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Fractional hyper-chaotic system with complex dynamics and high sensitivity: Applications in engineering

    https://doi.org/10.1142/S0217979224500127Cited by:7 (Source: Crossref)

    Abstract

    Hyper-chaotic systems have useful applications in engineering applications due to their complex dynamics and high sensitivity. This research is supposed to introduce and analyze a new noninteger hyper-chaotic system. To design its fractional model, we consider the Caputo fractional operator. To obtain the approximate solutions of the extracted system under the considered fractional-order derivative, we employ an accurate nonstandard finite difference (NSFD) algorithm. Moreover, the existence and uniqueness of the solutions are provided using the theory of fixed-point. Also, to see the performance of the utilized numerical scheme, we choose different values of fractional orders along with various amounts of the initial conditions (ICs). Graphs of solutions for each case are provided.

    PACS: 07.05.Tp, 44.05.+e, 98.80.Cq, 05.45.Df, 05.45.−a

    1. Introduction

    The inquiry of hyperchaos is a concept based on chaotic dynamical systems. These systems have more complex dynamical behavior with at least two positive Lyapunov exponents compared with the chaotic system. Complex dynamics and high sensitivity of hyper-chaotic systems (HCS) have revealed the significance of such systems in communication technologies. These features were observed when the first were introduced.1,2,3 Owning at least four dimensions and a couple of unstable directions can be regarded as the main aspects of such systems. Many studies have been worked on such systems. For example, in Ref. 4, Chaotic digital cryptosystem has been reported. New parallel image encryption with chaotic windows was done in Ref. 5. Also. hyperchaotic attractor in a new hyperjerk model was studied in Ref. 6. Moreover, a novel hypersensitive hyperchaotic system equilibria have been studied in Ref. 7. Also, the fractional frame of a no-equilibrium hyperchaotic system can be seen in Ref. 8. Another study on a new 5D hyperchaotic system can be studied in Ref. 9. A new set of 4D hyperchaotic and chaotic systems with quadric surfaces was studied in Ref. 10. We consider a new hyper-chaotic system as follows11 :

    dxdt=a1x(t)+a2y(t),dydt=a3x(t)a2y(t)z(t)sgnz(t),dzdt=a5y(t)+a6z(t)+a7w(t)dwdt=a7z(t)a9w(t)+a10x(t).(1)
    In the above system, x, y, z, and w are dependent variables and a1 to a10 are constants. Noninteger mathematical systems permit to do a more accurate investigation instead of classic counterparts, which is essential for the proceeding of quantitative expression of different models and more acceptable simulation of the behaviors of systems and the process of foretelling the characteristics cannot be obviously observed with the experimental data. A remarkable amount of investigations implied that noninteger-order differential equations (FDEs) can manifest complex dynamical aspects more correctly than integer-order differential equations.12,13,14,15,16 Because fractional-order derivatives have the ability to determine the attributes of memory effects as an important feature of real-world phenomena.17,18,19,20,21,22,23,24,25 Many researchers have paid considerable attention to using fractional calculus for modeling and analysis of dynamical systems with chaotic aspects. A fractional study of the COVID-19 model was reported in Ref. 26. Moreover, a new study of the complex Layla and Majnun love story using fractional operator can be seen in Ref. 27. Also, another work on the multi-strain TB model using a noninteger derivative was reported in Ref. 28. Fractional investigation of a new hyper-chaotic model involving single nonlinearity can be read in Ref. 29. Numerical solutions of the fractal-fractional Klein–Gordon equation have been done in Ref. 30. A new fractal fractional investigation of the computer viruses model was reported in Ref. 31. A numerical study of the fractional crime model can be seen in Ref. 32. Also, numerical investigation of the fractal-fractional modeling of the immune-Tumor problem can be read in Ref. 33. Fractional investigation of Platelet-Poor Plasma Arising in a Blood Coagulation System can be observed in Ref. 34. In Ref. 35, modeling and control analysis of the circumscribed self-excited spherical strange attractor were reported. More studies can be found in Refs. 36, 37, 38.

    Motivated by the aforesaid argument, the primary purpose of this study is to process a new and effective mathematical model for the hyper-chaotic system using fractional Caputo derivative (1) as

    C0𝒟αtx(t)=a1x(t)+a2y(t),C0𝒟αty(t)=a3x(t)a2y(t)z(t)sgnz(t),C0𝒟αtz(t)=a5y(t)+a6z(t)+a7w(t)C0𝒟αtw(t)=a7z(t)a9w(t)+a10x(t).(2)
    In the above relation, C0𝒟αt is the αth-order Caputo derivative expressed in Ref. 39
    𝒞𝒟αu(t)=1Γ(1α)t0˙u(τ)(tτ)αdτ.(3)
    Now, the Grünwald–Letnikov (GL) approximation for the following problem is
    {𝒞𝒟αu(t)=f(x(t),t),t[0,tf],u(0)=u0,(4)
    where tf is the final time. We discretize the above problem as
    n+1j=0sαjun+1j=f(un+1,tn+1),n=0,1,2,,(5)
    where un is the approximation of u(tn), tn=nh and sαj is the GL coefficient calculated as
    sα0=hα,sαj=(11+αj)sαj1,j=1,2,3,,(6)
    and h is the time step size.

