ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

The Thomas Attractor with and without Delay: Complex Dynamics to Amplitude Death

Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 27--45 | DOI:10.5890/DNC.2020.03.003

Brayden McDonald$^{1}$, S. Roy Choudhury$^{2}$

$^{1}$ Department of Computer Science, University of Central Florida

$^{2}$ Department of Mathematics, University of Central Florida

Abstract

Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system, leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed point of the original system.

Acknowledgments

The author wishes to express his very sincere gratitude to the Editor Professor Volchenkov and/or an anonymous referee for several comments which improved the accuracy of the paper, including specifying the time intervals on all the numerical plots, as well as the comments regarding hidden attractors in Section 4 which have, in fact, also triggered several other investigations unrelated to the current work.

References

1.  [1] Ramussen, S., Knudsen, C., Feldberg, R., and Hindsholm, M. (1990), The coreworld: emergence and evolution of cooperative structures in computational chemistry, Physica, 42D, 111-134.
2.  [2] Deneubourg, J.L. and Goss, S. (1989), Collective patterns and decision-making, Ethology, Ecology and Evolution, 1(4), 295-312.
3.  [3] Thomas, R. (1999), Deterministic Chaos seen in terms of feedback circuits, Int. J. Bifurcations and Chaos, 9, 1889- 1905.
4.  [4] Kauffman, S. (1993), The Origins of Order, (Oxford University Press, Oxford).
5.  [5] Sprott, J.C. and Chlouverakis, K.E. (2007), Labyrinth chaos, International Journal of Bifurcation and Chaos, 17, 2097-2108.
6.  [6] Krise, S. and Choudhury, S.R. (2003), Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations, Chaos, Solitons and Fractals, 16, 59-77.
7.  [7] Nayfeh, A.H. and Balachandran, B. (1995), Applied Nonlinear Dynamics, (Wiley, New York).
8.  [8] Nayfeh, A.H. (2011), The Method of Normal Forms, (Wiley, New York).
9.  [9] Abarbanel, H.D.I., Brown, R., Sidorowich, J.J., and Tsimring, L.S. (1993), The analysis of observed chaotic data in physical systems, Rev. Mod. Physics, 65, 1331-1392.
10.  [10] Leonov, G.A., Kuznetsov, N.V., and Mokaev, T.N. (2015), Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion, European Physical Journal Special Topics, 224, 1421-1458.
11.  [11] Leonov, G.A. and Kuznetsov,N.V. (2013), Hidden attractors in dynamical systems. From hidden oscillations in Hilbert- Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, International Journal of Bifurcation and Chaos, 23, 1330002.
12.  [12] Stankevich, N.V., et al, (2017), Scenario of the birth of hidden attractors in the Chua circuit, International Journal of Bifurcation and Chaos, 27, 1730038.
13.  [13] Nazarimehr, F., Saedi, B., Jafari, S., and Sprott, J.C. (2017), Are perpetual points sufficient for locating hidden attractors, International Journal of Bifurcation and Chaos, 27, 1750037.
14.  [14] Dudkowski, D., Prasad, A., and Kapitaniak., T. (2018), Describing chaotic attractoors: regular and perpetual points, Chaos, 28, 033604.
15.  [15] Ray, A., Saha, P., and Roy Chowdhury, A. (2017), Competitive mode and topological properties of nonlinear systems with hidden attractor, Nonlinear Dynamics, 88(2017), 1989-2001.
16.  [16] Davis, H.T. (1962), Introduction to Nonlinear Differential and Integral Equations, (Dover, New York).
17.  [17] Cushing, J.M. (1977), Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, vol. 20, (Springer, Berlin, 1977).
18.  [18] MacDonald, N. (1978), Time Lage in Biological Models, Lecture Notes in Biomathematics, vol. 27, (Springer, Berlin, 1978).
19.  [19] Prasad, A., Dhamala, M., Adhikari, B.M., and Ramaswamy, R. (2010, 2016), Amplitude death in nonlinear oscillators with nonlinear coupling, Phys. Rev. E81 (2010) 027201; Illing, L, Amplitude death of identical oscillators in networks with direct coupling, Phys. Rev. E94 (2016) 022215