Some open problems in real and complex dynamical systems

Abstract

Theory of dynamical systems may be split into two parts. The larger one, dealing with multidimensional systems: flows in dim 3 and higher, diffeomorphisms in dim 2 and higher, may be called the realm of chaos. The smaller one, dealing with planar differential equations, may be called the realm of order. The problems below deal with both parts.

Different Complex ZFs Leading to Different Complex ZNN Models for Time-Varying Complex Generalized Inverse Matrices

“…Reformulating (19), we have the following dynamic equation of the new complex ZNN model aiming at solving the timevarying complex generalized inverse: (20) which is termed complex ZNN-V model (20) based on complex ZF (8). Note that ZNN-V model is also the G-M dynamic system [33] for the time-varying complex generalized inverse.…”

“…Following literature [33], we have the theoretical results about the convergence performance of complex ZNN-V model. That is, given a smoothly timevarying complex matrix C(t) ∈ C m×n of full rank, if initial state Z (0) satisfies Z (0) − C + (0) F β < ∞ and β ∈ R is sufficiently small, then C(t)Z (t) − I → 0 as t → ∞, i.e., the state matrix Z (t) ∈ C n×m of complex ZNN-V model (20) exponentially converges to the time-varying theoretically generalized inverse of matrix C(t). It is worth noting that the initial condition of the complex ZNN-V model should be chosen as Z (0) ≈ C + (0) [in other words, Z (0) should be sufficiently close to C + (0)].…”

“…In summary, we have constructed five different complex ZNN models, i.e., complex ZNN-I model (9), ZNN-II model (14), ZNN-III model (16), ZNN-IV model (18), and ZNN-V model (20), by defining five different complex ZFs [i.e., complex ZFs (4)-(8)] for solving online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem. For readers' convenience and also for comparison purpose, we summarize these complex ZFs and the corresponding complex ZNN models in Table I.…”

“…Note that, based on the above three examples, we can conclude that the convergence time of complex ZNN models does not increase as the matrix dimension increases. Comparison on the residual errors R(t) = C(t)Z (t) − I F synthesized by complex ZNN-V model (20) with the design parameter γ = 100, 200, and 500 for the pseudoinverse of (28).…”

“…For illustrative purpose, we only exploit complex ZNN-V model (20) to solve the generalized inverse of complex matrix (28).…”