Analysis of a discrete time fractional-order Vallis system

2.1 Vallis system

In 1986, Vallis was awarded three nonlinear systems for the identification of temperature fluctuations in the western and eastern parts of the equatorial ocean, which have a strong impact on the global climate of the earth (Vallis, 1988; Magnitskii et al., 2007).

Here $x$ the speed of water at the apparent surface of the ocean, $y=\frac{T_w-T_e}{2},z=\frac{T_w+T_e}{2}$ is and $T_w$ and $T_e$ are the temperature in the western and eastern parts of the sea, $\mu ,$ a are non-negative constants. This study focuses on the discrete Vallis system and the components of the basic three-component model are defined separately as $x(t),y(t),z(t)$ (Vallis, 1986).

2.2 Stability of the Vallis system

In this section, the stability conditions of the equilibrium points of the model (6) will be calculated. Let us consider an $n$ -dimensional nonlinear discrete-time system of equations:

Here, $i=1,2,3,\cdots ,n\mu ={\mu }_1,{\mu }_2,\cdots ,{\mu }_m$ are $t-$ independent parameters and $x(t)=x_1,x_2,\cdots ,x_n$ are variables.

Let the characteristic polynomial of (7) at some steady state $x_0$ has the form $$F\left(\lambda \right)=a_0{\lambda }^n+a_1{\lambda }^{n-1}+\cdots +a_{n-1}\lambda +a_n=0,a_0=1,a_i\left(\mu \right),i=0,\cdots ,n$$

$i=1,...,n$ is. Let's determine the stability of the equilibrium state by following the theorem (Li et al., 2011).

1. If all the eigenvalues ${\lambda }_i$ of the $n\times n$ Jacobian matrix $J({\mu }_0,x_0)$ of (7) lie in the open unit disk, i.e. $|{\lambda }_i|<1$ for all $i$ , then $x_0$ is asymptotically stable.

2. If the matrix $J({\mu }_0,x_0)$ has at least one eigenvalue ${\lambda }_0$ outside the open unit disk, i.e. | ${\lambda }_i$ | > 1, then $x_0$ is unstable.

1. $F\left(1\right)>0$ and ${\left(-1\right)}^{n}F\left(-1\right)>0,$

2. ${\mathrm{\Delta }}_1^{\pm }>0,{\mathrm{\Delta }}_3^{\pm }>0,\cdots ,{\mathrm{\Delta }}_{n-3}^{\pm }>0,{\mathrm{\Delta }}_{n-1}^{\pm }>0$ (when $n$ is even) or

${\mathrm{\Delta }}_2^{\pm }>0,{\mathrm{\Delta }}_4^{\pm }>0,\cdots ,{\mathrm{\Delta }}_{n-3}^{\pm }>0,{\mathrm{\Delta }}_{n-1}^{\pm }>0$ (when $n$ is odd) or.

Here, $$\begin{array}{c}{b}_n=1-a_0^2,b_{n-1}=a_{n-1}-a_1{a}_0,\cdots ,\hfill \\ b_j=a_j-a_{n-j}{a}_0,\cdots ,b_1=a_1-a_{n-1}{a}_0,b_0=0,\hfill \\ c_n=b_n-b_1\frac{b_1}{b_n},c_{n-1}=b_{n-1}-b_2\frac{b_1}{b_n},\cdots ,c_1=c_0=0,\hfill \\ ⋮\hfill \\ w_n=t_n-t_{n-1}\frac{t_{n-1}}{t_n},w_{n-1}=w_{n-2}=\cdots =0.\hfill \end{array}$$

in addition, the following Lemma is also mentioned.

Accordingly, for the model (6), to obtain the Jacobian matrix around any point $(x^{\ast },y^{\ast },z^{\ast })$ is done as follows:

Then the eigenvalues of $J(E_0)$ become ${\lambda }_1=1-\frac{h^{\alpha }}{\mathrm{\Gamma }\left(1+\alpha \right)},{\lambda }_2=1-\frac{h^{\alpha }}{2\mathrm{\Gamma }\left(1+\alpha \right)}\left(1-a\right)-\frac{\sqrt{{\left(\frac{h^{\alpha }}{\mathrm{\Gamma }\left(1+\alpha \right)}\right)}^2\left[{\left(1-a\right)}^2+4\mu \right]}}{2}$ and ${\lambda }_3=1-\frac{h^{\alpha }}{2\mathrm{\Gamma }\left(1+\alpha \right)}\left(1-a\right)+\frac{\sqrt{{\left(\frac{h^{\alpha }}{\mathrm{\Gamma }\left(1+\alpha \right)}\right)}^2\left[{\left(1-a\right)}^2+4\mu \right]}}{2}$ .

