Phenomenological approach to describe oscillatory growth or decay in different dynamical systems

Abstract

The approach of the phenomenological universalities of growth is considered to describe the behaviour of a system showing an oscillatory growth. Two phenomenological classes are proposed to consider the oscillatory behaviour of a system. One of them is showing oscillatory nature with constant amplitude and the other represents oscillatory nature with a change in amplitude. The term responsible for decay (or growth) in amplitude in the proposed class is also been identified. The variations in the nature of oscillation with the dependent parameters are studied in this communication. In this connection, the variation of a specific growth rate is also been considered. The significance of the presence and the absence of each term involved in the phenomenological description are also taken into consideration. These proposed classes might be useful for the experimentalists to extract a characteristic feature from the data set and to develop a suitable model consistent with their data set.

1 Introduction

In the field of science and technology, it is found that many characteristic patterns have been discovered that are remarkably similar in nature, though they are related to totally different phenomenologies [1–5]. It is because of this fact that the background mathematical formalism remains the same. Therefore, an universality in terms of mathematical formalism is observed, in which a mathematical equation is showing universality in the sense that the variables involved in the equation may represent any scientific discipline, from social science to engineering. A search for such universality in different phenomenologies is very important from the point of view of applications. A cross-fertilization among different fields might be accelerated when different fields share a common mathematical formalism or concept. One of such approaches is the classification scheme of phenomenological universalities [6].

Fig. 1

a Plots of oscillatory nature of the variable y with $b_1=0$ and $b_2=2$. The values of the parameter $b_0$ are 0.02, 0.82 and 1.62 from the top to the bottom. b Curves of specific growth rate with $b_1=0$ and $b_2=2$. The values of the parameter $b_0$ are 1.62 and 0.82 from left to right

Fig. 2

a Plots of oscillatory nature of the variable y with $b_1=0$ and $b_0=0.2$. The values of the parameter $b_2$ are 0.42, 1.42 and 2.42 from the top to the bottom . b Curves of specific growth rate with $b_0=0.02$ and $b_1=0$. The values of the parameter $b_2$ are 3.6, 2.4 and 1.2 from left to right

Fig. 3

Plots of damped-oscillatory nature of the variable y with $b_0=4.0$ and $b_2=0.4$. The values of the parameter $b_1$ are 0.8, 1.2 and 1.6 from the top to the bottom

In the phenomenological approach, the universality class at the level $n=0$ corresponds to Malthusian or autocatalytic processes. At the level $n=1$ (class U1), it produces Gompertz law of growth [7] that was being used for more than a century to describe growth phenomena in diversified fields. A variation of the class, termed as involuted Gompertz function, is used to describe the evolution and involution of organs undergoing atrophy [8]. The class (U2) corresponding to the level $n=2$ leads to the growth equation describing different biological growth patterns [9–13]. The classes corresponding to $n=1$ and $n=2$ have been applied in diversified and unrelated fields [14–17]. An extension of this approach, termed as complex universality, is characterized [18] and applied to explain the concurrent growth of two phenotypic features [19]. A vector generalization of the phenomenological universality is used to describe the interactive growth of two or more organisms in a given environment [20, 21]. The approach of vector universality is successfully employed to extract different features of tumour growth [22].

Fig. 4

Plots of damped-oscillatory nature of the variable y with $b_1=0.9$ and $b_2=0.4$. The values of the parameter $b_0$ are 2.0, 4.0, 6.0 and 8.0 from the top to the bottom

Fig. 5

Plots of damped-oscillatory nature of the variable y with $b_0=2.0$ and $b_1=0.9$. The values of the parameter $b_2$ are 0.3, 0.4, 0.5 and 0.6 from the top to the bottom

Fig. 6

Plots of  critically damped nature of the variable y with $b_0=2.0$ and $b_2=0.4$. The values of the parameter $b_1$ are 1.6 and 1.8 from the top to the bottom. The condition defined by $b_0=2.0$, $b_1=1.8$ and $b_2=0.4$ is just above the threshold of the condition $b_1^2=4b_0b_2$ and represents a behaviour similar to critically damped nature of a classical oscillator

The formalism of phenomenological approach [6] of classes n = 0, 1, 2 lead to monotonous growth of the system with no scope of oscillations, though they are an essential feature of different complex systems of nature. In some cases they are not observed, might be due to the fact that they were overlooked, or masked by large experimental errors or missed due to the inadequate range of variation of the variables being measured. Apart from these factors, oscillations are observed in different types of complex systems [23–33]. The existence of volume oscillation, of relatively large amplitude, in the growth of multicellular tumour spheroids is experimentally found [34, 35]. A variation of the phenomenological approach in the complex field instead of the real one, termed as complex universalities, is applied to analyze two correlated oscillatory features of the growing systems [18]. In the same framework, concurrent growth of phenotypic features are also explained [19]. One of the essential features of the complex universalities approach is that the oscillatory growth could be addressed in case of two coupled variables of a system. Therefore, it is not possible to explain the oscillating behaviour of a system in which such coupling is not observed.

