Research article

Global weighted regularity for the 3D axisymmetric non-resistive MHD system

  • We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in R3. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time T if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.

    Citation: Wenjuan Liu, Zhouyu Li. Global weighted regularity for the 3D axisymmetric non-resistive MHD system[J]. AIMS Mathematics, 2024, 9(8): 20905-20918. doi: 10.3934/math.20241017

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  • We consider the regularity criteria of axisymmetric solutions to the non-resistive MHD system with non-zero swirl in R3. By applying a new anisotropic Hardy-Sobolev inequality in mixed Lorentz spaces, we show that strong solutions to this system can be smoothly extended beyond the possible blow-up time T if the horizontal angular component of the velocity belongs to anisotropic Lorentz spaces.


    Fractional calculus is a subdivision of classical calculus that concerns itself with derivatives as well as integrals of nearly any fractional order. This mathematical field has gained significant attention due to its ability to model complex phenomena more accurately than integer-order calculus. Fractional calculus offers robust methods for characterizing memory and hereditary features of different materials and processes, rendering it essential in disciplines such as control theory [14], signal processing [4], and differential equations [11,12,13]. The foundations of fractional calculus were laid by early mathematicians such as Leibniz, Liouville, and Riemann, who explored the concept of generalizing the order of differentiation and integration beyond integers [3,7]. Modern advancements have further developed these ideas, leading to a robust theoretical framework and numerous practical applications.

    The nabla difference operator is a discrete counterpart of the continuous derivative, employed in discrete calculus, specifically in the context of time-scale calculus. This operator, often denoted by , operates on functions defined on discrete domains, such as sequences or time scales. The nabla difference operator plays a crucial role in various fields, including exact analysis, discrete dynamic systems, and operational calculus in [9,21,22]. It facilitates the formulation and solution of difference equations, which are the discrete counterparts of differential equations. The papers "On the definitions of nabla fractional operators" and "Discrete fractional calculus includes the nabla operator" go into great detail about what nabla fractional operators are and how they can be used in real life. They stress how important they are in the field of discrete mathematics [1,2]. Recently, the nabla difference operator has been seen in sequential differences in the nabla fractional calculus, the combined delta-nabla sum operator in discrete fractional calculus, and discrete fractional calculus consisting of the nabla operator in [2,20]. These studies contribute to a deeper understanding of the interplay between discrete and continuous analysis. In research [15], such as on K¨othe-Toeplitz duals of generalized difference sequence spaces and Laplace transforms for the nabla-difference operator, investigations into the theoretical underpinnings and applications of these operators in discrete calculus [6].

    The Fibonacci sequence is a widely recognized integer sequence defined by the recurrence relation Fr=Fr1+Fr2, where F0 is 0 and F1 is 1. This sequence appears in various natural phenomena and has numerous applications in computer science, mathematics, and financial modeling. The study of sequence spaces derived from difference operators has gained significant attention, particularly with generalized Fibonacci difference operators. The derivation of sequence spaces arising from triple-band generalized Fibonacci difference operator [18,24], investigates the structural properties of these spaces, while generalized Fibonacci difference sequence spaces and compact operators [17,23] explores their impact on the boundedness and compactness of sequences. These works extend classical sequence space theory by integrating the recursive properties of Fibonacci sequences, highlighting new connections in functional analysis.

    The motivation for this study arises from the need to further explore and expand the theoretical systems and applications of the nabla difference operator in generating and analyzing Fibonacci sequences. We present the trigonometric nabla difference operator of order 2 and its discrete integral, analyzing the ¯θ(t)-sequences, their summation, and the proportional derivative of the ¯θ(t)-polynomials.

    The contributions of the research are delineated as follows:

    1. By utilizing various trigonometric coefficients in the ¯θ(t)-Fibonacci equation and its inverses, we formulate new sequences and analyze their characteristics.

