We often use differential equations to represent continuous dynamical systems and difference equations to represent discrete dynamical systems. In general, discrete dynamical systems have rich dynamic behaviors. Bifurcation problems of differential systems have been extensively studied.The dynamics of a discrete-time Thomas type system is investigated in the closed first quadrant. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior by using a center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit much more interesting dynamical behavior, including orbits of period 2,4,8 and chaotic sets. These results show far richer dynamics of the discrete model compared with the continuous model.

Weixuan Xiao. Bifurcation Analysis of a Class of Discrete Thomas Type System. Academic Journal of Mathematical Sciences (2023) Vol. 4, Issue 5: 62-73. https://doi.org/10.25236/AJMS.2023.040509.

[1] M. Inoue, H. Kamifukumoto, Scenarios leading to chaos in a forced Lotka–Volterra model, Progr. Theoret. Phys. 71 (1984) 930–937.

[2] Y.A. Kuznetsov, S. Muratori, S. Rinaldi, Bifurcation and chaos in a periodic predator–prey model, Int. J. Bifurcations and Chaos 2 (1992) 117–128.

[3] S. Rinaldi, S. Muratori, Y. Kuznetsov, Multiple attractors, catastrophes and chaos in seasonally perturbed predator–prey communities, Bull. Math. Biol.55 (1993) 15–35.

[4] W.M. Schaffer, B.S. Pederson, B.K. Moore, O. Skarpaas, A.A. King, T.V. Bronnikova, Sub-harmonic resonance and multi-annual oscillations in northern mammals: a non-linear dynamical systems perspective, Chaos Solitons Fractals 12 (2001) 251–264.

[5] G.C.W. Sabin, D. Summers, Chaos in periodically forced predator–prey ecosystem model, Math. Biosci. 113 (1993) 91–113.

[6] D. Summers, J.G. Cranford, B.P. Healey, Chaos in periodically forced discrete-time ecosystem models, Chaos Solitons Fractals 11 (2000) 2331–2342.

[7] J. Vandermeer, Seasonal isochronic forcing of Lotka–Volterra equations, Progr. Theoret. Phys. 96 (1996) 13–28.

[8] J. Vandermeer, L. Stone, B. Blasius, Categories of chaos and fractal boundaries in forced predator–prey models, Chaos Solitons Fractals 12 (2001) 265–276.

[9] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.

[10] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd Ed., Boca Raton, London, New York, 1999.

[11] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag, New York, 2003.