In the phenomenological approach, it is expected that the values of the coefficients are governed by the particular growth mechanism involved in the system [6]. The biological and environmental constraints might induce a new growth processes or cease a growth mechanism which is already existing in the system. Therefore, the inclusion or withdrawal of a growth mechanism from a system may be considered as the deviation from the normal growth process due to some constraints. A deviation from the normal biological growth in terms of oscillation which is observed in the growth of multicellular tumour [34, 35], may be treated as adaptation or withdrawal of a new growth mechanism. Phenomenologically the new growth mechanism may be treated as inclusion and/or exclusion of a new phenomenological term in the class U2 [6]. The growth of a tumour may be treated as a deviation from the normal growth due to some biological and perhaps environmental constraints. The coexistence of tumour growth along with normal growth of tissues within the same biological system and the beginning of tumour growth from normal growth may be treated as an indication of such deviation. The growth of a tumour shows such oscillatory nature in terms of its volume with relatively large amplitude [34, 35]. This communication is motivated by these phenomenological observation and aimed to describe phenomenologically oscillatory behavior of a growing system.

In the present communication, phenomenological approach is considered to describe the behaviour of a system showing oscillatory growth. Here, we propose two different phenomenological classes. One of them shows oscillation with constant amplitude (class-I). The other shows growth or decay in the amplitude of oscillation (class-II). The emphasis is given to find out the role of coefficients involved in the phenomenological description of a dynamical system. Even a linear growth feature of the system may also be observed in absence of a phenomenological terms related to normal growth processes. The nature of variation of specific growth rates with different parameters are also studied. The phenomenological term responsible for decay (or growth) in amplitude in the proposed class is also identified. The proposed phenomenological classes might be helpful to address the periodic variation of the different physical properties of complex systems. This analysis is focused on helping the experimentalists to decide whether their data set, showing oscillatory nature, might be considerably expected to fit with these phenomenological classes.

This paper is organized as follows: In Sect. 2, we would propose two phenomenological classes presenting oscillatory nature. In this connection, the classification scheme of phenomenological universalities found in literature [6] is discussed in brief. Different aspects of proposed phenomenological classes would be considered in Sect. 3. Finally, we would conclude with our results in Sect. 4.

2 Phenomenological approach to oscillatory dynamical systems

A string of data $Y_j(t_j)$ showing a particular phenomenon can be described by an ordinary differential equation and is represented in the phenomenological approach in the following manner [6],

$$\begin{aligned} \frac{dY(t)}{dt}=\alpha (t)Y(t) \end{aligned}$$

(1)

where $\alpha (t)$ stands for specific growth rate of the variable Y(t). Equation (1) can also be expressed in terms of adimensional variable as,

$$\begin{aligned} \frac{dy(\tau )}{d\tau }=a(\tau )y(\tau ) \end{aligned}$$

(2)

where, $\tau =\alpha (0)t$, $y(\tau )=Y(t)/Y(0)$ and $a(\tau )=\alpha (t)/\alpha (0)$ are adimensional time, adimensional variable and adimensional specific growth rate respectively. It is assumed that $a(0)=y(0)=1$. An adimensional variable (z) is introduced in this connection so that,

$$\begin{aligned} z=\ln y \end{aligned}$$

(3)

The time variation of z is described by the following relations,

$$\begin{aligned} a=\frac{dz}{d\tau } \end{aligned}$$

(4)

and,

$$\begin{aligned} \varphi (a)=-\frac{d^2z}{d\tau ^2} \end{aligned}$$

(5)

Again, $\varphi (a)$ can be expressed in terms of power series as,

$$\begin{aligned} \varphi (a)={\sum _0^\infty } b_n a^n \end{aligned}$$

(6)

Thus, from Eqs. (3) and (4), the following expression can be derived,

$$\begin{aligned} \ln y=\int ad\tau + constant \end{aligned}$$

(7)

