysis and numerical simulations, the rich dynamical phenomena such as boundedness, dissipativity and invariance, equilibria and their stability, chaos and hyperchaos are studied. In addition, the Hopf bifurcation at the zero equilibrium point of the 4D Rabinovich system is investigated. The numerical simulations, including phase diagrams, Lyapunov exponent spectrum, bifurcations, power spectrum and Poincaré maps, are carried out in order to analyze and verify the complex phenomena of the 4D Rabinovich system.

    Citation: Junhong Li, Ning Cui. A hyperchaos generated from Rabinovich system[J]. AIMS Mathematics, 2023, 8(1): 1410-1426. doi: 10.3934/math.2023071

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  • In this paper, we present a 4D hyperchaotic Rabinovich system which obtained by adding a linear controller to 3D Rabinovich system. Based on theoretical analysis and numerical simulations, the rich dynamical phenomena such as boundedness, dissipativity and invariance, equilibria and their stability, chaos and hyperchaos are studied. In addition, the Hopf bifurcation at the zero equilibrium point of the 4D Rabinovich system is investigated. The numerical simulations, including phase diagrams, Lyapunov exponent spectrum, bifurcations, power spectrum and Poincaré maps, are carried out in order to analyze and verify the complex phenomena of the 4D Rabinovich system.



    When studying various applied problems related to the properties of media with a periodic structure, it is necessary to study differential equations with rapidly changing coefficients. Equations of this type are often found, for example, in electrical systems under the influence of high frequency external forces. The presence of such forces creates serious problems for the numerical integration of the corresponding differential equations. Therefore, asymptotic methods are usually applied to such equations, the most famous of which are the Feshchenko – Shkil – Nikolenko splitting method [9,10,11,12,23] and the Lomov's regularization method [18,20,21]. The splitting method is especially effective when applied to homogeneous equations, and in the case of inhomogeneous differential equations, the Lomov regularization method turned out to be the most effective. However, both of these methods were developed mainly for singularly perturbed equations that do not contain an integral operator. The transition from differential equations to integro-differential equations requires a significant restructuring of the algorithm of the regularization method. The integral term generates new types of singularities in solutions that differ from the previously known ones, which complicates the development of the algorithm for the regularization method. The splitting method, as far as we know, has not been applied to integro-differential equations. In this article, the Lomov's regularization method [1,2,3,4,5,6,7,8,13,14,15,16,17,19,24] is generalized to previously unexplored classes of integro-differential equations with rapidly oscillating coefficients and rapidly decreasing kernels of the form

    Lεz(t,ε)εdzdtA(t)zεg(t)cosβ(t)εB(t)ztt0e1εtsμ(θ)dθK(t,s)z(s,ε)ds=h(t),z(t0,ε)=z0,t[t0,T], (1.1)

    where z={z1,z2},h(t)={h1(t),h2(t)},μ(t)<0(t[t0,T]), g(t) is the scalar function, A(t) and B(t) are (2×2)-matrices, moreover A(t)=(01ω2(t)0), ω(t)>0,β(t)>0 is the frequency of the rapidly oscillating cosine, z0={z01,z02},ε>0 is a small parameter. It is precisely such a system in the case β(t)=2γ(t),B(t)=(0010) and in the absence of an integral term was considered in [18,20,21].

    The functions λ1(t)=iω(t), λ2(t)=+iω(t) form the spectrum of the limit operator A(t), the function λ5(t)=μ(t) characterizes the rapid change in the kernel of the integral operator, and the functions λ3(t)=iβ(t), λ4(t)=+iβ(t) are associated with the presence of a rapidly oscillating cosine in the system (1.1). The set {Λ}={λ1(t),...,λ5(t)} is called the spectrum of problem (1.1). Such systems have not been considered earlier and in this paper we will try to generalize the Lomov's regularization method [18] to systems of type (1.1).

    We introduce the following notations:

    λ(t)=(λ1(t),...,λ5(t)),

    m=(m1,...,m5) is a multi-index with non-negative components mj,j=¯1,5,

    |m|=5j=1mj is the height of multi-index m,

    (m,λ(t))=5j=1mjλj(t).

    The problem (1.1) will be considered under the following conditions:

    1)ω(t),μ(t),β(t)C([t0,T],R),ω(t)β(t)t[t0,T],

    g(t)([t0,T],C1),h(t)C([t0,T],C2),

    B(t)C([t0,T],C2×2),K(t,s)C({t0stT},C2×2);

    2) the relations (m,λ(t))=0,(m,λ(t))=λj(t),j{1,...,5} for all multi-indices m with |m|2 or are not fulfilled for any t[t0,T], or are fulfilled identically on the whole segment [t0,T]. In other words, resonant multi-indices are exhausted by the following sets:

    Γ0={m:(m,λ(t))0,|m|2,t[t0,T]},Γj={m:(m,λ(t))λj(t),|m|2,t[t0,T]},j=¯1,5.

    Note that by virtue of the condition ω(t)β(t), the spectrum {Λ} of the problem (1.1) is simple.

    Denote by σj=σj(ε) independent on t the quantities σ1=eiεβ(t0), σ2=e+iεβ(t0) and rewrite the system (1.1) in the form

    Lεz(t,ε)εdzdtA(t)zεg(t)2(eiεtt0β(θ)dθσ1+e+iεtt0β(θ)dθσ2)B(t)ztt0e1εtsμ(θ)dθK(t,s)z(s,ε)ds=h(t),z(t0,ε)=z0,t[t0,T]. (2.1)

    We introduce regularizing variables (see [18])

    τj=1εtt0λj(θ)dθψj(t)ε,j=¯1,5 (2.2)

    and instead of the problem (2.1) we consider the problem

    Lε˜z(t,τ,ε)ε˜zt+5j=1λj(t)˜zτjA(t)˜zεg(t)2(eτ3σ1+eτ4σ2)B(t)˜z
    tt0e1εtsμ(θ)dθK(t,s)˜z(s,ψ(s)ε,ε)ds=h(t),˜z(t,τ,ε)|t=t0,τ=0=z0,t[t0,T] (2.3)

    for the function ˜z=˜z(t,τ,ε), where it is indicated (according to (2.2)): τ=(τ1,...,τ5),ψ=(ψ1,...,ψ5). It is clear that if ˜z=˜z(t,τ,ε) is the solution of the problem (2.3), then the vector function z=˜z(t,ψ(t)ε,ε) is the exact solution of the problem (2.1), therefore, the problem (2.3) is expansion of the problem (2.1). However, it cannot be considered completely regularized, since the integral term

    J˜zJ(˜z(t,τ,ε)|t=s,τ=ψ(s)/ε)=tt0e1εtsμ(θ)dθK(t,s)˜z(s,ψ(s)ε,ε)ds

    has not been regularized in (2.3).