    1.1. Nonstandard finite difference scheme (NFDS)

    Now, we use the NFDS to get the approximate solutions of the considered system. The primary basis of the NFDS method comes from exact finite difference methods. These algorithms are well designed.40,41 Such methods are designed for recompensing deficiencies, for example for numerical instabilities that can be generated by standard finite difference schemes. With regard to the positiveness, boundedness, and monosyllabicity of solutions, NFDS methods own a more satisfactory application compared to the standard finite difference methods because of the adaptability that they own to create an NSFD method that is able to maintain certain features. Some applications of NFDS can be read in Refs. 4248. We consider the following problem :

    {du(t)dt=f(u(t),t,ζ),t[0,tf],x(0)=x0,(7)
    where ζ is a parameter. We discretize at tn=nh.

    (i)

    Suppose un as the approximation of u(tn), so for (7) we have

    du(t)dtun+1unϕ(h,ζ)(8)

    (ii)

    Also, in a similar way for f(x(t),t,ζ), we have f(un+1,xn,,t,ζ)

    un+1unp(h,ζ)=f(un+1,un,,t,ζ).(9)

    We set p(h,ζ)=h for Eq. (8), a classic discretization can be derived, so (8) can be regarded as a generalization for its classic one. Moreover, the next consistency condition must be held with the denominator p(h,ζ) :

    p(h,ζ)=h+O(h2),(10)
    which guarantees (8) to have convergency to the related continuous derivative when h approaches zero. For example, see the satisfying condition (10)49,50 :
    1eh,1eζhζ,sinh, h.(11)
    Mickens51 provided a method to assign p(h,ζ). For example
    du(t)dt=ζu(t)+(NL).(12)
    In the above relation, NL indicates the nonlinear terms, Mickens51 suggested the NFDS method
    un+1unp(h,ζ)=ζun+(NL)n,(13)
    where p(h,ζ)=eζh1ζ for ζ0 and p(h,ζ)=h for ζ=0. Also, the NSFD algorithm needs discrete-time computational grids on which the dependent functions should be modeled. For example47,48
    xy2un+1ynun+1yn+1,uyunyn+1,x3(un+1+un12)u2n,u22unun+1+un2.(14)
    In this stage, we develop the suggested method for the following FDE :
    {𝒞𝒟αu(t)=f(u(t),t,ζ),t[0,tf],x(0)=x0.(15)
    To do this, the discretized GL approximation formula by (5) can be used (5). So, we develop the NSFD method in the noninteger frame
    un+1=n+1j=1sαjun+1j+f(un+1,tn+1)cα0,n=0,1,2,,(16)
    with
    sα0=(p(h,ζ))α,sαj=(11+αj)sαj1,j=1,2,3,.(17)

    2. Numerical Method

    Now we use the NSFD method presented in the previous section to show the results. Recall that Mickens suggested writing a general multistep numerical method to approximate the simulation of (7) by (9), where p(h,ζ) holds h+O(h2), and f(un+1,un,,t,ζ) is a discretization f(u(t),t,ζ). The examples of how the rules are employed to develop the discretization was reported in the previous section.49,50 Using the above-mentioned procedure, the next discretization can be written for (1)

    xn+1xnp1(h)=a1xn+1+a2yn,yn+1ynp2(h)=a3xna2yn+1znsgnznzn+1znp3(h)=a5yn+a6zn+1+a7wn,wn+1wnp4(h)=a7zna9wn+1+a10xn,(18)
    where p1(h)=ea1h+1a1, p2(h)=1ePhv, p3(h)=1ea2ha2, p4(h)=1ea9ha9, v=a2znsignzn.