Different forms of $\varphi$ would generate different types of growth equations. Each term in the right hand side of the Eq. (6) represents different types of growth mechanisms involved in the system [6]. These mechanisms might be independent with respect to each other or they might be mutually dependent. In case of Gompertz-type growth, $b_0=0$ and $b_n=0$ for $n\ge 2$. It indicates that the growth mechanism corresponding to the term $b_1a$ is not dependent on the other growth mechanisms found in nature. In case of West-type allometry based biological growth process, $b_0=0$ and $b_n=0$ for $n\ge 3$. The growth mechanism corresponding to the terms $b_1a$ and $b_2a^2$ are simultaneously found in this type of system. Thus, the sole existence of the growth mechanism corresponding to $b_1a$ is reported and co-existence of growth mechanisms corresponding to the terms $b_1a$ and $b_2a^2$ are also found in nature. The sole existence of the growth mechanism corresponding to the term $b_2a^2$ or a coexistence with the terms other than $b_1a$ is still not reported or not even considered from the theoretical point of view in the phenomenological approach. Here, such a possibility is explored from the phenomenological point of view.

In the phenomenological class (class-I) represented by $\varphi =b_0+b_2a^2$ with $b_1=0$ and all $b_n=0$ for $n>2$, the analytical solution of y is expressed as,

$$\begin{aligned} y=A^{\frac{1}{\alpha \beta }}\mid cos[\alpha \tau - \theta ]\mid ^{\frac{1}{\alpha \beta }} \end{aligned}$$

(8)

where, $A=\frac{1}{\mid cos \theta \mid }$, $\theta = \arctan \beta$, $\alpha = \sqrt{b_0b_2}$, $\beta =\sqrt{\frac{b_2}{b_0}}$. Here, $\alpha$ might be treated as the frequency of oscillation.

The solution of the class $\varphi =b_0+b_2a^2$ can also be achieved in terms of sine function as,

$$\begin{aligned} y=A^{\frac{1}{\alpha \beta }}\mid sin[\alpha \tau + \xi ]\mid ^{\frac{1}{\alpha \beta }} \end{aligned}$$

(9)

where, $A=\frac{1}{\mid sin \xi \mid }$, $\xi = \cot ^{-1} \beta$.

Though the solution can be expressed in terms of sine function as well as cosine function, the cosine function will be considered to describe the behaviour of the system in this communication. It is verified that the same features of the system could be extracted in both cases.

The specific growth rate of this phenomenological class (class-I) is expressed as,

$$\begin{aligned} a=\frac{1}{\beta }\tan (\theta -z) \end{aligned}$$

(10)

where, $z=\sqrt{b_0b_2}\tau$.

It is not possible to describe the behaviour of the system with the help of a single differential equation. It is because of the fact that y is not differentiable when $[\alpha \tau - \theta ]=\frac{\pi }{2}, \frac{3\pi }{2}, \frac{5\pi }{2},\ldots$. Therefore, the formation of segmented or piece-wise differential equations are possible that represent the behaviour of the system. The differential equation given below,

$$\begin{aligned} \frac{1}{y}\frac{d^2y}{d\tau ^2}+\frac{b_2-1}{y^2}\left( \frac{dy}{d\tau }\right) ^2=b_0 \end{aligned}$$

(11)

is valid for the interval $(2q+1)\frac{\pi }{2}<[\alpha \tau - \theta ]<(2q+3)\frac{\pi }{2}$, where $q=0, 2, 4,\ldots$  . For the interval $0<[\alpha \tau - \theta ]<\frac{\pi }{2}$ and $(2m+1)\frac{\pi }{2}<[\alpha \tau - \theta ]<(2m+3)\frac{\pi }{2}$ (where, $m=1, 3, 5,\ldots$), the differential equation is given by,

$$\begin{aligned} \frac{1}{y}\frac{d^2y}{d\tau ^2}+\frac{b_2-1}{y^2}\left( \frac{dy}{d\tau }\right) ^2=-b_0 \end{aligned}$$

(12)

For the phenomenological class (class-II) corresponding to $\varphi =b_0+b_1a+b_2a^2$ with all $b_n=0$ for $n>2$ and satisfying the condition $b_1^2<4b_0b_2$; i.e; zeros of the function $\varphi$ is complex; the system is governed by the following expression,

$$\begin{aligned} y=Pexp(-\sigma \tau )\mid \cos (\omega \tau - \gamma )\mid ^{\frac{1}{b_2}} \end{aligned}$$