    To regularize the integral term, we introduce a class Mε, asymptotically invariant with respect to the operator J˜z (see [18], p. 62]). We first consider the space of vector functions z(t,τ), representable by sums

    z(t,τ,σ)=z0(t,σ)+5i=1zi(t,σ)eτi+2|m|Nzzm(t,σ)e(m,τ),z0(t,σ),zi(t,σ),zm(t,σ)C([t0,T],C2),i=¯1,5,2|m|Nz, (2.4)

    where the asterisk above the sum sign indicates that in it the summation for |m|2 occurs only over nonresonant multi-indices m=(m1,...,m5), i.e. over m5i=0Γi. Note that in (2.4) the degree of the polynomial with respect to exponentials eτj depends on the element z. In addition, the elements of the space U depend on bounded in ε>0 constants σ1=σ1(ε) and σ2=σ2(ε), which do not affect the development of the algorithm described below, therefore, henceforth, in the notation of element (2.4) of this space U, we omit the dependence on σ=(σ1,σ2) for brevity. We show that the class Mε=U|τ=ψ(t)/ε is asymptotically invariant with respect to the operator J.

    The image of the operator J on the element (2.4) of the space U has the form:

    Jz(t,τ)=tt0e1εtsλ5(θ)dθK(t,s)z0(s)ds+5i=1tt0e1εtsλ5(θ)dθK(t,s)zi(s)e1εst0λi(θ)dθds
    +2|m|Nztt0e1εtsλ5(θ)dθK(t,s)zm(s)e1εst0(m,λ(θ))dθds
    =tt0e1εtsλ5(θ)dθK(t,s)z0(s)ds+e1εtt0λ5(θ)dθtt0K(t,s)z5(s)ds
    +5i=1,i5e1εtt0λ5(θ)dθtt0K(t,s)zi(s)e1εst0(λi(θ)λ5(θ))dθds
    +2|m|Nze1εtt0λ5(θ)dθtt0K(t,s)zm(s)e1εst0(me5,λ(θ))dθds.

    Integrating in parts, we have

    J0(t,ε)=tt0K(t,s)z0(s)e1εtsλ5(θ)dθds=εtt0K(t,s)z0(s)λ5(s)de1εtsλ5(θ)dθ
    =εK(t,s)z0(s)λ5(s)e1εtsλ5(θ)dθ|s=ts=t0εtt0(sK(t,s)z0(s)λ5(s))e1εtsλ5(θ)dθds
    =ε[K(t,t0)z0(t0)λ5(t0)e1εtt0λ5(θ)dθK(t,t)z0(t)λ5(t)]+εtt0(sK(t,s)z0(s)λ5(s))e1εtsλ5(θ)dθds.

    Continuing this process further, we obtain the decomposition

    J0(t,ε)=ν=0εν+1[(Iν0(K(t,s)z0(s)))s=t0e1εtt0λ5(θ)dθ(Iν0(K(t,s)z0(s)))s=t],I00=1λ5(s),Iν0=1λ5(s)sIν10(ν1).

    Next, apply the same operation to the integrals:

    J5,i(t,ε)=e1εtt0λ5(θ)dθtt0K(t,s)zi(s)e1εst0(λi(θ)λ5(θ))dθds
    =εe1εtt0λ5(θ)dθtt0K(t,s)zi(s)λi(s)λ5(s)de1εst0(λi(θ)λ5(θ))dθ
    =εe1εtt0λ5(θ)dθ[K(t,s)zi(s)λi(s)λ5(s)e1εst0(λi(θ)λ5(θ))dθ|s=ts=t0εtt0(sK(t,s)zi(s)λi(s)λ5(s))e1εst0(λi(θ)λ5(θ))dθds]
    =ε[K(t,t)zi(t)λi(t)λ5(t)e1εtt0λi(θ)dθK(t,t0)zi(t0)λi(t0)λ5(t0)e1εtt0λ5(θ)dθ]
    εe1εtt0λ5(θ)dθtt0(sK(t,s)zi(s)λi(s)λ5(s))e1εst0(λi(θ)λ5(θ))dθds
    =ν=0(1)νεν+1[(Iνi(K(t,s)zi(s)))s=te1εtt0λi(θ))dθ(Iνi(K(t,s)zi(s)))s=t0e1εtt0λ5(θ)dθ],
    I0i=1λi(s)λ5(s),Iνi=1λi(s)λ5(s)sIν1i(ν1),i=¯1,4;
    Jm(t,ε)=e1εtt0λ5(θ)dθtt0K(t,s)zm(s)e1εst0(me5,λ(θ))dθds
    =εe1εtt0λ5(θ)dθtt0K(t,s)zm(s)(me5,λ(s))de1εst0(me5,λ(θ))dθ=εe1εtt0λ5(θ)dθ[K(t,s)zm(s)(me5,λ(s))e1εst0(me5,λ(θ))dθ|s=ts=t0
    εtt0(sK(t,s)zm(s)(me5,λ(s)))e1εst0(me5,λ(θ))dθds]
    =ε[K(t,t)zm(t)(me5,λ(t))e1εtt0(m,λ(θ))dθK(t,t0)zm(t0)(me5,λ(t0))e1εtt0λ5(θ)dθ]
    εe1εtt0λ5(θ)dθtt0(sK(t,s)zm(s)(me5,λ(s)))e1εst0(me5,λ(θ))dθ
    =ν=0(1)νεν+1[(Iν5,m(K(t,s)zm(s)))s=te1εtt0(m,λ(θ))dθ(Iν5,m(K(t,s)zm(s)))s=t0e1εtt0λ5(θ)dθ],
    I05,m=1(me5,λ(s)),Iν5,m=1(me5,λ(s))sIν15,m(ν1),2|m|Nz.

