Abstract and Applied Analysis

Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model

Marluci Cristina Galindo, Cristiane Nespoli, and Marcelo Messias

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study a cancer model given by a three-dimensional system of ordinary differential equations, depending on eight parameters, which describe the interaction among healthy cells, tumour cells, and effector cells of immune system. The model was previously studied in the literature and was shown to have a chaotic attractor. In this paper we study how such a chaotic attractor is formed. More precisely, by varying one of the parameters, we prove that a supercritical Hopf bifurcation occurs, leading to the creation of a stable limit cycle. Then studying the continuation of this limit cycle we numerically found a cascade of period-doubling bifurcations which leads to the formation of the mentioned chaotic attractor. Moreover, analyzing the model dynamics from a biological point of view, we notice the possibility of both the tumour cells and the immune system cells to vanish and only the healthy cells survive, suggesting the possibility of cure, since the interactions with the immune system can eliminate tumour cells.

Article information

Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 354918, 11 pages.

First available in Project Euclid: 15 April 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Galindo, Marluci Cristina; Nespoli, Cristiane; Messias, Marcelo. Hopf Bifurcation, Cascade of Period-Doubling, Chaos, and the Possibility of Cure in a 3D Cancer Model. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 354918, 11 pages. doi:10.1155/2015/354918. https://projecteuclid.org/euclid.aaa/1429104593

Export citation


  • R. P. Araujo and D. L. McElwain, “A history of the study of solid tumour growth: the contribution of mathematical modelling,” Bulletin of Mathematical Biology, vol. 66, no. 5, pp. 1039–1091, 2004.
  • R. Chammas, D. Silva, A. Wainstein, and K. Abdallah, “Imunologia clinica das neoplasias,” in Imunologia Clínica na Prática Médica, pp. 447–460, Atheneu, São Paulo, Brazil, 2009.
  • W. Chang, L. Crowl, E. Malm, K. Todd-Brown, L. Thomas, and M. Vrable, Analyzing Immunotherapy and Chemotherapy of Tumors Through Mathematical Modeling, Department of Mathematics, Harvey-Mudd University, Claremont, Calif, USA, 2003.
  • D. Kirschner and J. C. Panetta, “Modeling immunotherapy of the tumor–-immune interaction,” Journal of Mathematical Biology, vol. 37, no. 3, pp. 235–252, 1998.
  • M. Galach, “Dynamics of the tumor-immune system competition–-the effect of time delay,” International Journal of Applied Mathematics and Computer Science, vol. 13, no. 3, pp. 395–406, 2003.
  • M. Itik and S. P. Banks, “Chaos in a three-dimensional cancer model,” International Journal of Bifurcation and Chaos, vol. 20, no. 1, pp. 71–79, 2010.
  • L. G. de Pillis and A. Radunskaya, “The dynamics of an optimally controlled tumor model: a case study,” Mathematical and Computer Modelling, vol. 37, no. 11, pp. 1221–1244, 2003.
  • Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, NY, USA, 2004.
  • M. Itik and S. P. Banks, “On the structure of periodic orbits on a simple branched manifold,” International Journal of Bifurcation and Chaos, vol. 20, no. 11, pp. 3517–3528, 2010.
  • L. S. Pontryagin, Ordinary Differential Equations, Addison-Wesley, New York, NY, USA, 1962.
  • C. Letellier, F. Denis, and L. A. Aguirre, “What can be learned from a chaotic cancer model?” Journal of Theoretical Biology, vol. 322, pp. 7–16, 2013. \endinput