Research Article The Dynamics Behavior of Coupled Generalized van der Pol Oscillator with Distributed Order Asma Al Themairi 1 and Ahmed Farghaly 2,3 1 Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt 3 Department of Basic Science, College of Computer and Information Sciences, Majmaah University, Al-Majmaah 11952, Saudi Arabia Correspondence should be addressed to Asma Al emairi; aialthumairi@pnu.edu.sa Received 3 April 2020; Revised 18 June 2020; Accepted 6 July 2020; Published 28 July 2020 Academic Editor: Isabel S. Jesus Copyright©2020AsmaAlemairiandAhmedFarghaly.isisanopenaccessarticledistributedundertheCreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, we presented different behaviors such as chaotic and hyperchaotic of the generalized van der Pol oscillator with distributed order. We introduced the parameter intervals of these behaviors by computing the Lyapunov exponents of the oscillator, which is a good test for classifying the dynamical systems’ solutions. e active control approach with the Laplace transform technique was used to realize the antisynchronization and control of the proposed oscillator. Finally, numerical investigations have been carried out on the dynamics of the proposed oscillator to verify the reliability of our analytical results. 1. Introduction In 1920, van der Pol invented the van der Pol oscillator [1]. It describes the oscillation of a triode in an electrical circuit. It is a fundamental mathematical model, where it has many numerous applications and exciting features. is oscillator is used in designing many biological models such as heartbeats [2], designing physical models such as mobile and phone oscillators [3], and modeling of electrical systems [4]. Mathematically, there are many versions of the van der Pol oscillator like € x + μ x 2 − 1   _ x + x � 0, (1) € x + x 3 + ax 2 − 1 (  _ x � b(sin wt + y), € y + y 3 + ay 2 − 1 (  _ y � b(sin wt + x),  (2) where (1) was introduced in 1927 by Van der Pol and Van Der Mark [4], and (2) was submitted in 1991 by Kapitaniak and Steeb [5]. In this paper, we will introduce the sinusoidal forcing with amplitude b and frequency w as a fractional version of the generalized van der Pol of the form € x + x 3 + ε x 2 − 1   _ x � b sin(wt), (3) where μ, ε, a, b, and w are constants. Fractional calculus plays an essential role in modern science. It is a different and distinct method for dealing with nonlinear systems along with the integer order. Fractional order models are adequate for the description of dynamical systems rather than integer order models. We can recognize, describe, and know dynamic phenomena such as chaos, hyperchaos, synchronization, and some other aspects of fractional order models faster and more accurately than those of the integer order of nonlinear systems. At present, the application of fractional calculus in most scientific fields has attracted much attention. So, the fractional calculus on the dynamical system was essential and exciting, which had been investigated recently by many researchers [6–10]. Here, in our paper, we used distributed order as a type of fractional calculus to study the dynamic behavior of nonlinear gen- eralized van der Pol oscillator. Distributed order calculus has been investigated for the first time as the extinction of fractional order calculus by Caputo [11]. Caputo et al. introduced useful properties for the distributed order calculus [12–15]. Fern´ andez-Anaya Hindawi Mathematical Problems in Engineering Volume 2020, Article ID 5670652, 10 pages https://doi.org/10.1155/2020/5670652