    Here it is taken into account that (me5,λ(s))0, since by definition of the space U, multi-indices mΓ5. This means that the image of the operator J on the element (2.4) of the space U is represented as a series

    Jz(t,τ)=e1εtt0λ5(θ))dθtt0K(t,s)z5(s)ds+ν=0(1)νεν+1[(Iν0(K(t,s)z0(s)))s=t0e1εtt0λ5(θ))dθ
    (Iν0(K(t,s)z0(s)))s=t]+5i=1,i5ν=0(1)νεν+1[(Iνi(K(t,s)zi(s)))s=te1εtt0λi(θ))dθ
    (Iνi(K(t,s)zi(s)))s=t0e1εtt0λ5(θ)dθ]
    +2|m|Nzν=0(1)νεν+1[(Iν5,m(K(t,s)zm(s)))s=te1εtt0(m,λ(θ))dθ(Iν5,m(K(t,s)zm(s)))s=t0e1εtt0λ5(θ)dθ].

    It is easy to show (see, for example, [22], pages 291–294) that this series converges asymptotically as ε+0 (uniformly in t[t0,T]). This means that the class Mε is asymptotically invariant (as ε+0) with respect to the operator J.

    We introduce the operators Rν:UU, acting on each element z(t,τ)U of the form (2.4) according to the law:

    R0z(t,τ)=eτ5tt0K(t,s)z5(s)ds, (2.50)
    R1z(t,τ)=[(I00(K(t,s)z0(s)))s=t0eτ5(I00(K(t,s)z0(s)))s=t]
    +4i=1[(I0i(K(t,s)zi(s)))s=teτi(I0i(K(t,s)zi(s)))s=t0eτ5] (2.51)
    +2|m|Nz[(I05,m(K(t,s)zm(s)))s=te(m,τ)(I05,m(K(t,s)zm(s)))s=t0eτ5],
    Rν+1z(t,τ)=ν=0(1)ν[(Iν0(K(t,s)z0(s)))s=t0eτ5(Iν0(K(t,s)z0(s)))s=t]
    +4i=1ν=0(1)ν[(Iνi(K(t,s)zi(s)))s=teτi(Iνi(K(t,s)zi(s)))s=t0eτ5] (2.5v+1)
    +2|m|Nzν=0(1)ν[(Iν5,m(K(t,s)zm(s)))s=te(m,τ)(Iν5,m(K(t,s)zm(s)))s=t0eτ5].

    Let now ˜z(t,τ,ε) be an arbitrary continuous function in (t,τ)[t0,T]×{τ:Reτj0,j=¯1,5} with asymptotic expansion

    ˜z(t,τ,ε)=k=0εkzk(t,τ),zk(t,τ)U (2.6)

    converging as ε+0 (uniformly in (t,τ)[t0,T]×{τ:Reτj0,j=¯1,5}). Then the image J˜z(t,τ,ε) of this function is decomposed into an asymptotic series

    J˜z(t,τ,ε)=k=0εkJzk(t,τ)=r=0εrrs=0Rrszs(t,τ)|τ=ψ(t)/ε.

    This equality is the basis for introducing an extension of the operator J on series of the form (2.6):

    ˜J˜z(t,τ,ε)˜J(k=0εkzk(t,τ))def=r=0εrrs=0Rrszs(t,τ). (2.7)

    Although the operator ˜J is formally defined, its usefulness is obvious, since in practice it is usual to construct the N-th approximation of the asymptotic solution of the problem (2.1), in which only N-th partial sums of the series (2.6) will take part, which have not formal, but true meaning. Now we can write down a problem that is completely regularized with respect to the original problem (2.1):

    Lε˜z(t,τ,ε)ε˜zt+5j=1λj(t)˜zτjA(t)˜zεg(t)2(eτ3σ1+eτ4σ2)B˜z˜J˜z=h(t),˜z(t,τ,ε)|t=t0,τ=0=z0,t[t0,T], (2.8)

    were the operator ˜J has the form (2.7).

    Substituting the series (2.6) into (2.8) and equating the coefficients for the same powers of ε, we obtain the following iterative problems:

    Lz0(t,τ)5j=1λj(t)z0τjA(t)z0R0z0=h(t),z0(t0,0)=z0; (3.10)
    Lz1(t,τ)=z0t+g(t)2(eτ3σ1+eτ4σ2)B(t)z0+R1z0,z1(t0,0)=0; (3.11)
    Lz2(t,τ)=z1t+g(t)2(eτ3σ1+eτ4σ2)B(t)z1+R1z1+R2z0,z2(t0,0)=0; (3.12)
    Lzk(t,τ)=zk1t+g(t)2(eτ3σ1+eτ4σ2)B(t)zk1+Rkz0+...+R1zk1,zk(t0,0)=0,k1. (3.1k)

    Each of the iterative problem (3.1k) can be written as

    Lz(t,τ)5j=1λj(t)zτjA(t)zR0z=H(t,τ),z(t0,0)=z, (3.2)

    where H(t,τ)=H0(t)+5j=1Hj(t)eτj+2|m|NHHm(t)e(m,τ) is a well-known vector-function of the space U, z is a well-known constant vector of a complex space C2, and the operator R0 has the form (see (2.50))

    R0zR0(z0(t)+5j=1zj(t)eτj+2|m|Nzzm(t)e(m,τ))=eτ5tt0K(t,s)z5(s)ds.

    In the future we need the λj(t)-eigenvectors of the matrix A(t):

    φ1(t)=(1iω(t)),φ2(t)=(1+iω(t)),

    as well as ˉλj(t)-eigenvectors of the matrix A(t):

    χ1(t)=12(1iω(t)),χ2(t)=12(1+iω(t)).

    These vectors form a biorthogonal system, i.e.

    (φk(t),χj(t))={1,k=j,0,kj(k,j=1,2).

    We introduce the scalar product (for each t[t0,T]) in the space U:

    <z,w>≡<z0(t)+5j=1zj(t)eτj+2|m|Nzzm(t)e(m,τ),w0(t)+5j=1wj(t)eτj
    +2|m|Nwwm(t)e(m,τ)>def=(z0(t),w0(t))+5j=1(zj(t),wj(t))+2|m|min(Nz,Nw)(zm(t),wm(t)),

    where we denote by (,) the ordinary scalar product in a complex space C2. We prove the following statement.

    Theorem 1. Let conditions 1) and 2) are satisfied and the right-hand side H(t,τ)=H0(t)+5j=1Hj(t)eτj+2|m|NHHm(t)e(m,τ) of the system (3.2) belongs to the space U. Then for the solvability of the system (3.2) in U it is necessary and sufficient that the identities

    <H(t,τ),χk(t)eτk>≡0,k=1,2,t[t0,T], (3.3)

    are fulfilled.