    So, from (18), the following relation can be obtained (1) :

    xn+1=a2p1(h)yn+xn1a1p1(h),yn+1=a3p2(h)xn+yn1+a2p2(h)znsignzn,zn+1=a5p3(h)+a7p3(h)wn+zn1a6p3(h),zn+1=a7p4(h)zn+a10p4(h)xn+wn1a9p4(h).(19)
    Employing the GL for (2) we gain
    n+1j=0sα1jxn+1j=a1xn+1+a2yn,n+1j=0sα2jyn+1j=a3xna2yn+1znsgnzn,n+1j=0sα3jzn+1j=a5yn+a6zn+1+a7wn,n+1j=0sα3jwn+1j=a7zna9wn+1+a10xn.(20)
    Finally, for (2) we have
    xn+1=n+1j=1sα1jxn+1j+a2ynsα10+a1,yn+1=n+1j=1sα2jyn+1j+a3xnsα20+a2znsgnzn,zn+1=n+1j=1sα3jzn+1j+a7zn+a10xnsα30+a6,wn+1=n+1j=1sα3jwn+1j+a7zn+a7zn+a10xnsα40a9,(21)
    where43,44
    sα10=(ea1h+1a1)α1,sα20=(1evhv)α2,sα30=(1ea2ha2)α3,sα40=(1ea9ha9)α4(22)
    and sαij are calculated by (17).

    3. Existence and Uniqueness

    Now, we examine the existence and uniqueness of the solution of the considered fractional system employing the theory of fixed-point. Assume (J) be a Banach space for the continuous real-valued functions on J=[0,a] and 𝒬=(J)×(J)×(J) with the norm (x,y,w,z)=x+y+w+z, x=suptJ|x|, y=suptJ|y|, w=suptJ|w|, z=suptJ|z|. By employing the Caputo operator for relation (2), one can obtain

    x(t)x(0)=C𝒟α[a1x(t)+a2y(t)],y(t)y(0)=C𝒟α[a3x(t)a2y(t)z(t)sgnz(t)],z(t)z(0)=C𝒟α[a5y(t)+a6z(t)+a7w(t)],w(t)w(0)=C𝒟α[a7z(t)a9w(t)+a10x(t)],(23)
    By supposing
    K1=a1x(t)+a2y(t),K2=a3x(t)a2y(t)z(t)sgnz(t),K3=a5y(t)+a6z(t)+a7w(t),K4=a7z(t)a9w(t)+a10x(t),(24)
    we can write the relation (23) as
    x(t)x(0)=(α)t0K1(ρ,Θ,x(Θ))(tΘ)αdΘ,y(t)y(0)=(α)t0K2(ρ,Θ,y(Θ))(tΘ)αdΘ,z(t)z(0)=(α)t0K3(ρ,Θ,z(Θ))(tΘ)αdΘ,w(t)w(0)=(α)t0K3(ρ,Θ,w(Θ))(tΘ)αdΘ.(25)
    Observe K1(x,Θ), K2(y,Θ), K2(z,Θ), K3(w,Θ) and (α)=1Γ(1α) satisfy the Lipschitz condition if and only if x(t), y(t), z(t) and w(t) own an upper bound. For x(t) and x(t), we have
    K1(α,t,x(t))K(α,t,x(t))=a1(x(t)x(t)).(26)
    Suppose v1:=a1, we get
    K1(α,t,x(t))K1(α,t,x(t))v1x(t)x(t),(27)
    also
    K2(α,t,y(t))K2(α,t,y(t))v2y(t)y(t),K3(α,t,z(t))K3(α,t,z(t))v3z(t)z(t),K3(α,t,w(t))K3(α,t,w(t))v4w(t)w(t),(28)
    where v2=a2z(t)sgnz(t), v3=a6 and v4=a9. So, this implies that the Lipschitz condition is held for K1, K2, K3 and K4. Also, we can write relation (25)
    xn(t)=(α)t0K1(α,Θ,xn1(Θ))(tΘ)αdΘ,yn(t)=(α)t0K2(α,Θ,yn1(Θ))(tΘ)αdΘ,zn(t)=(α)t0K3(α,Θ,zn1(Θ))(tΘ)αdΘ,wn(t)=(α)t0K4(α,Θ,wn1(Θ))(tΘ)αdΘ,(29)
    with x0(t)=x(0), y0(t)=y(0), z0(t)=z(0) and w0(t)=w(0), and by subtracting the above successive terms, it results
    Ψx,n(t)=xn(t)xn1(t)=(α)t0K1(α,Θ,xn1(Θ))K1(α,Θ,xn2(Θ))(tΘ)dΘ,Ψy,n(t)=yn(t)yn1(t)=(α)t0K2(α,Θ,yn1(Θ))K2(α,Θ,yn2(Θ))(tΘ)dΘ,Ψz,n(t)=zn(t)zn1(t)=(α)t0K3(α,Θ,zn1(Θ))K3(α,Θ,zn2(Θ))(tΘ)dΘ,Ψw,n(t)=wn(t)wn1(t)=(α)t0K4(α,Θ,wn1(Θ))K4(α,Θ,wn2(Θ))(tΘ)dΘ,(30)
    and consider
    xn(t)=nj=0Ψx,j(t),yn(t)=nj=0Ψy,j(t),zn(t)=nj=0Ψz,j(t),wn(t)=nj=0Ψw,j(t),(31)
    and using (27) and (28) and by applying Ψx,n1(t)=xn1(t)xn2(t), Ψy,n1(t)=yn1(t)yn2(t), Ψz,n1(t)=zn1(t)zn2(t) and Ψw,n1(t)=wn1(t)wn2(t), yields
    Ψx,n(t)=(α)v1t0Ψx,n1(Θ)(tΘ)αdΘ,Ψy,n(t)=(α)v2t0Ψy,n1(Θ)(tΘ)αdΘ,Ψz,n(t)=(α)v3t0Ψz,n1(Θ)(tΘ)αdΘ,Ψw,n(t)=(α)v3t0Ψw,n1(Θ)(tΘ)αdΘ.(32)
    Hence, we can prove the next theorem.