(13)

where, $P=[\frac{b_1^2+4b_2(b_2+b_1+b_0)}{4b_0b_2-b_1^2}]^{\frac{1}{2b_2}}$, $\sigma =\frac{b_1}{2b_2}$, $\omega =\sqrt{b_0b_2-\frac{b_1^2}{4}}$, $\tan \gamma =\frac{b_1+2b_2}{\sqrt{4b_0b_2-b_1^2}}$ [Derivation of Eq. (13) is given in Appendix]. In this case, $\sigma = \frac{b_1}{2b_2}$, may be termed as the damping factor of the oscillating system, and $\omega =\sqrt{b_0b_2-\frac{b_1^2}{4}}$, may be termed as the angular frequency of the oscillation.

3 Results and discussions

In the phenomenological class-I, defined by $\varphi =b_0+b_2a^2$, $\alpha$ and $A^{\frac{1}{\alpha \beta }}$ (both of them are function of $b_0$ and $b_2$) can be treated as the angular frequency and the amplitude of oscillation respectively. It is found that the amplitude of oscillation of the system decreases with the increase of $b_0$ or $b_2$. In case of change in frequency of oscillation, $b_0$ and $b_2$ play the same role, but the change in frequency is more effected by the change in $b_0$ than the change in $b_2$. The frequency of oscillation increases with the increase in $b_0$ and $b_2$. The change in nature of oscillation with the change in $b_0$ and $b_2$ is represented in Figs. 1(a) and 2(a). The nature of oscillation can also be characterized by the sharpness of oscillation which does not depend on $b_0$. It is totally controlled by $b_2$ [as observed in Figs. 1(a) and 2(a)]. The sharpness of the characteristic curve increases with the decrease in $b_2$ and it is shown in Fig. 2(a). No such behaviour is observed in case of $b_0$. The proposed phenomenological class might be useful to phenomenologically describe the sustained oscillations observed in the different physical systems [28, 31–33].

The change in specific growth rate of class-I with the change in $b_0$ and $b_2$ are represented in Figs. 1(b) and 2(b). It is found that the frequency of oscillation of specific growth rate increases with the increase in $b_0$ and $b_2$. As a consequence, the time period corresponding to the change in specific growth rate decreases with the increase of values of the phenomenological coefficients. The frequency of oscillation of y is identical with the frequency of oscillation of specific growth rate, as expected from Eqs. (8) and (10).

The phenomenological class-II corresponding to $\varphi =b_0+b_1a+b_2a^2$ with the condition $b_1^2<4b_0b_2$ leads to damped oscillatory nature of a growing system. The amplitude of oscillation is given by $P exp(-\sigma \tau )$. It is a function of time in class-II whereas it is independent of time in class-I. The exponential decay in amplitude of class-II is governed by the magnitude of $b_1$ and $b_2$, as it is expected from Eq. (13). The decay [represented by the term $exp(-\sigma \tau )$] in amplitude increases with increase in $b_1$ and with the decrease in $b_2$, as shown in the Figs. 3 and 5. The decay does not depend on $b_0$ as it is not function of $b_0$. Thus, it can be concluded that the damping of the system depends on $b_1$ and $b_2$. An undamped system may be considered for either $b_1=0$ or $b_2\rightarrow \infty$. It is not possible to attain the second condition in reality but it is possible to consider a situation for which $b_1=0$ or $b_1\rightarrow 0$. Therefore it can be concluded that decay in amplitude could easily be controlled by $b_1$. The magnitude of P increases with the increase in $b_1$. But, the same decreases with the increase in $b_0$ and $b_2$.

The frequency of the damped oscillatory system (phenomenological class-II) may be represented by $\omega$, as it is found in Eq. (13). It increases with the increase in $b_0$ and $b_2$ (shown in Figs. 4 and 5 respectively) but the same decreases with the increase in $b_1$ (shown in Fig. 5). It is interesting to note that the damped frequency ($\omega$) of oscillation is less than the undamped frequency ($\alpha$) of oscillation. The difference between $\alpha$ and $\omega$ depends on $b_1$. The amplitude of oscillation shows exponential growth for $b_1<0$ and $b_2>0$. This proposed phenomenological class might be helpful to describe oscillatory behaviour of a system accompanied by damping (may be positive or negative).