    Proof. We will determine the solution to the system (3.2) in the form of an element (2.4) of the space U:

    z(t,τ)=z0(t)+5j=1zj(t)eτj+2|m|NHzm(t)e(m,τ). (3.4)

    Substituting (3.4) into the system (3.2), we have

    5j=1[λj(t)IA(t)]zj(t)eτj+2|m|NH[(m,λ(t))IA(t)]zm(t)e(m,τ)
    A(t)z0(t)eτ5tt0K(t,s)z5(s)ds=H0(t)+5j=1Hj(t)eτj+2|m|NHHm(t)e(m,τ).

    Equating here separately the free terms and coefficients at the same exponents, we obtain the following systems of equations:

    A(t)z0(t)=H0(t), (3.50)
    [λj(t)IA(t)]zj(t)=Hj(t),j=¯1,4, (3.5j)
    [λ5(t)IA(t)]z5(t)tt0K(t,s)z5(s)ds=H5(t), (3.55)
    [(m,λ(t))IA(t)]zm(t)=Hm(t),2|m|Nz,m5j=0Γj. (3.5m)

    Due to the invertibility of the matrix A(t), the system (3.50) has the solution A1(t)H0(t). Since λ5(t)=μ(t) is a real function, and the eigenvalues of the matrix A(t) are purely imaginary, the matrix λ5(t)IA(t) is invertible and therefore the system (3.55) can be written as

    z5(t)=tt0([λ5(t)IA(t)]1K(t,s))z5(s)ds+[λ5(t)IA(t)]1H5(t). (3.6)

    Due to the smoothness of the kernel ([λ5(t)IA(t)]1K(t,s)) and the heterogeneity [λ5(t)IA(t)]1H5(t), this Volterra integral system has a unique solution z5(t)C([t0,T],C2). Systems (3.53) and (3.54) also have unique solutions

    zj(t)=[λj(t)IA(t)]1Hj(t)C([t0,T],C2),j=3,4, (3.7)

    since λ3(t),λ4(t) do not belong to the spectrum of the matrix A(t). Systems (3.51) and (3.52) are solvable in the space C([t0,T],C2) if and only if the identities (Hj(t),χj(t))0t[t0,T],j=1,2 hold.

    It is easy to see that these identities coincide with the identities (3.3).

    Further, since multi-indices m5j=0Γj in systems (3.5m), then these systems are uniquely solvable in the space C([t0,T],C2) in the form of functions

    zm(t)=[(m,λ(t))IA(t)]1Hm(t),0|m|NH. (3.8)

    Thus, condition (3.3) is necessary and sufficient for the solvability of the system (3.2) in the space U. The theorem 1 is proved.

    Remark 1. If identity (3.3) holds, then under conditions 1) and 2) the system (3.2) has (see (3.6) – (3.8)) the following solution in the space U:

    z(t,τ)=z0(t)+5j=1zj(t)eτj+2|m|NHzm(t)e(m,τ)z0(t)+2k=1αk(t)φk(t)eτk
    +h12(t)φ2(t)eτ1++h21(t)φ1(t)eτ2+z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ), (3.9)

    where αk(t)C([t0,T],C1) are arbitrary functions, k=1,2,z0(t)=A1H0(t),z5(t) is the solution of the integral system (3.6) and the notations are introduced:

    h12(t)(H1(t),χ2(t))λ1(t)λ2(t),h21(t)(H2(t),χ1(t))λ2(t)λ1(t),Pj(t)[λj(t)IA(t)]1Hj(t),
    Pm(t)[(m,λ(t))IA(t)]1Hm(t).

    We proceed to the description of the conditions for the unique solvability of the system (3.2) in the space U. Along with the problem (3.2), we consider the system

    Lw(t,τ)=zt+g(t)2(eτ3σ1+eτ4σ2)B(t)z+R1z+Q(t,τ), (4.1)

    where z=z(t,τ) is the solution (3.9) of the system (3.2), Q(t,τ)U is the known function of the space U. The right-hand side of this system:

    G(t,τ)zt+g(t)2(eτ3σ1+eτ4σ2)B(t)z+R1z+Q(t,τ)
    =t[z0(t)+5j=1zj(t)eτj+2|m|NHzm(t)e(m,τ)]
    +g(t)2(eτ3σ1+eτ4σ2)B(t)[z0(t)+5j=1zj(t)eτj+2|m|NHzm(t)e(m,τ)]+R1z+Q(t,τ),

    may not belong to the space U, if z=z(t,τ)U. Since zt,R1z,Q(t,τ)U, then this fact needs to be checked for the function

    Z(t,τ)g(t)2(eτ3σ1+eτ4σ2)B(t)[z0(t)+5j=1zj(t)eτj
    +2|m|NHzm(t)e(m,τ)]=g(t)2B(t)z0(t)(eτ3σ1+eτ4σ2)
    +5j=1g(t)2B(t)zj(t)(eτj+τ3σ1+eτj+τ4σ2)+g(t)2(eτ3σ1+eτ4σ2)B(t)2|m|NHzm(t)e(m,τ).

    Function Z(t,τ)U, since it has resonant exponents

    eτ3+τ4=e(m,τ)|m=(0,0,1,1,0),eτ3+(m,τ)(m3+1=m4,m1=m2=m5=0),
    eτ4+(m,τ)(m4+1=m3,m1=m2=m5=0),

    therefore, the right-hand side G(t,τ)=Z(t,τ)zt+R1z+Q(t,τ) of the system (19) also does not belong to the space U. Then, according to the well-known theory (see [18], p. 234), it is necessary to embed :G(t,τ)ˆG(t,τ) the right-hand side G(t,τ) of the system (4.1) in the space U. This operation is defined as follows.

    Let the function G(t,τ)=N|m|=0wm(t)e(m,τ) contain resonant exponentials, i.e. G(t,τ) has the form

    G(t,τ)=w0(t)+5j=1wj(t)eτj+5j=0N|mj|=2:mjΓjwmj(t)e(mj,τ)+N|m|=2,mmj,j=¯0,5wm(t)e(m,τ).