    Theorem 3.1. The non-integer model (2) owns a unique solution with

    (α)αrαv1<1,i=1,2,3(33)
    when t[0,r].

    Proof. As reported, z(t), y(t), z(t) and w(t) are bounded and K1, K2, K3 and K4 hold the Lipschitz condition. Hence, using and (32) we have

    Ψx,n(t)x0(t)((α)αrαv1)n,Ψy,n(t)y0(t)((α)αrαv2)n,Ψz,n(t)z0(t)((α)αrαv3)n,Ψw,n(t)w0(t)((α)αrαv3)n,(34)
    so, we have Ψx,n(t)0, Ψy,n(t)0, Ψz,n(t), Ψw,n(t)0 when n. Moreover, using triangle inequality and (34) for any p results in the following relation :
    xn+p(t)xn(t)n+pj=n+1κj1=κn+11κn+p+111κ1,yn+p(t)yn(t)n+pj=n+1κj2=κn+12κn+p+121κ2,zn+p(t)zn(t)n+pj=n+1κj3=κn+13κn+p+131κ3,wn+p(t)wn(t)n+pj=n+1κj3=κn+13κn+p+131κ3,(35)
    where κi=(α)αrαvi. So, xn, yn, zn, wn are Cauchy sequences in (J). So, they converge. By the limit theorem, sequence (29) is the unique solution of the system (2). So, the proof is obvious. □

    4. Results and Discussion

    In this paper, we used an accurate nonstandard finite difference for the first time for solving a new noninteger hyper-chaotic system. Generating difference methods that preserve the stability behavior of the equilibrium points is crucial in numerical simulation. Now, we use the presented nonstandard finite difference to see the numerical behavior of the proposed fractional system. We show the accuracy and efficiency of the proposed numerical scheme for solving the studied fractional-order system under different fractional orders along with initial conditions. The values of parameters in the considered systems are as follows: a1=1.05, a2=1.2, a31=2, a4=0.1, a5=1, a6=0.01, a7=5, a8=10, a9=0.8 and a10=0.1. We provide the approximate solutions of the system under different values of α as 0.65,0.75,0.85 and initial conditions 0.95 and x(0)=1, y(0)=1, z(0)=1 and w(0)=1 by Fig. 1. Figure 1 shows how dependent variables behave under the selected orders. Figures 2 and 3 display the behavior of the solutions in 2D under the considered fractional orders using the numerical scheme reported in Sec. 2. Moreover, chaotic behaviors of the approximate solutions can be seen in Fig. 4.

    Fig. 1.

    Fig. 1. (Color online) Approximate solutions for different values of α as 0.65, 0.75, 0.85 and initial conditions 0.95 and x(0)=1, y(0)=1, z(0)=1 and w(0)=1.

    Fig. 2.

    Fig. 2. (Color online) Approximate solutions for different values of α as 0.65, 0.75, 0.85 and initial conditions 0.95 and x(0)=1, y(0)=1, z(0)=1 and w(0)=1.

    Fig. 3.

    Fig. 3. (Color online) Approximate solutions for different values of α as 0.65, 0.75, 0.85 and initial conditions 0.95 and x(0)=1, y(0)=1, z(0)=1 and w(0)=1.

    Fig. 4.

    Fig. 4. (Color online) Approximate solutions for different values of α as 0.65, 0.75, 0.85 and initial conditions 0.95 and x(0)=1, y(0)=1, z(0)=1 and w(0)=1.

    5. Conclusion

    This work was dedicated to using the Caputo fractional derivative to design a new model for a new chaotic system. The existence and uniqueness of the solutions were provided successfully. Also, a numerical algorithm called a nonstandard finite difference scheme was used to obtain the approximate solutions. To see the performance of the suggested method, different values of initial conditions were considered and the solutions were shown through some figures. Also, to see how the studied fractional system behaves, we selected various amounts of fractional orders.

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