1. $E_0$ is called a sinking point and if $h<$ min $\left\{\sqrt[\alpha ]{2\mathrm{\Gamma }\left(1+\alpha \right),}\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a+\sqrt{\psi }}}\right\},$

2. $E_0$ is called a source point and if $h>$ max $\left\{\sqrt[\alpha ]{2\mathrm{\Gamma }\left(1+\alpha \right),}\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a-\sqrt{\psi }}}\right\},$

3. $E_0$ is saddle min $\left\{\sqrt[\alpha ]{2\mathrm{\Gamma }\left(1+\alpha \right),}\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a+\sqrt{\psi }}}\right\}<\mathrm{m}\mathrm{a}\mathrm{x}\left\{\sqrt[\alpha ]{2\mathrm{\Gamma }\left(1+\alpha \right),}\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a-\sqrt{\psi }}}\right\},$

4. $E_0$ is non-hyperbolic if $h=\sqrt[\alpha ]{2\mathrm{\Gamma }\left(1+\alpha \right),}$ or $h=\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a+\sqrt{\psi }}}$ or $h=\sqrt[\alpha ]{\frac{4\mathrm{\Gamma }\left(1+\alpha \right)}{1+a-\sqrt{\psi }}},$ where $\psi ={\left(a-1\right)}^2+4\mu$ .

Proof If $R_0\le 1$ , then the model (6) is seen to have $E_0$ . The Jacobian matrix in $E_0$ is.

Let's consider it as follows:

Let's write the characteristic equation of $J(E_1,E_2)$

Where

and $H=\frac{h^{\alpha }}{\mathrm{\Gamma }\left(1+\alpha \right)}$ .

According to theorem 2, the roots of the equation (11) are with respect to the unit disk if and only if

and.

$\mathrm{\Delta }=4b_1^3{b}_3-b_1^2{b}_2^2-18b_1{b}_2{b}_3+27b_3^2+4b_2^3$ . Otherwise $E_1$ is unstable.

Proof if we substitute (12) for (13),

According to the Jury criterion, the model is asymptotically stable when (6), (14a)-(14d) are arranged. It can be concluded that for positive parameters $a,\mu ,R_0>0,b_1>0,b_2>0,b_3>0$ is positive. Thus, if $R_0>1,$ (14a) is held. From (14b)-(14d) inequalities, let us consider the following equations:

Equation (15a) has a real root and a double conjugate complex root with the negative coefficient of $H={\xi }_1$ and $H^3$ . Therefore, when $H\in \xi (0,{\xi }_1),$ the inequality (14a) is provided. By solving the equation (15b), these four real roots and a pair of conjugate complex roots are obtained. The four real roots are repeated three roots at $H=0$ and the fourth at $H={\xi }_2$ . Therefore, the interval of the solution is $(0,{\xi }_1)$ at (14c). When equation (15c) is $0<H<{\xi }_3$ by solving, (14d) is violated. Therefore, the fact that (14a)-(14d) has at least one solution is obtained from the intersection of $H$ , ${\xi }_1-{\xi }_3$ i.e. $0<H<$ min $\{{\xi }_1,{\xi }_2,{\xi }_3\},$ .

2.3 Hopf bifurcation

In this section, we will briefly review the discrete Hopf bifurcation criterion. First, let's consider a two-dimensional parameterized system:

For a real variable parameter $\mu \in R$ and an equilibrium point $(x^{\ast },y^{\ast })$ , simultaneously $x=f(x^{\ast },y^{\ast };\mu )$ and $y=g(x^{\ast },y^{\ast };\mu )$ is the $\mu$ value that provides. With the eigenvalues ${\lambda }_{1,2}(\mu )$ providing ${\lambda }_2\left(\mu \right)={\overline{\lambda }}_1(\mu )$ becomes a Jacobian matrix $J(\mu )$ in this equilibrium. Also for some small details,

Then the system undergoes a Hopf bifurcation at the bifurcation point $(x^{\ast },y^{\ast },{\mu }^{\ast })$ . More precisely, for any left neighbor ${\mu }^{\ast }$ (i.e. $\mu <{\mu }^{\ast })$ , ${(x}^{\ast },y^{\ast })$ is a constant focus, and for any right neighbor of ${\mu }^{\ast }$ (i.e. $\mu >{\mu }^{\ast }$ ) this usually changes to be unstable surrounded by a limit cycle (Chen et al., 1999).