    Then

    ˆG(t,τ)=w0(t)+5j=1wj(t)eτj+5j=0N|mj|=2:mjΓjwmj(t)eτj+N|m|=2,mmj,j=¯0,5wm(t)e(m,τ).

    Therefore, the embedding operation acts only on the resonant exponentials and replaces them with a unit or exponents eτj of the first dimension according to the rule:

    (e(m,τ)|mΓ0)=e0=1,(e(m,τ)|mΓj)=eτj,j=¯1,5.

    We now turn to the proof of the following statement.

    Theorem 2. Suppose that conditions 1) and 2) are satisfied and the right-hand side H(t,τ)=H0(t)+5j=1Hj(t)eτj+2|m|NHHm(t)e(m,τ)U of the system (3.2) satisfies condition (3.3). Then the problem (3.2) under additional conditions

    <ˆG(t,τ),χk(t)eτk>≡0t[t0,T],k=1,2, (4.2)

    where Q(t,τ)=Q0(t)+5k=1Qk(t)eτk+2|m|NQQm(t)e(m,τ) is the well-known vector function of the space U, is uniquely solvable in U.

    Proof. Since the right-hand side of the system (3.2) satisfies condition (3.3), this system has a solution in the space U in the form (3.9), where αk(t)C([t0,T],C1), k=1,2 are arbitrary functions so far. Subordinate (3.9) to the initial condition z(t0,0)=z. We obtain 2k=1αk(t0)φk(t0)=z, where is indicated

    z=z+A1(t0)H0(t0)[λ5(t0)IA(t0)]1H5(t0)4j=3[λj(t0)IA(t0)]1Hj(t0)
    (H1(t0),χ2(t0))λ1(t0)λ2(t0)φ2(t0)(H2(t0),χ1(t0))λ2(t0)λ1(t0)φ1(t0)2|m|NHPm(t0).

    Multiplying scalarly the equality 2k=1αk(t0)φk(t0)=z by χj(t0) and taking into account the biorthogonality of the systems {φk(t)} and {χj(t)}, we find the values αk(t0)=(z,χk(t0)),k=1,2. Now we subordinate the solution (3.9) to the orthogonality condition (4.2). We write in more detail the right-hand side G(t,τ) of the system (4.1):

    G(t,τ)t[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1
    +h21(t)φ1(t)eτ2+z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]
    +g(t)2(eτ3σ1+eτ4σ2)B(t)[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1
    +h21(t)φ1(t)eτ2+z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]
    +R1[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2
    +z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]+Q(t,τ).

    Putting this function into the space U, we will have

    ˆG(t,τ)t[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1
    +h21(t)φ1(t)eτ2+z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]
    +{g(t)2(eτ3σ1+eτ4σ2)B(t)(z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1
    +h21(t)φ1(t)eτ2+z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)}
    +R1[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2
    +z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]+Q(t,τ)
    =t[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2
    +z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]   ()
    +{12g(t)B(t)(eτ3σ1z0(t)+eτ3+τ1σ1α1(t)φ1(t)+eτ3+τ2σ1α2(t)φ2(t)
    +eτ3+τ1σ1h12(t)φ2(t)+eτ3+τ2σ1h21(t)φ1(t)+eτ3+τ5σ1z5(t)
    +e2τ3σ1P3(t)+eτ3+τ4σ1P4(t)+eτ4σ2z0(t)+eτ4+τ1σ2α1(t)φ1(t)
    +eτ4+τ2σ2α2(t)φ2(t)+eτ4+τ1σ2h12(t)φ2(t)+eτ4+τ2σ2h21(t)φ1(t)
    +eτ4+τ5σ2z5(t)+eτ3+τ4σ2P3(t)+e2τ4σ2P4(t)
    +12g(t)B(t)2|m|NHPm(t)(emτ+τ3σ1+emτ+τ4σ2)}
    +R1[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2
    +z5(t)eτ5+4i=3Pi(t)eτi+2|m|NHPm(t)e(m,τ)]+Q(t,τ).

    Given that the expression R1(z0(t,τ)) linearly depends on α1(t) and α2(t) (see the formula (2.51)):

    R1(z0(t,τ))R1[z0(t)+2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2
    +z5(t)eτ5+4j=3Pj(t)eτj+2|m|NHPm(t)e(m,τ)]2j=1Fj(α1(t),α2(t),t)eτj+˜R1(z0(t,τ)),

    (here Fj(α1(t),α2(t),t) are linear functions of α1(t),α2(t), and the expression ˜R1(z0(t,τ)) does not contain linear terms of α1(t),α2(t)), we conclude that, after the embedding operation, the function ˆG(t,τ) will linearly depend on scalar functions α1(t) and α2(t).

    Taking into account that under conditions (4.2), scalar multiplication by vector functions χk(t)eτk, containing only exponentials eτk, k=1,2, it is necessary to keep in the expression ˆG(t,τ) only terms with exponents eτ1 and eτ2. Then it follows from (**) that conditions (4.2) are written in the form

    <t(2k=1αk(t)φk(t)eτk+h12(t)φ2(t)eτ1+h21(t)φ1(t)eτ2)
    +(F1(α1(t),α2(t),t)+N|m1|=2:m1Γ1wm1(α1(t),α2(t),t))eτ1+(F2(α1(t),α2(t),t)N|m2|=2:m2Γ2wm2(α1(t),α2(t),t))eτ2
    +Q1(t)eτ1+Q2(t)eτ2,χk(t)eτk>≡0,t[t0,T],k=1,2,

    where the functions wmj(α1(t),α2(t),t),j=1,2, depend on α1(t) and α2(t) in a linear way. Performing scalar multiplication here, we obtain linear ordinary differential equations with respect to the functions αk(t),k=1,2, involved in the solution (3.9) of the system (3.2). Attaching the initial conditions αk(t0)=(z,χk(t0)), k=1,2, calculated earlier to them, we find uniquely functions αk(t), and, therefore, construct a solution (3.9) to the problem (3.2) in the space U in a unique way. The theorem 2 is proved.