    2. The nabla difference operator of order 2 facilitates the generation of ¯θ(t)-Fibonacci sequences and their exact and numerical solutions.

    3. We provide chaotic behavior of the generating function of the second-order Fibonacci sequence with the coefficients of trigonometric functions.

    4. The study includes MATLAB examples to demonstrate the practical applications of our theoretical findings.

    Throughout this article, we make use of the notations and elucidations from the following Table 1.

    Table 1.  The notations and elucidations.
    ξ Shift (translation) value (i.e., ξ[0,))
    t(x) t(tξ)(t2ξ)(t3ξ)...(t(x1)ξ)
    t(n+r)ξ tn,r, where n, r are integers and t(,).
    E-solution Exact solution
    N-solution Numerical solution
    ¯θ(t)-sequence Fibonacci sequence derived from general difference equation (Nabla) with trigonometric coefficients of order 2.
    ¯θ(t)-equation General difference equation (Nabla) with trigonometric coefficients of order 2.

     | Show Table
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    Here, we formulate a generic nabla-difference operator that includes a trigonometric co-efficient ¯θ(t)v(t)=v(t)α1sin(b1t)v(t0,1)α2sin(b2t)v(t0,2) which generates second-order ¯θ(t)-Fibonacci sequence and its sum.

    Definition 2.1. Let t be any positive real number and n2; a second-order generic ¯θ(t)-sequence is defined recurrently as Ft,0=1,  Ft,1=α1sin(b1t), and

    Ft,n=α1sin(b1tn,1)Ft,n1+α2sin(b2tn,2)Ft,n2. (2.1)

    Definition 2.2. For any positive real number t, a generic nabla difference operator of order two using sine (any trigonometric) function coefficients on v(t), denoted as ¯θ(t)v(t), is defined as

    ¯θ(t)v(t)=v(t)α1sin(b1t)v(t0,1)α2sin(b2t)v(t0,2), (2.2)

    and the inverse of ¯θ(t)v(t)=u(t) is defined as

    v(t)=¯θ(t)1u(t). (2.3)

    Definition 2.3. [16] A proportional α- derivative is defined by

    Dαf(t)=αf(t)+(1α)f(t), α[0,1], (2.4)

    is the proportional α-derivative of f(t). Here Dα is proportional α- provided the function f(t) is differentiable at t.

    Lemma 2.1. For any real number t,v(t) is a function. Then it leads

    ¯θ(t)1ast[1α1sin(b1t)asξα2sin(b2t)a2sξ]=ast. (2.5)

    Proof: By doing v(t)=at0,0 in (2.2), we observe that

    ¯θ(t)ast=ast[1α1sin(b1t)asξα2sin(b2t)a2sξ].

    Now, the function v(t) follows from the Definition 2.2.

    Remark 2.1. When α1=1=α2 in lemma 2.1, then we observe that

    ¯θ(t)1ast[1sin(b1t)asξsin(b2t)a2sξ]=ast. (2.6)

    Corollary 2.1. For any real number t, est is a function. Then we observe

    ¯θ(t)1est[1α1sin(b1t)esξα2sin(b2t)e2sξ]=est. (2.7)

    Proof: By doing a=e in (2.5), we conclude the proof.

    Corollary 2.2. For any real number t, {est} is a function. Then we observe

    ¯θ(t)1est[1α1sin(b1t)esξα2sin(b2t)e2sξ]=est. (2.8)

    Proof: By doing a=e1 in (2.5), we conclude the proof.

    Corollary 2.3. For any real number t, {est} is a function. Then we observe

    ¯θ(t)1est[1sin(b1t)esξsin(b2t)e2sξ]=est. (2.9)

    Proof: By doing α1=α2=1 in (2.8), we conclude the proof.