The characteristic equation of undamped oscillation (phenomenological class-I) could be obtained from the characteristic solution of damped oscillation (phenomenological class-II) by considering $b_1=0$ (the derivation of phenomenological class-I from the phenomenological class-II for $b_1\rightarrow 0$ is given in Appendix). In a similar fashion, the undamped frequency ($\alpha$) is obtained by considering $b_1=0$ in the expression of damped frequency ($\omega$) which is less than the undamped frequency. Therefore, the change in frequency due to damping is solely determined by the term $b_1$, which also controls the decay in amplitude due to damping. Hence, it can be concluded that the mechanism involved in the system represented by the term $b_1a$ is independent with respect to other mechanisms and solely responsible for damping. In absence of the terms $b_0$ and $b_1$, the phenomenological class is represented by $\varphi =b_2a^2$ and corresponding growth function is given by,

$$\begin{aligned} y=(b_2\tau +1)^{\frac{1}{b_2}} \end{aligned}$$

(14)

The same can also be derived from Eq. (13). In this case, $b_2=1$ leads to a linear growth of the system. A system showing always linear growth with respect to time is not normally observed in nature.

When the values of $b_0$, $b_1$ and $b_2$ are very close to the threshold condition $b_1^2=4b_0b_2$, the oscillatory nature along with damping is about to cease, as represented in the Fig. 6. It is also found that the behaviour of the system is similar to the critically damped oscillatory nature of a classical oscillator when the values of the phenomenological coefficients are just above the threshold condition, $b_1^2=4b_0b_2$. This nature is represented in the Fig. 6. The behaviours of the damped oscillation and undamped oscillation of the phenomenological classes are very similar to the behaviours observed in case of classical oscillations. In case of a classical oscillator, the undamped oscillation is characterised by the mass of the system, the stiffness factor or force constant and the damping is determined by the interaction of the system with the environment. The withdrawal of that interaction causing damping generates an undamped oscillation. The damping of a classical oscillator depends on the property of the surrounding medium. Mass of a classical oscillator contributes in determining the damping and frequency of oscillation. Therefore, as an analogy, it can be concluded that the term $b_1$ plays the same role as the interaction of the system with the environment in case of classical oscillator. The terms, $b_0$ and $b_2$, determine undamped frequency of oscillation that is governed by force constant and mass of a classical oscillator.

A comparison between the proposed phenomenological approach (showing oscillatory nature) and the others found in different literature [18, 36, 37] is essential to understand the importance and applicability of this approach. One of the phenomenological approaches has considered that specific growth rate a [of a characteristic variable y] could be treated as an addition of time-dependent specific growth rate $a_t$ and y-dependent specific growth rate $a_y$ [36, 37]. In this approach, the time derivative $\dot{ a_y }$ could be expressed in terms of power series of $a_y$ (as normally considered in phenomenological approach) whereas time derivative $\dot{ a_t }$ could be expressed in terms of power series of E, where E is expressed in terms of harmonic functions. As a result, the behavior of y is oscillatory that is governed by the properties of these two power series. The contribution coming from $\dot{ a_y }$ acts as amplitude of oscillation which is a function of time. The contribution of $\dot{ a_t }$ is responsible for oscillatory nature of y. Different phenomenological classes showing oscillatory nature could be produced by truncating those power series up to different terms. The contribution coming from $\dot{ a_y }$ attains saturation level (for sigmoidal growth) with time and the system shows sustained oscillation in the steady state [37]. This approach is used to study oscillations in the growth of multicellular tumour spheroids [37]. In another approach, the coupling between two characteristic variables of a growing system is addressed with the help of a complex variable in the phenomenological approach [18]. This approach leads to different complex universality classes [CUN]. Characteristic variables and corresponding specific growth rates of different CUN exhibit oscillatory nature with respect to time. One of the essential features of this approach is that two characteristic variables of a growing system must be coupled via a complex interaction (presented by a complex variable). In the newly proposed phenomenological approach, no such explicit dependence of specific growth rate on time in terms of harmonic function (as reported in [36, 37]) or a coupling between two variables (as reported in [18]) are considered. In this approach, it is assumed that specific growth rate of class-I depends explicitly on y only. It does not have a direct time dependence (though it depends indirectly on time because y is time-dependent). As a result, the system (class-I) proposed here is endogenous by nature [18]. Class-II may be a non-endogenous system. Therefore, it can be concluded that the proposed phenomenological approach is applicable for investigating oscillatory growth (dependent on single phase variable only) of a complex system.