    As mentioned above, the right-hand sides of iterative problems (3.1k) (if them solve sequentially) may not belong to the space U. Then, according to [18] (p. 234), the right-hand sides of these problems must be embedded into the U, according to the above rule. As a result, we obtain the following problems:

    Lz0(t,τ)5j=1λj(t)z0τjA(t)z0R0z0=h(t),z0(t0,0)=z0; (¯3.10)
    Lz1(t,τ)=z0t+[g(t)2(eτ3σ1+eτ4σ2)B(t)z0]+R1z0,z1(t0,0)=0; (¯3.11)
    Lz2(t,τ)=z1t+[g(t)2(eτ3σ1+eτ4σ2)B(t)z1]+R1z1+R2z0,z2(t0,0)=0; (¯3.12)
    Lzk(t,τ)=zk1t+[g(t)2(eτ3σ1+eτ4σ2)B(t)zk1]+Rkz0+...+R1zk1,zk(t0,0)=0,k1, (¯3.1k)

    (images of linear operators t and Rν do not need to be embedded in the space U, since these operators act from U to U). Such a replacement will not affect the construction of an asymptotic solution to the original problem (1.1) (or its equivalent problem (2.1)), so on the narrowing τ=ψ(t)ε the series of problems (3.1k) will coincide with the series of problems (¯3.1k) (see [18], pp. 234–235].

    Applying Theorems 1 and 2 to iterative problems (¯3.1k), we find their solutions uniquely in the space U and construct series (2.6). As in [18] (pp. 63-69), we prove the following statement.

    Theorem 3. Let conditions 1)–2) be satisfied for the system (2.1). Then, for ε(0,ε0](ε0>0 is sufficiently small) system (2.1) has a unique solution z(t,ε)C1([t0,T],C2); at the same time there is the estimate

    ||z(t,ε)zεN(t)||C[t0,T]cNεN+1,N=0,1,2,...,

    where zεN(t) is the restriction on τ=ψ(t)ε of the N -th partial sum of the series (2.6) (with coefficients zk(t,τ)U, satisfying the iterative problems (¯3.1k)) and the constant cN>0 does not depend on ε at ε(0,ε0].

    Using Theorem 1, we try to find a solution to the first iterative problem (¯3.10). Since the right-hand side h(t) of the system (¯3.10) satisfies condition (3.3), this system (according to (3.9)) has a solution in the space U in the form

    z0(t,τ)=z(0)0(t)+2k=1α(0)k(t)φk(t)eτk, (5.1)

    where α(0)k(t)C([t0,T],C1) are arbitrary functions, k=1,2,z(0)0(t)=A1(t)h(t). Subordinating (4.2) to the initial condition z0(t0,0)=z0, we have

    2k=1α(0)k(t0)φk(t0)=z0+A1(t0)h(t0).

    Multiplying this equality scalarly χj(t0) and taking into account biorthogonality property of the systems {φk(t)} and {χj(t)}, find the values

    α(0)k(t0)=(z0+A1(t0)h(t0),χk(t0)),k=1,2. (5.2)

    For a complete calculation of the functions α(0)k(t), we proceed to the next iterative problem (¯3.11). Substituting the solution (5.1) of the system (¯3.10) into it, we arrive at the following system:

    Lz1(t,τ)=ddtz(0)0(t)2k=1ddt(α(0)k(t)φk(t))eτk+K(t,t)z(0)0(t)λ5(t)eτ5K(t,t0)z(0)0(t0)λ5(t0)
    +[g(t)2(eτ3σ1+eτ4σ2)B(t)(z(0)0(t)+2k=1α(0)k(t)φk(t)eτk)] (5.3)
    +2j=1[(K(t,t)α(0)j(t)φj(t))λj(t)eτj(K(t,t0)α(0)j(t0)φj(t0))λj(t0)],

    (here we used the expression (2.51) for R1z(t,τ) and took into account that when z(t,τ)=z0(t,τ) in the sum (2.51) only terms with eτ1, eτ2 and eτ5 remain). We calculate

    M=[g(t)2(eτ3σ1+eτ4σ2)B(t)(z(0)0(t)+2k=1α(0)k(t)φk(t)eτk)]
    =12g(t)B(t)[α(0)1σ1φ1(t)eτ3+τ1+σ2α(0)2(t)φ1(t)eτ4+τ1
    +σ1α(0)2(t)φ2(t)eτ3+τ2+σ2α(0)2(t)φ2(t)eτ4+τ2+eτ3σ1z0(t)+eτ4σ2z0(t))].

    Let us analyze the exponents of the second dimension included here for their resonance:

    eτ3+τ1|τ=ψ(t)/ε=e1εtt0(iβiω)dθ,iβiω=[0,iω,+iω,,iβiω=[iβ,+iβ,μ;
    eτ4+τ1|τ=ψ(t)/ε=e1εtt0(+iβiω)dθ,+iβiω=[(0),iω,(+iω),[β=ω,β=2ω,+iβiω=[(iβ),+iβ,μ,2β=ω[^eτ4+τ1=e0=1(β=ω),^eτ4+τ1=eτ2(β=2ω),^eτ4+τ1=eτ3(2β=ω);
    eτ3+τ2|τ=ψ(t)/ε=e1εtt0(iβ+iω)dθ,iβ+iω=[(0),(iω),+iω,[β=ω,β=2ω;iβ+iω=[iβ,(+iβ),μ,2β=ω[^eτ3+τ2=e0=1(β=ω),^eτ3+τ2=eτ1(β=2ω),^eτ3+τ2=eτ4(2β=ω);
    eτ4+τ2|τ=ψ(t)/ε=e1εtt0(+iβ+iω)dθ,+iβ+iω=[0,iω,+iω,,+iβ+iω=[iβ,+iβ,μ,.

    Thus, the exponents eτ3+τ1and eτ4+τ2 are not resonant, and the exponents eτ4+τ1 and eτ3+τ2 are resonant at certain ratios between frequencies β(t), and ω(t), moreover, their embeddings are carried out as follows:

    [^eτ4+τ1=e0=1(β=ω),^eτ4+τ1=eτ2(β=2ω),^eτ4+τ1=eτ3(2β=ω),[^eτ3+τ2=e0=1(β=ω),^eτ3+τ2=eτ1(β=2ω),^eτ3+τ2=eτ4(2β=ω).