    Propsition 2.1. If the function v(t)=1¯θ(t)u(t) is a E-solution of (2.2), Ft,0=1,Ft,1=α1sin(b1t) and

    Ft,n+1=α1sin(b1tn,0)Ft,n+α2sin(b2tn,1)Ft,n1, for i=0,1,2,... then, the E-solution is equal to a N-solution, which is given by

    v(t)Ft,n+1v(tn,1)α2sin(b2tn,0)Ft,nv(tn,2)=ni=0Ft,iu(t0,i). (2.10)

    Proof: For the function v(t), the Definition 2.2 yields

    v(t)=u(t)+α1sin(b1t)v(t0,1)+α2sin(b2t)v(t0,2). (2.11)

    By changing k by k0,1 and then put the value of v(t0,1) into (2.11), we get

    v(t)=u(t)+Ft,1u(t0,1)+[Ft,1α1sin(b1t0,1)+

    α2sin(b2t)]v(t0,2)+Ft,1α2sin(b2t0,1)v(t0,3), (2.12)

    which leads to v(t)=Ft,0u(t)+Ft,1u(t0,1)+

    Ft,2v(t0,2)+α2sin(b2t0,1)Ft,1v(t0,3), (2.13)

    where Ft,0=0, Ft,1=α1sin(b1t) and Ft,2=Ft,n+1=α1sin(b1t1,0)Ft,1+α2sin(b2t1,1)Ft,0. By changing k by k0,2 in (2.11) and then putting the value of v(t0,2) into (2.13), we observe

    v(t)=Ft,0u(t)+Ft,1u(t0,1)+Ft,2u(t0,2)+Ft,3v(t0,3)+α2sin(b2t0,2)Ft,2v(t0,4),

    where Ft,3=Ft,3=α1sin(b1t2,0)Ft,2+α2sin(b2t2,1)Ft,1.

    By carrying out this procedure repeatedly (2.10).

    Corollary 2.4. If the function 1¯θ(t)u(t)=v(t), Ft,0=1,Ft,1=sin(b1t) and

    Ft,n+1=sin(b1tn,0)Ft,n+sin(b2tn,1)Ft,n1, for i=0,1,2,... then

    v(t)Ft,n+1v(t0,1)sin(b2tn,0)Ft,nv(t0,2)=ni=0Ft,iu(t0,i). (2.14)

    Proof: The proof follows by doing α1=1=α2 in Theorem 2.1.

    Corollary 2.5. Let v(t) be any function and

    ¯θ(t)v(t)=ast[1α1sin(b1t)asξα2sin(b2t)a2sξ],

    then we observe that

    astFt,n+1as(tn,1)α2sin(b2tn,0)Ft,nas(tn,2)

    =ni=0Ft,iast0,i[1α1sin(b1t0,i)asξα2sin(b2t0,i)a2sξ]. (2.15)

    Proof: By doing v(t)=ast and applying (2.5) in (2.10), we observed the proof.

    The following illustration proves the significance of corollary 2.5.

    Corollary 2.6. Let α1=α2=1 in (2.15). Then we have

    at0,0Ft,n+1atn,1sin(b2tn,0)Ft,natn,2

    =ni=0Ft,iat0,i[1sin(b1t0,i)aξsin(b2t0,i)a2ξ]. (2.16)

    Proof: By doing v(t)=at0,0 and putting (2.5) in (2.10), we complete the proof.

    The following illustration proves the significance of corollary 2.6.

    Corollary 2.7. Let est be a function of t(,). Then

    estFt,n+1es(tn,1)α2sin(b2tn,0)Ft,nes(tn,2)

    =ni=0Ft,iest0,i[1α1sin(b1t0,i)eξα2sin(b2t0,i)e2ξ]. (2.17)

    Proof: By doing v(t)=est and applying (2.8) in (2.5), we obtain (2.17).

    Propsition 2.2. Let xN(0). Then, the E-solution of ¯θ(t)-equation

    v(t)α1sin(b1t)v(t0,1)α2sin(b2t)v(t0,2)=[tx0,0α1sin(b1t)tx0,1α2sin(b2t)tx0,2] is

    1¯θ(t)[tx0,0α1sin(b1t)tx0,1α2sin(b2t)tx0,2]=tx0,0 (2.18)

    Proof: By doing v(t)=tx0,0 with the equation (2.2) and using (2.3), we obtain (2.18).