4 Conclusions

Two phenomenological classes exhibiting oscillatory nature are proposed in the present communication. The variation in oscillating behaviour is analyzed with the variation of different phenomenological coefficients and is represented graphically. The variation in specific growth rate is also studied. The phenomenological class representing growth (or decay) in amplitude of oscillation is identified with the conditional relationship between the phenomenological coefficients. In this connection, an analogy between the behaviour of damped classical oscillation and the damped oscillation of a complex system in phenomenological approach is considered. The term solely responsible for growth (or decay) in amplitude is also pointed out. An effort is given to identify the physical significance of each term involved in the phenomenological description from this analogy. In a nutshell, the proposed phenomenological description is able to address the oscillation of a system with constant amplitude. The phenomenological classes may be useful to describe the behaviour of oscillatory physical systems [28, 32, 34, 35]. It may be beneficial to address oscillatory growth of different diseases. These proposed phenomenological classes might be helpful for the experimentalists to consider their data set from a totally different point of view and to extract a suitable model for describing the temporal evolution of the system.

Appendix

1.1 Appendix 1 : Derivation of Eq. (13)

Equation (5) can be expressed in the following way,

$$\begin{aligned} \varphi (a)=-\frac{da}{d\tau } \end{aligned}$$

(15)

When zeros of the function $\varphi =b_0+b_1a+b_2a^2$ is complex, specific growth rate can be calculated from (15) using the condition $a(0)=1$ at $\tau =0$ and is expressed as,

$$\begin{aligned} a=\frac{k-M\tan Q}{k+z_0\tan Q} \end{aligned}$$

(16)

where $M=\frac{b_0}{b_2}+\frac{b_1}{2b_2}$, $k^2=\frac{b_0}{b_2}-\frac{b_1^2}{4b_2^2}$, $Q=kb_2\tau$ and $z_0=1+\frac{b_1}{2b_2}$.

Now eliminating a from Eq. (7) with the help of Eq. (16) and using the condition $a(0)=y(0)=1$ at $\tau =0$, y can be expressed as,

$$\begin{aligned} y=\left| \cos Q+\frac{z_0}{k}\sin Q\right| ^{\frac{1}{b_2}}\exp \left( -\frac{b_1\tau }{2b_2}\right) \end{aligned}$$

(17)

By considering $p\cos \gamma =1$ and $p\sin \gamma =z_0/k$, Eq. (17) takes the form given by,

$$\begin{aligned} y=p^{\frac{1}{b_2}}\vert \cos (\omega \tau -\gamma )\vert ^{\frac{1}{b_2}}\exp \left( -\frac{b_1\tau }{2b_2}\right) \end{aligned}$$

(18)

where $p=\sqrt{\frac{b_1^2+4b_2(b_2+b_0+b_1)}{4b_0b_2-b_1^2}}$, $\tan \gamma =z_0/k=\frac{b_1+2b_2}{\sqrt{4}b_0b_2-b_1^2}$ and $\omega =kb_2$.

By replacing $p^{\frac{1}{b_2}}$ by P and $-b_1/2b_2$ by $\sigma$, Eq. (18) could be expressed in the form of Eq. (13).

1.2 Appendix 2 : Analysis of Eq. (13) when $b_1\rightarrow 0$:

When $b_1\rightarrow 0$,

• 1. $\sigma \rightarrow 0$

• 2. $\omega \rightarrow \sqrt{b_0b_2}=\alpha$

• 3. $z_0/k\rightarrow \sqrt{\frac{b_2}{b_0}}=\beta$

Therefore, for the condition $b_1\rightarrow 0$, Eq. (17) is given by,

$$\begin{aligned} y=\left| \cos \sqrt{b_0b_2}\tau +\sqrt{\frac{b_2}{b_0}}\sin \sqrt{b_0b_2}\tau \right| ^{\frac{1}{b_2}} \end{aligned}$$

(19)

By considering $A\sin \theta =\sqrt{\frac{b_2}{b_0}}$ and $A\cos \theta =1$, Eq. (19) can be expressed as,

$$\begin{aligned} y=\vert A\cos (\sqrt{b_0b_2}\tau -\theta )\vert ^{\frac{1}{b_2}} \end{aligned}$$

(20)

where $\tan \theta =\sqrt{\frac{b_2}{b_0}}=\beta$ and $A=\sqrt{1+\frac{b_2}{b_0}}=\frac{1}{\cos \theta }$.

Therefore, Eq. (20) is expressed as,

$$\begin{aligned} y=A^{\frac{1}{\alpha \beta }}\vert \cos (\alpha \tau -\theta )\vert ^{\frac{1}{\alpha \beta }} \end{aligned}$$

(21)

Equation (21) is identical with Eq. (8).