Abstract
This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.
Highlights
An important task in nonlinear science is to determine the nature of the bifurcations the system under study exhibits
The Euclidean distance between the actual parameter values and those of the closest bifurcation can be used to characterize the robustness of the system
The problem of estimating a measure of criticality of the Hopf bifurcation is treated as the distance evaluation between a point and a quadric in a ten-dimensional parameter space
Summary
An important task in nonlinear science is to determine the nature of the bifurcations the system under study exhibits. It should be noted that the bifurcation conditions arising in applications are frequently represented implicitly via a nonlinear polynomial equation G(u) = 0 For this case, an alternative approach for the pointto-manifold distance evaluation problem is based on symbolic algorithms of elimination of variables in the system of algebraic equations. These algorithms implement either the computation of the resultants or, generally, the construction of Gröbner bases [18,19] and result in an univariate algebraic equation This variable might be a particular coordinate of the nearest point in the manifold or even the distance to this point. The present paper is devoted to application of the distance equation for solving the problem of finding the distance to the Hopf bifurcation manifold in the parameter space. Accuracy all the approximate computations have been performed within the accuracy 10−40, the final results are rounded to 10−4
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