Control and numerical analysis for cancer chaotic system
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T E C H N I C A L N OT E
Javaria Iqbal · Salman Ahmad · Muhammad Marwan · Mansoor Shaukat
Control and numerical analysis for cancer chaotic system
Received: 24 December 2019 / Accepted: 6 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This article investigates the problem of control of chaotic dynamics of tumor cells, immune system cells, and healthy tissue cells in a three-dimensional cancer model by adaptive control technique. Adaptive control law is derived such that the trajectory of controlled system asymptotically approaches equilibrium point with estimated parameter converges to stabilizing values. A nonlinear control law is designed which change the original chaotic system into a controlled one linear system. In addition, we present and analyze numerical solution of the cancer dynamical system with the help of a discretization technique. Achieved solutions show a comparable results with Runge–Kutta methods. The reliability and accuracy of the proposed technique is presented by comparing numerical results. The used technique has displayed a brilliant prospective in dealing with the numerical solutions of nonlinear dynamical systems. Keywords Cancer cells · Chaos · Discretization · Control functions
1 Introduction Cancer has become one of the main research areas to focus for scientists with the growth of mortality due to cancer in the world. Tumor follows different stages to grow enough large that it becomes reason to loss life or permanently reduce the quality of life for patient. Advanced and developed countries are investing a large amount of money into tumor research to focus on cure and treatment to improve the quality of human life [1]. Scientists have developed mathematical models to describe the tumor growth [2]. In last ten to twenty years, many mathematical models have been presented to study the interaction of tumor cells with other cells such as healthy cells, immune system and viruses. For instance, Kuznetsov et al. [3] gave a second-order model, which examined the interaction among the tumor cell and immune system cells. This model has been the foundation for many advance studies. Itik and Banks proposed [4] a new cancer growth model that investigated the chaotic growth of tumor cells when interacted with healthy cells and immune system cells. Another interesting model is proposed by Malinzi et al. [5] for the treatment of cancer using chemovirotherapy. J. Iqbal (B) · S. Ahmad · M. Marwan Department of Applied Mathematics and Statistics, Institute of Space Technology, Islamabad 44000, Pakistan J. Iqbal E-mail: [email protected] M. Shaukat Department of Mathematics, COMSATS University Islamabad, Islamabad 45550, Pakistan S. Ahmad · M. Marwan Institute of Space Technology, Islamabad 44000, Pakistan
J. Iqbal et al.
Chaos theory is one of the fascinating fields in nonlinear sciences [6]. Chaos owns certain unusual aspects such as enormously sensitive to minor modification in initial state. Chaos theory has various applications in several areas [7]. Since
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