Chaos and bifurcation analysis of tumor-immune controlled system with time delay

This paper establishes a delayed discrete tumor-immune model with feedback controller. The results indicate that the controlled system exhibits a trend from chaotic state to stability. The Flip bifurcation and Neimark–Sacker bifurcation of system are proved by the normal form method, central manifold theorem, and bifurcation theory. The Gram–Schmidt orthonormalization (GSO) method is used to interpret the attractor properties of uncontrolled and controlled systems, it is revealed that the attractors of controlled systems are smaller than those of uncontrolled systems, and the theoretical and numerical results are consistent. In addition, this research indicates that noise can cause significant changes in the amplitude of time-delay free system oscillations, with little impact on the period. Interestingly, the dynamic characteristics of noise in time-delay systems are not significantly affected by noise.