Narrow Results

In the paper we propose a new synchronization principle. To guarantee synchronization between coupled chaotic oscillators, proper coupling constants are selected by the Liapunov stability theory and Hurwitz Theorem. As an example and application, we prove the conjecture [Wu & Chua, 1994] that synchronization between two chaotic Chua's circuits can be achieved by using the second state as feedback variable for sufficiently large coupling constant.

In this paper, the linearity of the dimensional-decrease effect of the Riemann–Liouville fractional integral is mainly explored. It is proved that if the Box dimension of the graph of an $\alpha$-Hölder continuous function is greater than one and the positive order $\nu$ of the Riemann–Liouville fractional integral satisfies $\alpha +\nu <1$, the upper Box dimension of the Riemann–Liouville fractional integral of the graph of this function will not be greater than $2-\alpha -\nu$. Furthermore, it is proved that the Riemann–Liouville fractional integral of a Lipschitz continuous function defined on a closed interval is continuously differentiable on the corresponding open interval.

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.

This paper concentrates on discussing the properties of Riemann–Liouvile fractional (RLF) calculus of two special continuous functions. The first type proves the non-differentiability of a special continuous function that does not satisfy Hölder condition, and the second type uses fractal iteration to construct a fractal function defined on $[0,1]$ with unbounded variation. Then we calculate RLF integral and RLF derivative of this special function, and give the corresponding numerical calculation results and the corresponding function image.

In this paper, we mainly research on fractional differentiability of certain continuous functions with fractal dimension one. First, Riemann–Liouville fractional differential of differentiable functions must exist. Then, we prove the existence of Riemann–Liouville fractional differential of continuous functions satisfying the Lipschitz condition, which means that all of their Riemann–Liouville fractional integral of any positive orders in $(0,1)$ are differentiable. For continuous functions which do not satisfy the Lipschitz condition, we give counterexamples of certain continuous functions whose Riemann–Liouville fractional differential does not exist of certain positive order in $(0,1)$. Riemann–Liouville fractional differentiability of other one-dimensional continuous functions has also been investigated elementary. Fractional differentiability takes interpretation on physical problems like moving particle and transports through porous and percolation medium with residual memory.

In this paper, we study opposite diagonal sections of quasi-copulas and copulas. The best-possible upper bound for the set of copulas with a given opposite diagonal section being known, we focus on the best-possible lower bound, which in general is a quasi-copula. Moreover, it exhibits an interesting type of bivariate symmetry called opposite symmetry.

The random Dirichlet type functions on the unit ball of Cⁿ are studied. Sufficient conditions of the multipliers of for 0 < μ ≤ 1, if n = 1 or 0 < μ < 2 if n > 1 are given. The smoothness of random Dirichlet type functions is discussed.