    So, resonances are possible only in the following cases of relations between frequencies: a) β=2ω, b) β=ω,c) 2β=ω. Case b) is not considered (see condition (1)). We consider cases a) and c).

    a) β=2ω. In this case, the system (5.3) after embedding takes the form

    Lz1(t,τ)=ddtz(0)0(t)2k=1ddt(α(0)k(t)φk(t))eτk+K(t,t)z(0)0(t)λ5(t)eτ5
    K(t,t0)z(0)0(t0)λ5(t0)+12g(t)B(t)[σ1α(0)1(t)φ1(t)eτ3+τ1+σ2α(0)1(t)φ1(t)eτ2
    +σ1α(0)2(t)φ2(t)eτ1+σ2α(0)2(t)φ2(t)eτ4+τ2+eτ3σ1z0(t)+eτ4σ2z0(t))]
    +2j=1[(K(t,t)α(0)j(t)φj(t))λj(t)eτj(K(t,t0)α(0)j(t0)φj(t0))λj(t0)].

    This system is solvable in the space U if and only if the conditions of orthogonality are satisfied:

    2k=1ddt(α(0)k(t)φk(t))eτk+12g(t)B(t)[σ1α(0)2(t)φ2(t)eτ1
    +σ2α(0)1(t)φ1(t)eτ2]+2i=1(K(t,t)α(0)i(t)φi(t))λi(t)eτi,χj(t)eτj0,j=1,2.

    Performing scalar multiplication here, we obtain a system of ordinary differential equations:

    dα(0)1(t)dt(˙φ1(t),χ1(t))α(0)1(t)+12g(t)σ1(B(t)φ2(t),χ1(t))α(0)2(t)+(K(t,t),χ1(t))λ1(t)α(0)1(t)0,dα(0)2(t)dt(˙φ2(t),χ2(t))α(0)2(t)++12g(t)σ2(B(t)φ1(t),χ2(t))α(0)1(t)+(K(t,t),χ2(t)(t))λ2(t)α(0)2(t)0. (5.4)

    Adding the initial conditions (5.2) to this system, we find uniquely functions α(0)k(t), k=1,2, and, therefore, uniquely calculate the solution (5.1) of the problem (¯3.10) in the space U. Moreover, the main term of the asymptotic solution of the problem (2.1) has the form

    zε0(t)=z(0)0(t)+2k=1α(0)k(t)φk(t)e1εtt0λk(θ)dθ, (5.5)

    where the functions α(0)k(t0) satisfy the problem (5.2), (5.4), z(0)0(t)=A1(t)h(t). We draw attention to the fact that the system of equations (5.4) does not decompose into separate differential equations (as was the case in ordinary integro-differential equations). The presence of a rapidly oscillating coefficient in the problem (1.1) leads to more complex differential systems of type (5.4), the solution of which, although they exist on the interval [t0,T], is not always possible to find them explicitly. However, in third case this it manages to be done.

    c) 2β=ω. In this case, the system (5.3) after embedding takes the form (take into account that ^eτ4+τ1=eτ3^,eτ3+τ2=eτ4)

    Lz1(t,τ)=ddtz(0)0(t)2k=1ddt(α(0)k(t)φk(t))eτk+K(t,t)z(0)0(t)λ5(t)eτ5K(t,t0)z(0)0(t0)λ5(t0)
    +12g(t)B(t)[σ1α(0)1(t)φ1(t)eτ3+τ1+σ2α(0)1(t)φ1(t)eτ2
    +σ1α(0)2(t)φ2(t)eτ1+σ2α(0)2(t)φ2(t)eτ4+τ2+eτ3σ1z0(t)+eτ4σ2z0(t))]
    +2j=1[(K(t,t)α(0)j(t)φj(t))λj(t)eτj(K(t,t0)α(0)j(t0)φj(t0))λj(t0)].

    This system is solvable in the space U if and only if the conditions of orthogonality

    2k=1ddt(α(0)k(t)φk(t))eτk+2i=1(K(t,t)α(0)i(t)φi(t))λi(t)eτi,χj(t)eτj0,

    j=1,2, are satisfied. Performing scalar multiplication here, we obtain a system of diverging ordinary differential equations

    dα(0)1(t)dt(˙φ1(t),χ1(t))α(0)1(t)+(K(t,t),χ1(t))λ1(t)α(0)1(t)0,dα(0)2(t)dt(˙φ2(t),χ2(t))α(0)2(t)+(K(t,t),χ2(t)(t))λ2(t)α(0)2(t)0.

    Together with the initial conditions (5.2), it has a unique solution

    α(0)k(t)=(z0+A1(t0)h(t0),χk(t0))exp{tt0(K(θ,θ)˙φk(θ),χk(θ))λk(θ)dθ},

    k=1,2, and therefore, the solution (5.1) of the problem (¯3.10) will be found uniquely in the space U. In this case the leading term of the asymptotics has the form (5.5), but with functions α(0)k(t), explicitly calculated. Its influence is revealed when constructing the asymptotics of the first and higher orders.

    All authors declare no conflicts of interest in this paper.