    Corollary 2.8. By doing x=2 with the Theorem 2.2, we observed

    1¯θ(t)[t20,0α1sin(b1t)t20,1α2sin(b2t)t20,2]=t20,0, (2.19)

    which is the E-solution of the nabla difference equation of order two

    ¯θ(t)v(t)=t20,0α1sin(b1t)t20,1α2sin(b2t)t20,2.

    Proof: From (2.18), replacing x=2, we obtain (2.19).

    Corollary 2.9. If v(t)=1¯θ(t)[tx0,0α1sin(b1t)tx0,1α2sin(b2t)tx0,2] is a E-solution of the equation (2.18), then

    v(t)Ft,n+1v(t0,1)α2sin(b2tn,0)Ft,nv(t0,2)

    =ni=0Ft,i[tx0,iα1sin(b1t0,i)txi,1α1sin(b2t0,i)txi,2]. (2.20)

    Proof: Taking u(t)=tx0,0α1sin(b1t)tx0,1α2sin(b2t)tx0,2 in (2.10), we have (2.20).

    Propsition 2.3. If v(t) is a E-solution of sine-coefficients ¯θ(t)-equation

    v(t)α1sin(b1t)v(t0,1)α2sin(b2t)v(t0,2)=tx0,0at0,0α1sin(b1t)tx0,1at0,1α2sin(b2t)tx0,2at0,2,

    then we have

    v(t)Ft,n+1(tn,1)α2sin(b2tn,0)v(tn,2)

    =ni=0Ft,itx0,iat0,i[1α1sin(b1t0,i)txi,1aξα2sin(b2t0,i)txi,2a2ξ]. (2.21)

    Proof: By doing u(t)=[tx0,0at0,0α1sin(b1t)tx0,1at0,1α2sin(b2t)tx0,2at0,2] in (2.10) and using (2.5), we get (2.21).

    Corollary 2.10. A E-solution of ¯θ(t)-equation is

    ¯θ(t)v(t)=t30,0at0,0α1sin(b1t)t30,1at0,1α2sin(b2t)t30,2at0,2  is  t3at0,0

    and hence we have

    t30,0at0,0Ft,n+1t3n,1atn,1α2sin(b2tn,0)t3n,2atn,2

    =ni=0Ft,it30,iat0,iα1sin(b1t0,i)t3i,1ati,1α2sin(b2t0,i)t3i,2ati,2. (2.22)

    Proof: The proof is completed by doing x=3 in Theorem 2.3.

    Propsition 2.4. Let v(t) be a solution of ¯θ(t)-equation

    v(t)α1sin(b1t)v(t0,1)α2sin(b2t)v(t0,2)=t(x)at0,0α1sin(b1t)t(x)0,1at0,1α2sin(b2t)t(x)0,2at0,2, then we have

    v(t)Ft,n+1v(tn,1)α2sin(b2tn,0)v(tn,2)

    =ni=0Ft,iat0,i[t(x)0,iα1sin(b1t0,i)t(x)i,1aξα2sin(b2t0,i)t(x)i,2a2ξ]. (2.23)

    Proof: By doing v(t)=t(x)0,0at0,0 in (2.10) and using (2.5), we get (2.23).

    Corollary 2.11. Let v(t) be a solution of ¯θ(t)-equation

    v(t)sin(b1t)v(t0,1)sin(b2t)v(t0,2)=t(x)at0,0sin(b1t)t(x)0,1at0,1sin(b2t)t(x)0,2at0,2, then we have

    v(t)Ft,n+1v(tn,1)sin(b2tn,0)v(tn,2)

    =ni=0Ft,iat0,i[t(x)0,isin(b1t0,i)t(x)i,1aξsin(b2t0,i)t(x)i,2a2ξ]. (2.24)

    Proof: By doing α1=α2=1 in (2.23), we obtain (2.24).

