IMPLICIT CRITERIA OF EIGENVALUE ASSIGNMENT AND TRANSVERSALITY FOR BIFURCATION CONTROL IN FOUR-DIMENSIONAL MAPS

Abstract

In anti-control of bifurcations, it is common to create different types of bifurcations by adjusting the control parameters. For maps, the type of bifurcation is determined by the eigenvalue assignment on the unit circle at the bifurcation parameter point. Thus, an unavoidable problem in the creation of bifurcations is to desirably assign some eigenvalues at the specified locations on the unit circle, and the others inside the unit circle. However, for relatively complicated and high dimensional maps, the explicit expressions of eigenvalues are usually not available so that the implementation of the eigenvalue assignment becomes very difficult. To solve this problem, we proposed the new criteria of eigenvalue assignment without using eigenvalues. The criteria give implicit conditions to specify the eigenvalue assignments in terms of some simple algebraic equalities and inequalities associated with the elements of Jacobian matrix, i.e. eventually associated with the control parameters. Bifurcation occurs with another critical condition, the transversality condition. The computation of the transversality condition is usually nontrivial in high dimensional maps because it is related to the partial differentiation of the eigenvalues on the unit circle. We also present the implicit expression of the transversality condition in the form of the derivative of the Jacobian matrix and its eigenvectors that are computable at the bifurcation point. The proposed criteria cover most known types of bifurcations in four-dimensional maps and serve as the preferable methods for designing the critical bifurcation conditions in anti-control of bifurcations. The application to a modified Hénon map is illustrated in conjunction with the use of the delayed-feedback control and the washout-filter-aided feedback control.