    [1] O. E. Rössler, An equation for hyperchaos, Phys. Lett. A, 71 (1979), 155–157. https://doi.org/10.1016/0375-9601(79)90150-6
    [2] C. Xiu, R. Zhou, S. Zhao, G. Xu, Memristive hyperchaos secure communication based on sliding mode control, Nonlinear Dyn., 104 (2021), 789–805. https://doi.org/10.1007/s11071-021-06302-9 doi: 10.1007/s11071-021-06302-9
    [3] M. Boumaraf, F. Merazka, Secure speech coding communication using hyperchaotic key generators for AMR-WB codec, Multimedia Syst., 27 (2021), 247–269. https://doi.org/10.1007/s00530-020-00738-6 doi: 10.1007/s00530-020-00738-6
    [4] D. Jiang, L. Liu, L. Zhu, X. Wang, X. Rong, H. Chai, Adaptive embedding: A novel meaningful image encryption scheme based on parallel compressive sensing and slant transform, Signal Proc., 188 (2021), 108220. https://doi.org/10.1016/j.sigpro.2021.108220 doi: 10.1016/j.sigpro.2021.108220
    [5] P. C. Rech, Chaos and hyperchaos in a Hopfield neural network, Neurocomputing, 74 (2011), 3361–3364. https://doi.org/10.1016/j.neucom.2011.05.016 doi: 10.1016/j.neucom.2011.05.016
    [6] H. Li, Z. Hua, H. Bao, L. Zhu, M. Chen, B. Bao, Two-dimensional memristive hyperchaotic maps and application in secure communication, IEEE T. Ind. Electron., 68 (2020), 9931–9940. https://doi.org/10.1109/TIE.2020.3022539 doi: 10.1109/TIE.2020.3022539
    [7] Y. Su, X. Wang, Characteristic analysis of new four-dimensional autonomous power system and its application in color image encryption, Multimedia Syst., 28 (2022), 553–571. https://doi.org/10.1007/s00530-021-00861-y doi: 10.1007/s00530-021-00861-y
    [8] Y. Si, H. Liu, Y. Chen, Constructing a 3D exponential hyperchaotic map with application to PRNG, Int. J. Bifurcat. Chaos, 32 (2022), 2250095. https://doi.org/10.1142/S021812742250095X doi: 10.1142/S021812742250095X
    [9] A. Chen, J. Lu, J. Lü, S. Yu, Generating hyperchaotic Lü attractor via state feedback control, Phys. A, 364 (2006), 103–110. https://doi.org/10.1016/j.physa.2005.09.039 doi: 10.1016/j.physa.2005.09.039
    [10] Z. Yan, Controlling hyperchaos in the new hyperchaotic Chen system, Appl. Math. Comput., 168 (2005), 1239–1250. https://doi.org/10.1016/j.amc.2004.10.016 doi: 10.1016/j.amc.2004.10.016
    [11] T. Gao, Z. Chen, Q. Gu, Z. Yuan, A new hyper-chaos generated from generalized Lorenz system via nonlinear feedback, Chaos Soliton. Fract., 35 (2008), 390–397. https://doi.org/10.1016/j.chaos.2006.05.030 doi: 10.1016/j.chaos.2006.05.030
    [12] H. Wang, X. Li X, A novel hyperchaotic system with infinitely many heteroclinic orbits coined, Chaos Soliton. Fract., 106 (2018), 5–15. https://doi.org/10.1016/j.chaos.2017.10.029 doi: 10.1016/j.chaos.2017.10.029
    [13] N. Nguyen, T. Bui, G. Gagnon, P. Giard, G. Kaddoum, Designing a pseudorandom bit generator with a novel five-dimensional-hyperchaotic system, IEEE T. Ind. Electron., 69 (2022), 6101–6110. https://doi.org/10.1109/TIE.2021.3088330 doi: 10.1109/TIE.2021.3088330
    [14] S. Emiroglu, A. Akgül, Y. Adıyaman, T. E. Gümüş, Y. Uyaroglu, M. A. Yalçın, A new hyperchaotic system from T chaotic system: Dynamical analysis, circuit implementation, control and synchronization, Circuit World, 48 (2021), 265–277. https://doi.org/10.1108/CW-09-2020-0223 doi: 10.1108/CW-09-2020-0223
    [15] A. S. Pikovski, M. I. Rabinovich, V. Y. Trakhtengerts, Onset of stochasticity in decay confinement of parametric instability, Sov. Phys. JETP, 7 (1978), 715–719.
    [16] J. Llibre, M. Messias, P. D. Silva, On the global dynamics of the Rabinovich system, J. Phys. A, 41 (2008), 275210. https://doi.org/10.1088/1751-8113/41/27/275210 doi: 10.1088/1751-8113/41/27/275210
    [17] V. A. Boichenko, G. A. Leonov, V. Reitmann, Dimension theory for ordinary differential equations, Vieweg+Teubner Verlag, Wiesbaden, 2005.
    [18] Y. Liu, Q. Yang, G. Pang, A hyperchaotic system from the Rabinovich system, J. Comput. Appl. Math., 234 (2010), 101–113. https://doi.org/10.1016/j.cam.2009.12.008 doi: 10.1016/j.cam.2009.12.008
    [19] Y. Liu, Circuit implementation and finite-time synchronization of the 4D Rabinovich hyperchaotic system, Nonlinear Dyn., 67 (2012), 89–96. https://doi.org/10.1007/s11071-011-9960-2 doi: 10.1007/s11071-011-9960-2
    [20] Z. Wei, P. Yu, W. Zhang, M. Yao, Study of hidden attractors, multiple limit cycles from Hopf bifurcation and boundedness of motion in the generalized hyperchaotic Rabinovich system, Nonlinear Dyn., 82 (2015), 131–141. https://doi.org/10.1007/s11071-015-2144-8 doi: 10.1007/s11071-015-2144-8
    [21] X. Tong, Y. Liu, M. Zhang, H. Xu, Z. Wang, An image encryption scheme based on hyperchaotic Rabinovich and exponential chaos maps, Entropy, 17 (2015), 181–196. https://doi.org/10.3390/e17010181 doi: 10.3390/e17010181
    [22] Z. Zhang, L. Huang, A new 5D Hamiltonian conservative hyperchaotic system with four center type equilibrium points, wide range and coexisting hyperchaotic orbits, Nonlinear Dynam., 108 (2022), 637–652. https://doi.org/10.1007/s11071-021-07197-2 doi: 10.1007/s11071-021-07197-2
    [23] S. Yan, X. Sun, Z. Song, Y. Ren, Dynamical analysis and bifurcation mechanism of four-dimensional hyperchaotic system, Eur. Phys. J. Plus, 137 (2022), 734. https://doi.org/10.1140/epjp/s13360-022-02943-w doi: 10.1140/epjp/s13360-022-02943-w
    [24] Z. Li, F. Zhang, X. Zhang, Y. Zhao, A new hyperchaotic complex system and its synchronization realization, Phys. Scripta, 96 (2021), 045208. https://doi.org/10.1088/1402-4896/abdf0c doi: 10.1088/1402-4896/abdf0c
    [25] X. D. Edmund, K. Charles, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A, 35 (1987), 5288–5290. https://doi.org/10.1103/PhysRevA.35.5288 doi: 10.1103/PhysRevA.35.5288
    [26] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, J. Appl. Mech., 51 (1984), 947. https://doi.org/10.1115/1.3167759 doi: 10.1115/1.3167759
    [27] B. Elizabeth, J. Gayathri, S. Subashini, A. Prakash, Hide: Hyperchaotic image encryption using DNA computing, J. Real-Time Image Proc., 19 (2022), 429–443. https://doi.org/10.1007/s11554-021-01194-9 doi: 10.1007/s11554-021-01194-9
    [28] S. Sajjadi, D. Baleanu, A. Jajarmi, H. Pirouz, A new adaptive synchronization and hyperchaos control of a biological snap oscillator, Chaos Soliton. Fract., 138 (2020), 109919. https://doi.org/10.1016/j.chaos.2020.109919 doi: 10.1016/j.chaos.2020.109919
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