    The objective of this section is to demonstrate the efficacy of the primary findings presented in this paper by employing precise examples from the literature. Also, we have investigated graphical representations of Fibonacci sequences with the co-efficients of the Sine, Cosine and Co-secant functions for different values of t in the following Figures 13.

    Figure 1.  Sine and Cosine-Fibonacci Sequences for t=10, ξ=0.4, and t=15, ξ=0.5, respectively.
    Figure 2.  Sine and Cosine-Fibonacci Sequences for t=11, ξ=0.3.
    Figure 3.  Sine and Co-secant-Fibonacci Sequences for t=10, ξ=0.4, and t=0.2, ξ=1.8, respectively.

    The validity of the definition 2.1 is confirmed by the subsequent illustrative example 3.1.

    Example 3.1. (i) By doing t=10, b1=2, b2=1, ξ=0.4, α1=2, and α2=1 in (2.1), we obtain a Sine-Fibonacci sequence {1,1.8259,0.7097,0.9351,1.9327,3.9778,}.

    (ii) When t=15, ξ=0.5, α1=0.2, α2=0.1, r1=3, and r2=1 in (2.1), we have a Cosine-Fibonacci sequence {1.0000,1.5194,1.9182,3.9008,3.4203,0.8856,5.1684,}.

    Also, the sine Fibonacci polynomials are observed by

    F0(t)=1,

    F1(t)=2sin(2t),

    F2(t)=4sin(2t)sin(2t0.8)+sin(t),

    F3(t)=8sin(2t)sin(2t1.6)sin(2t0.8)+2sin(t)sin(2t1.6)+2sin(t0.4)sin(2t), etc.

    Furthermore, using (2.4), we have the following proportional α-derivative of the sine Fibonacci polynomials;

    Fα0(t)=0,

    Fα1(t)=4αcos(2t)+2(1α)sin(2t),

    Fα2(t)=8α[sin(2t0.8)cos2t+cos(2t0.8)sin2t+cost]

    +(1α)[4sin(2t0.8)sin2t+sint],

    Fα3(t)=α[16cos(2t1.6)sin(2t0.8)sin(2t)+16sin(2t1.6)cos(2t0.8)sin(2t)+

    16sin(2t1.6)sin(2t0.8)cos(2t)2cos(t)sin(2t1.6)+2sin(t)cos(2t1.6)+2cos(t0.4)sin(2t)+4sin(t0.4)cos(2t)]+(1α)[8sin(2t1.6)sin(2t0.8)sin(2t)+2sin(t)sin(2t1.6)],and etc.

    One can derive second-order Fibonacci polynomials and sequences for each pair ¯θ(t)R2. The validity of the result 2.5 is confirmed by the subsequent illustrative example 3.2.

    Example 3.2. Setting t=11, b1=2, α1=3, s=2, α2=2, n=2, ξ=0.3, b2=1, and a=3 in (2.15), we obtain the following:

    (i) The sum of the sine-Fibonacci series

    322Ft,3310.12Ft,2sin(10.4)319.6=2i=0Ft,i3(220.6i)[1sin(220.6i)30.6sin(330.9i)31.2]=48298205761.12,

    where Ft,0=1, Ft,1=0.03, Ft,2=2.04, Ft,3=5.65.

