Corrigendum to “Fractional nonlinear dynamics of learning with memory” nonlinear dynamics. 2020. Vol.100. P.1231–1242.

Abstract

In this Comment, we correct misprints in equations of the paper “Fractional Nonlinear Dynamics of Learning with Memory” Nonlinear Dynamics. 2020. Vol.100. P.1231–1242. We also give conditions of the existence of solutions for nonlinear fractional differential equation, which correct the conditions given in Propositions 3.8 and 3.9 of the book (Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006). These conditions impose restrictions on the existence of solutions of nonlinear equations that describe the dynamics of learning with memory. Using these conditions, we give the correct formulations of the principle of inevitability of growth for process with memory and the principle of changing growth rates by memory, which are proposed in the commented article.

Appendix

In this Appendix, we give the proof of the form of the explicit solution of a nonlinear fractional differential equation, and then we obtain conditions for the existence of this solution.

Let us consider the nonlinear fractional differential equation

$$\begin{aligned} (D^{\alpha }_{C;t_0+} Z)(t) = n_0 \, (t-t_0)^{\delta } \, Z^a(t), \end{aligned}$$

(17)

where $n_0 > 0$, $t>t_0$, $\delta \in \mathbb {R}$ and $\alpha >0$ is non-integer positive parameter $\alpha \in (n-1,n)$, $n \in \mathbb {N}$.

We will seek a solution in the power-law form

$$\begin{aligned} Z(t) = C \, (t-t_0)^{b} , \end{aligned}$$

(18)

where the coefficients $C >0$, and b are real constants, $t>t_0$. Substituting equation (18) into fractional differential equation (17), we obtain the equality

$$\begin{aligned} C \, \left( D^{\alpha }_{C;t_0+} (\tau -t_0)^{b} \right) (t) = n_0 \, (t-t_0)^{\delta } \, C^a \, (t-t_0)^{ab} . \end{aligned}$$

(19)

We should consider the following three cases: (1) the real values $b>n-1$; (2) the non-negative integer values $b \le n-1$; (3) the other real values $b \le n-1$.

For $b>n-1$, we should use the equation of the Caputo fractional derivative of the power-law function in the form

$$\begin{aligned} \left( D^{\alpha }_{C,t_0+} (\tau -t_0)^{b} \right) (t) = \frac{\Gamma (b+1)}{\Gamma (b-\alpha +1)} \, (t-t_0)^{b-\alpha } \end{aligned}$$

(20)

for non-integer values of $\alpha \in (n-1,n)$, $n \in \mathbb {N}$. Using (20), equation (19) takes the form

$$\begin{aligned} C \, \frac{\Gamma (b+1)}{\Gamma (b-\alpha +1)} \, (t-t_0)^{b-\alpha } = n_0 \, (t-t_0)^{\delta } \, C^a \, (t-t_0)^{ab} . \end{aligned}$$

(21)

Using $C \ne 0$ and $t>t_0$, we get that equation (21) holds if the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\Gamma (b+1)}{\Gamma (b-\alpha +1)}-n_0 \, C^{a-1}=0 , \\ b-\alpha = \delta + ab , \end{array} \right. \end{aligned}$$

(22)

are satisfied. Equalities (22) give

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle C= \left( \frac{\Gamma (b+1)}{ n_0 \Gamma (b-\alpha +1)} \right) ^{1/(a-1)} , \\ \displaystyle b = \frac{\delta + \alpha }{1-a} . \end{array} \right. \end{aligned}$$

(23)

We proved that function (18) in the form

$$\begin{aligned} Z(t) = \left( \frac{\Gamma (b+1)}{ n_0 \Gamma (b-\alpha +1)} \right) ^{1/(a-1)} \, (t-t_0)^{(\delta + \alpha )/(1-a)} , \end{aligned}$$

(24)

can be considered as a solution of fractional differential equation (17) under some conditions (these conditions will be described below).

To simplify expressions, we will use the notation

$$\begin{aligned} \gamma (\alpha ) = \frac{\delta +a \alpha }{a-1} . \end{aligned}$$

(25)

Therefore,

$$\begin{aligned} \frac{\delta + \alpha }{1-a}= \alpha - \gamma (\alpha ) . \end{aligned}$$

(26)

For integer values $b\le n-1$, i.e., $b=0,1,2,...,n-1$, we should use the equality

$$\begin{aligned} \left( D^{\alpha }_{C,t_0+} (\tau -t_0)^{b} \right) (t) = 0 , \end{aligned}$$

(27)

for non-integer values of $\alpha \in (n-1,n)$, $n \in \mathbb {N}$. Substituting equation (27) into fractional differential equation (17), we obtain the equality

$$\begin{aligned} n_0 \, (t-t_0)^{\delta } \, C^a \, (t-t_0)^{ab} =0 . \end{aligned}$$

(28)

Therefore, there are no solutions to equation (28) for non-zero values of the parameters C, $n_0$, $t>t_0$.

For $b\le n-1$ such that $b \ne 0,1, ... , n-1$, the integral in the expression of the Caputo fractional derivative is improper and divergent.

