Dynamics of fractional order nonlinear system: A realistic perception with neutrosophic fuzzy number and Allee effect

Abstract

Introduction: The fusion of fractional order differential equations and fuzzy numbers has been widely used in modelling different engineering and applied sciences problems. In addition to these, the Allee effect, which is of high importance in field of biology and ecology, has also shown great contribution among other fields of sciences to study the correlation between density and the mean fitness of the subject.

Objectives: The present paper is intended to measure uncertain dynamics of an economy by restructuring the Cobb-Douglas paradigm of the renowned Solow-Swan model. The purpose of study is further boosted innovatively by subsuming the perception of logistic growth with Allee effect in the dynamics of physical capital and labor force.

Methods: Fractional order derivative and neutrosophic fuzzy (NF) theory are applied on the parameters of the Cobb-Douglas equation. Distinctively, cogitating fractional order derivative to study the change at each fractional stage; single-valued triangular neutrosophic fuzzy numbers (SVTNFN) to cope the uncertain situations; logistic growth function with Allee effect to analyze the factors in natural way, are the significant and novel features of this endeavor.

Results: The incorporation of the aforementioned theories and effects in the Cobb-Douglas equation, resulted in producing maximum sustainable capital investment and maximum capacity of labor force. The solutions in intervals located different possible solutions for different membership degrees, which accumulated the uncertain circumstances of a country.

Conclusion: Explicitly, these notions add new facts and figures not only in the dynamical study of capital and labor, which has been overlooked in classical models, but also left the door open for discussion and implementation on classical models of different fields.

Keywords: Caputo fractional derivative; Economic growth; Neutrosophic fuzzy numbers; Production function.

Graphical abstract

Fig. 1

The impact on the physical capital with truth membership degree $\alpha =0.35$ of fractional order parameter $\psi$.

Fig. 2

The impact on the physical capital with indeterminacy membership degree $\beta =0.35$ of fractional order parameter $\psi$.

Fig. 3

The impact on the physical capital with falsity membership degree $\gamma =0.35$of fractional order parameter $\psi$.

Fig. 4

The impact on the labour function with truth membership degree $\alpha =0.35$of fractional order parameter $\psi$.

Fig. 5

The impact on the labour function with indeterminacy membership degree $\beta =0.35$of fractional order parameter $\psi$.

Fig. 6

The impact on the labour function with falsity membership degree $\gamma =0.35$of fractional order parameter $\psi$.

Fig. 7

(a-c). Contours of capital with respect to truth, indeterminacy and falsity membership degrees for $s=100$.

Fig. 8

(a-c). Contours of labours with respect to truth, indeterminacy and falsity membership degrees for $s=100$.