    (ii) The sum of the cosine-Fibonacci series

    322Ft,3310.12Ft,2cos(10.4)319.6=2i=0Ft,i3(220.6i)[1cos(220.6i)30.6cos(330.9i)31.2]=78777462484.69,

    where Ft,0=1, Ft,1=3.00, Ft,2=7.48,Ft,3=6.57,

    and the cosine Fibonacci polynomials are given by

    F0(t)=1,

    F1(t)=3cos(2t),

    F2(t)=9cos(2t0.6)cos(2t)+2cos(t)

    F3(t)=27cos(2t1.2)cos(2t0.6)cos(2t)+6cos(2t1.2)cos(t)+2cos(t0.4)cos(2t), etc. Furthermore, using (2.4), we have the following proportional α-derivative of the co-secant Fibonacci polynomials;

    Fα0(t)=0,Fα1(t)=α[6sin(2t)]+3(1α)cos(2t),

    Fα2t)=α[18sin(2t0.6)cos(2t)18cos(2t0.6)

    sin(2t)2sint]+(1α)9cos(2t0.6)cos(2t)+2cos(t)], and etc.

    The validity of the definition 2.7 is confirmed by the subsequent illustrative example 3.3.

    Example 3.3. By doing α1=2, t=10, α2=1, ξ=0.4, b1=2, n=2, and b2=1 in (2.16), we have the following:

    (i) The sine-Fibonacci sum of the series is

    e9F4e5(0.3)62Ft,3e4=3i=0Ft,ie(9i)[1(0.8)(9i)3e(0.3)(9i)2e2]

    =523194317.45, where Ft,0=1, Ft,1=1.83, Ft,2=0.71, and Ft,3=0.23.

    (ii) The co-secant-Fibonacci sum of the series is

    e9F4e5(0.3)62Ft,3e4=3i=0Ft,ie(9i)[1(0.8)(9i)3e(0.3)(9i)2e2]=1.49, where Ft,0=1, Ft,1=5.31, Ft,2=0.71 and Ft,3=0.94.

    Also, taking α1=1, b1=1, ξ=0.5, α2=2, and b3=1 in (2.16), we have the following co-secant Fibonacci polynomials:

    F0(t)=1,  F1(t)=cosec(2t),  F2(t)=cosec(t0.5)cosec(2t)+cosec(3t)

    F3(t)=cosec(t1)cosec(t0.5)cosec(2t)+cosec(t1)cosec(3t)+cosec(t0.5)cosec(2t), etc. Further-more, using (2.4), we have the following proportional α-derivative of the co-secant Fibonacci polynomials;

    Fα0(t)=0,  Fα1(t)=α[2cosec(2t)cot(2t)]+(1α)cosec(2t),

    Fα2(t)=α[2cosec(t0.5)cot(t0.5)cosec(2t)2cosec(t0.5)cosec(2t)cot(2t)3cosec(3t)cot(3t)]+(1α)[cosec(t0.5)cosec(2t)+cosec(3t)], and etc.

    The objective of this section is to demonstrate the efficacy of the bifurcation analysis of the ¯θ(t)-Fibonacci generating function and primary findings presented in this paper by employing analysis from the following precise examples.

    A discrete one-dimensional dynamical system is a system subjected to a single equation of this type

    x(t+1)=f(t) (4.1)

    where xZ and f is a function of x. The variable t is in general considered as the time, but in discrete systems the time takes only discrete values, so that it is possible to take tZ.

    A generalized discrete two-dimensional dynamical system is a system subjected to a single equation of this type

    x(t+2ξ)=x(t+ξ)+x(t) (4.2)

    where tR. When we reformulate Eq (4.2), we obtain a two-dimensional discrete system

    x(t+ξ)=x1(t)+x2(t)x2(t+ξ)=x1(t).

    A trajectory is a set {x(t)}t=0 of points satisfying the above equation (4.1). It is evident that the initial point x0=x(0) determines the entire trajectory. The behaviour of the dynamical system is therefore given by all the trajectories {x(t):x(0)=x0} for all initial values x0I. When the value of the parameter changes continuously, the behaviour of the system may change in a discontinuous way. One says that a bifurcation occurs for an isolate value of the parameter at which the type of dynamic changes. In bifurcation analysis, the region of stable operation is determined through the search of Hopf bifurcation points. This gives an insight into how the variations in the system parameters influence region of stable operation. This knowledge can be effectively used by the system designers to ensure the stability of the actual system.