2.4 Grünwald-Letnikov algorithm (Binomial coefficients)

Equation obtained from the Grünwald-Letnikov limitation is used. The Grünwald-Letnikov derivative is a generalization of fractional orders of the higher-order derivative, the classical motivation of the derivative is that, from considering the limit form of the nth-order derivative, its similarity to the binomial theorem is seen, and a generalization is obtained in a similar way (Diaz et al., 1974). To study the behavior of a fractional-order chaotic system, the time approximation method GL is used. It has been found that nonlinearity has a great influence on the dynamics of the system. The Grünwald-Letnikov approximation is known that the two constraints are equivalent to the Grünwald-Letnikov constraint and the Caputo constraint. The $\mathit{mh}(m=1,2,...)$ node is associated for the numerical

Here K is “memory length”, calculation of time step $t_m=mh,h$ and binomial shape of ${(-1)}^i(\begin{array}{c}\phi \\ i\end{array})$ is $C_i^{\phi }(i=0,1,2,\cdots )$ . Let's calculate using the following expression (Vinagre et al., 2003): $$C_0^{\phi }=1,C_i^{\phi }=\left(1-\frac{1+\phi }{i}\right)C_{i-1}^{\phi },$$

Let's consider a nonlinear fractional system:

The initial values with are $\gamma (t_0)={\gamma }_0$ .

Here, ${{\alpha }^D}_t^{\phi }\gamma (t)$ , $g\left(t,\gamma \left(t\right)\right),g$ stands for the derivative of the $\gamma$ order of the function, $a$ and $t$ indicate the limits of the operation.

In this article, the fractional discrete-time equation using the fractional Caputo derivative with $0<\gamma <1$ and $\gamma$ is discussed.

If the Grünwald-Letnikov approach is applied to the equation (19) given above, binomial coefficients are preferred using the concept of memory, and a discrete equation is formed which is shortened by the length of memory.

this relationship (20) can be written as:

This relationship of GL is clear from (21). What is given in this relation is related to the total memory. If we substitute the first equation (20) in system (2), then $$\frac{1}{h^{{\phi }_1}}{\sum }_{i=0}^m{C}_i^{{\phi }_1}x\left(t_{m-i}\right)=\mu y\left(t_{-1+m}\right)-ax(t_{-1+m}),$$

If it is written this way, $$C_0^{{\phi }_1}x\left(t_m\right)+{\sum }_{i=1}^m{C}_i^{{\phi }_1}x\left(t_{m-i}\right)=(\mu y\left(t_{-1+m}\right)-ax(t_{-1+m}))h^{{\phi }_1},$$

Since $C_0^{{\phi }_1}=1$ , $$x\left(t_m\right)+{\sum }_{i=1}^m{C}_i^{{\phi }_1}x\left(t_{m-i}\right)=(\mu y\left(t_{-1+m}\right)-ax(t_{-1+m}))h^{{\phi }_1},$$

If it is written this way, $$x\left(t_m\right)=(\mu y\left(t_{-1+m}\right)-ax(t_{-1+m}))h^{{\phi }_1}-{\sum }_{i=1}^m{C}_i^{{\phi }_1}x\left(t_{m-i}\right),$$

Similarly, if the second and third equations of the system (2) are made, $$\begin{array}{c}y\left(t_m\right)=(-y\left(t_{-1+m}\right)+x(t_m)z(t_{-1+m}))h^{{\phi }_2}-{\sum }_{i=1}^m{C}_i^{{\phi }_2}y\left(t_{m-i}\right),\hfill \\ z\left(t_m\right)=(1-z\left(t_{-1+m}\right)+x(t_m)y\left(t_m\right))h^{{\phi }_3}-{\sum }_{i=1}^m{C}_i^{{\phi }_3}z\left(t_{m-i}\right),\hfill \end{array}$$ Here, using binomial coefficients with GL, $$\begin{array}{c}x\left(t_m\right)=(\mu y\left(t_{-1+m}\right)-ax(t_{-1+m}))h^{{\phi }_1}-{\sum }_{i=1}^m{C}_i^{{\phi }_1}x\left(t_{m-i}\right),\hfill \\ y\left(t_m\right)=(-y\left(t_{-1+m}\right)+x(t_m)z(t_{-1+m}))h^{{\phi }_2}-{\sum }_{i=1}^m{C}_i^{{\phi }_2}y\left(t_{m-i}\right),\hfill \\ z\left(t_m\right)=(1-z\left(t_{-1+m}\right)+x(t_m)y\left(t_m\right))h^{{\phi }_3}-{\sum }_{i=1}^m{C}_i^{{\phi }_3}z\left(t_{m-i}\right),\hfill \end{array}$$

is obtained.