As a result, solution (24) can be written in the form

$$\begin{aligned} Z(t) = \left( \frac{n_0 \Gamma (1-\gamma (\alpha ))}{\Gamma (\alpha -\gamma (\alpha )+1)} \right) ^{1/(1-a)} \, (t-t_0)^{\alpha -\gamma (\alpha )} , \end{aligned}$$

(29)

where $a>0$, $a \ne 1$, $n_0>0$, $b>n-1$, $\alpha \in (n-1,n)$, $n \in \mathbb {N}$, and we use using $1/(a-1)=-1/(1-a)$.

By definition of the Caputo fractional derivative, solution (29) must be continuous and hence bounded in a right neighborhood of $t_0$, i.e., $[t_0,t_0+\epsilon ]$ with $\epsilon >0$, which yields the condition $\alpha -\gamma (\alpha ) \ge 0$ and hence $Z(t) \in C[t_0,t_1]$.

Using expression (29), the right-hand side of equation (17) has the form

$$\begin{aligned}&F[t,Z(t)] = n(t) \, Z^a(t) \nonumber \\&\quad = n_0 \left( \frac{n_0 \Gamma (1-\gamma (\alpha ))}{\Gamma (\alpha -\gamma (\alpha )+1)} \right) ^{a/(1-a)} \, \nonumber \\&\qquad (t-t_0)^{a(\alpha -\gamma (\alpha ))+\delta } . \end{aligned}$$

(30)

Using equality (25), we get

$$\begin{aligned}&a \, (\alpha -\gamma (\alpha ))+\delta = a \, \left( \alpha - \frac{\delta +a \alpha }{a-1} \right) + \delta \nonumber \\&\quad = -\frac{\delta + a \alpha }{a-1} = - \gamma (\alpha ) . \end{aligned}$$

(31)

Therefore, expression (30) takes the form

$$\begin{aligned}&F[t,Z(t)] = n(t) \, Z^a(t) \nonumber \\&\quad = n_0 \left( \frac{n_0 \Gamma (1-\gamma (\alpha ))}{\Gamma (\alpha -\gamma (\alpha )+1)} \right) ^{a/(1-a)} \, (t-t_0)^{-\gamma (\alpha )} . \end{aligned}$$

(32)

The right-hand side of equation (17), which is given by expression (32), belongs to $C_{\gamma }[t_0,t_1]$ in the case $\gamma (\alpha ) \in (0,1)$ and function (32) belongs to $C[t_0,t_1]$ in the case $\gamma (\alpha ) \le 0$. Therefore, we have condition $\gamma (\alpha )<1$. Here $C_{\gamma }[t_0,t_1]$ is the weighted space of functions $Z(\tau )$ given on $(t_0,t_1]$, such that the function $(\tau -t_0)^{\gamma } A(\tau ) \in C[t_0,t_1]$, and

$$\begin{aligned} \Vert Z(\tau ) \Vert _{C_{\gamma }} = \Vert (\tau -t_0)^{\gamma } Z(\tau ) \Vert _{C} , \quad C_{0}[t_0,t_1]=C[t_0,t_1] . \end{aligned}$$

(33)

As a result, we have the conditions of the existence of solution (29) in the form

$$\begin{aligned} \left\{ \begin{array}{l} a>0, \quad a \ne 1, \\ \alpha \in (n-1,n) , \\ \gamma (\alpha ) < 1 , \\ b= \alpha - \gamma (\alpha ) >n-1 . \end{array} \right. \end{aligned}$$

(34)

The conditions $b>n-1$ of system (34) have the form $\alpha -\gamma (\alpha ) > n-1$. Then we have $\gamma (\alpha ) < \alpha -(n-1)$ that can be written as $\gamma (\alpha ) < \{\alpha \}$, where $\{\alpha \} \in (0,1)$ is the fractional part of $\alpha$, i.e., $\alpha = [\alpha ]+\{\alpha \}$, $[\alpha ]=n-1$. Therefore, if the condition $b= \alpha - \gamma (\alpha ) >n-1$ holds, then inequality $\gamma (\alpha ) <1$ is satisfied. This fact allows us to represent system (34) in the form

$$\begin{aligned} \left\{ \begin{array}{l} a>0, \quad a \ne 1 , \\ \alpha \in (n-1,n) ,\\ \displaystyle b= \frac{\delta +\alpha }{1-a} >n-1 . \end{array} \right. \end{aligned}$$

(35)

We can consider the following two cases

$$\begin{aligned}&\left\{ \begin{array}{l} a \in (0,1) , \\ \alpha \in (n-1,n) , \\ \delta + \alpha> (n-1)(1-a) , \end{array} \right. \quad or \nonumber \\&\quad \left\{ \begin{array}{l} a >1 , \\ \alpha \in (n-1,n) , \\ \delta + \alpha < (n-1)(1-a) . \end{array} \right. \end{aligned}$$

(36)

These systems give conditions (12) that are used in the Statement. The solution of nonlinear fractional differential Eq.(17) with $\alpha \in (n-1,n)$, $n \in \mathbb {N}$ is described by equations (29) if conditions (36) are satisfied. This completes the proof.