Study On Two Kinds Of High Co-dimensional Bifurcations For Discrete Dynamical Systems And The Anti-control Of Bifurcations

High-dimensional and multi-parameter nonlinear systems often exhibit complex high co-dimensional bifurcation behaviors.It is of great theoretical significance for practical engi-neering problems to study high co-dimensional bifurcation mechanism in depth and implement effective control.In this paper,the anti-control of codimension-2 bifurcation in the case of1:2 resonance and Hopf-Hopf-flip codimension-3 bifurcation of discrete dynamical systems are studied.The main contents are as follows:A set of nonlinear feedback control strategies were designed to realize the anti-control of codimension-2 bifurcation in discrete dynamical systems with 1:2 resonance from the per-spective of bifurcation anti-controlling.Firstly,a new explicit criterion for codimension-2bifurcation in 1:2 resonance was proposed.Based on this explicit criterion,the linear con-trol gain was designed to ensure the existence of such codimension-2 bifurcation.Then,the central manifold of 1:2 resonance was derived.Based on the normal form method,the types and stability of codimension-2 bifurcation solution in 1:2 resonance were analyzed through design of nonlinear control gain.Finally,an Arneodo-Coullet-Tresser mapping was taken as an example to realize various bifurcation solutions with 1:2 resonance bifurcation properties by control at the specified parameter points,which further validates the theoretical analysis.The Hopf-Hopf-flip bifurcation of n-dimensional discrete dynamical systems is studied.Firstly,aimed at the difficulty caused by the traditional bifurcation criterion in determining the high co-dimensional bifurcation point by direct calculation of eigenvalues,the explic-it Hopf-Hopf-flip bifurcation critical criteria,including eigenvalue assignment,transversality condition and non-resonance condition,were established.Then,the mapping is reduced to a five-dimensional mapping by the central manifold method,and the corresponding normal form and the analytical expressions of correlation coefficient were derived.Furthermore,the local dynamic behaviors of three-parameter unfolding near the Hopf-Hopf-flip bifurcation point were analyzed.Finally,the T~1,2T~1and 2T~2invariant toris of Hopf-Hopf-flip bifurcation in discrete dynamical system were revealed by numerical examples.