    A bifurcation diagram is a traditional and visual way to look into how dynamical systems, difference equations, and differential equations change over time [10]. This tool is excellent for looking at how the system reacts to changes in parameters [19]. This diagram illustrates the system's different dynamic patterns and phase transitions by plotting the link between the system reaction and parameters [5,8]. This section employs bifurcation theory to determine the existence of the period-doubling (flip) bifurcation. We discuss the ¯θ(t)-Fibonacci generating function and investigate the bifurcation analysis of the ¯θ(t)-Fibonacci sequences.

    ¯θ(t)-Fibonacci sequences are generated by

    f(t)=11θ1(t)tθ2(t)t2, (4.3)

    where θ1(t) and θ2(t) are any trigonometric and hyperbolic functions. If θ1(t)=sin(b1t) and θ2(t)=sin(b2t) in (4.3), then we have the Fibonacci generating function

    f(t)=11sin(b1t)tsin(b2t)t2. (4.4)

    (ii). If θ1(t)=cos(b1t) and θ2(t)=cos(b2t) in (4.3), then we have the Fibonacci generating function

    f(t)=11cos(b1t)tcos(b2t)t2. (4.5)

    In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory. With the doubled period, it takes twice as long (or, in a discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system. A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are a common route by which dynamical systems develop chaos.

    Now, we consider the recurrent form of (4.3)

    tr=11θ1(tr1)tr1θ2(tr1)t2r1, (4.6)

    and we consider the recurrent form of (4.4)

    tr=11sin(b1tr1)tr1sin(b2tr1)t2r1. (4.7)

    We consider the recurrent form of (4.5)

    tn=11cos(b1tn1)tn1cos(b2tn1)t2n1. (4.8)

    We examine the orbit {tr}r=0 for any point t0 within the domain of the map.

    Figure 4(a) displays a periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the sine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1[8,11]. Figure 4(b) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1[5,5]. Figure 4(c) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the tangent function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1[5,1]. Figure 4(d) displays periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1[10,5].

    Figure 4.  Period doubling bifurcation diagram for ¯θ(t) -Fibonacci generating function.

    Figure 5(a) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the sine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b=2 and b1[0,0.3]. Figure 5(b) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=2 and b1[0.58,2]. Figure 5(c) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the tangent function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1[2.2,1.8]. Figure 5(d) displays sub periodic doubling bifurcation diagram of the Fibonacci generating function of the co-efficient of the cosine function with the initial condition t0=0.8 at the intrinsic growth bifurcation parameter b2=1 and b1[3,2.1].

    Figure 5.  Sub period doubling cascade for ¯θ(t) -Fibonacci generating function.

    This paper deduced an inverse formula for the ¯θ(t)-Fibonacci sequence. The inverse of a generic difference (nabla) operator with trigonometric coefficients of order 2 was used to derive this formula. The results we have obtained regarding the E and N solutions, Fibonacci polynomials, and the proportional derivative of the generic difference equation with trigonometric coefficients of order 2 will be applied to our future research. Additionally, we have conducted a bifurcation analysis of the ¯θ(t)-Fibonacci generating function.

    R.P: Conceptualization; R.P, S.T.M.T, and I.K.: Data curation; R.P, S.T.M.T, I.K, and M.V.C.: Formal analysis; P.R, I.K, and M.V.C.: Investigation; R.P, S.T.M.T, and I.K.: Methodology; R.P and S.T.M.T.: Writing-original draft; R.P, I.K and M.V.C.: Writing-review and editing. All authors have read and agreed to the published version of the article.

    The authors declare they have not used artificial intelligence (AI) tools in the creation of this article.

    "La derivada fraccional generalizada, nuevos resultados y aplicaciones a desigualdades integrales" Cod UIO-077-2024. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).

    The authors declare that they have no conflicts of interest.



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