The double well function (DWF) is an important theoretical model originating from quantum physics and has been used to describe the energy constraint problem in quantum mechanics and structural chemistry. Although its form may vary, the DWF has two different local minima in the one-dimensional case, and the number of local minima increases as its dimension grows. As a multi-stable function, the DWF is assumed to be a potential candidate for testing the performance of the heuristic optimization algorithm, which aims to seek the global minimum. To verify this idea, a typical DWF is employed in this paper, and a mathematical analysis is presented herein, and its properties as a benchmark function are discussed in different cases. In addition, we conducted a set of experiments utilizing a few optimization algorithms, such as the multi-scale quantum harmonic oscillator algorithm and covariance matrix adaptation evolution strategy; thus the analysis has been illustrated by the results of the numerical experiment. Moreover, to guide the design of an ideal DWF used as a benchmark function in experiment, different values of the decisive parameters were tested corresponding to our analysis, and some useful rules were given based on the discussion of the results.

This paper studies topological entropy of switched nonlinear systems. We construct a general upper bound for the topological entropy in terms of an average of the asymptotic suprema of the measures of Jacobian matrices of individual modes, weighted by the corresponding active rates. A general lower bound is constructed in terms of an active-rate-weighted average of the asymptotic infima of the traces of these Jacobian matrices. For switched systems with diagonal modes, we construct upper and lower bounds that only depend on the eigenvalues of Jacobian matrices, their relative order among individual modes, and the active rates. For both cases, we also construct more conservative upper bounds that require less information on the switching, with their relations illustrated by numerical examples of a switched Lotka-Volterra ecosystem model.

This paper studies topological entropy and stability properties of switched linear systems. First, we show that the exponential growth rates of solutions of a switched linear system are essentially upper bounded by its topological entropy. Second, we estimate the topological entropy of a switched linear system by decomposing it into a part that is generated by scalar multiples of the identity matrix and a part that has zero entropy, and proving that the overall topological entropy is upper bounded by that of the former. Third, we prove that a switched linear system is globally exponentially stable if its topological entropy remains zero under a destabilizing perturbation. Finally, the entropy estimation via decomposition and the entropy-based stability condition are applied to three classes of switched linear systems to construct novel upper bounds for topological entropy and novel sufficient conditions for global exponential stability.