Research Article Adaptive Control of a New Chaotic Financial System with Integer Order and Fractional Order and Its Identical Adaptive Synchronization Paul Yaovi Dousseh, 1 Cyrille Ainamon, 1 Cl´ ement Hod´ ev` ewan Miwadinou , 1,2 Adjimon Vincent Monwanou, 1 and Jean Bio Chabi Orou 1 1 Laboratoire de M´ ecaniques des Fluides, de la Dynamique Non-lin´ eaire et de la Mod´ elisation des Syst` emes Biologiques (LMFDNMSB), Institut de Math´ ematiques et de Sciences Physiques (IMSP), Porto-Novo, Benin 2 D´ epartement de Physique, ENS-Natitingou, Universit´ e des Sciences, Technologies, Ing´ enierie et Math´ ematiques (UNSTIM), Abomey, Benin Correspondence should be addressed to Cl´ ement Hod´ ev` ewan Miwadinou; clement.miwadinou@imsp-uac.org Received 19 January 2021; Revised 13 February 2021; Accepted 26 February 2021; Published 13 March 2021 Academic Editor: Yi Qi Copyright © 2021 Paul Yaovi Dousseh et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, adaptive control and adaptive synchronization of an integer and fractional order new financial system with unknown constant parameters are studied. Based on Lyapunov’s stability theory, an adaptive control law is designed to as- ymptotically stabilize the state variables of the system to the origin in integer and fractional order cases. By the same theory, an adaptive synchronization law is designed to perform the identical synchronization of the new financial system in the cases of integer and fractional order with unknown constant parameters. Numerical simulations are carried out in order to show the efficiency of the theoretical results. 1. Introduction Fractional order derivatives are a subject over 300 years old, initiated by Leibniz's letter to L’Hospital [1, 2] and are a generalization of integer order derivatives But their appli- cations in scientific fields are very recent and this is due to the lack of their physical interpretation. e difference between these fractional order derivatives and the integer order derivatives is that fractional order derivatives have the memory that turns out to be very useful when it comes to describing systems with memory and heredity properties. In the literature, several systems have been described using fractional order derivatives, we can cite the fractional order Liu system [3], the fractional order financial system [4], the fractional order glucose-insulin regulatory system [5], the fractional order Chua system [6], etc. Chaotic dynamical systems are first of all nonlinear systems, depending on several parameters and having an extreme sensitivity to initial conditions. ese systems are found in many scientific fields including chemical, physical [6], economic [4], or biological [5]. is has led researchers from various horizons to take an interest in these types of systems, and especially the control of the chaos which intervenes and the syn- chronization of these systems with integer and fractional order. Chaos control in a dynamical system consists in designing a control law which stabilizes the system as- ymptotically on one of these unstable fixed points. In the literature, several methods have been proposed to achieve this goal. We have among others, the linear feedback control [7], adaptive control [8, 9], sliding mode control [10], Lyapunov-based nonlinear control [11], adaptive sliding mode control [12], etc. Recently, for the stabilization of dynamical systems, different results have been obtained in the literature in fields as diverse as varied. For example, see Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 5512094, 15 pages https://doi.org/10.1155/2021/5512094