Invariant measures for random expanding on average Saussol maps

In this paper, we investigate the existence of random absolutely continuous invariant measures (ACIP) for random expanding on average Saussol maps in higher dimensions. This is done by the establishment of a random Lasota–Yorke inequality for the transfer operators on the space of bounded oscillation. We prove that the number of ergodic skew product ACIPs is finite and will provide an upper bound for the number of these ergodic ACIPs. This work can be seen as a generalization of the work in [F. Batayneh and C. González-Tokman, On the number of invariant measures for random expanding maps in higher dimensions, Discrete Contin. Dyn. Syst. 41 (2021) 5887–5914] on admissible random Jabłoński maps to a more general class of higher-dimensional random maps.

ITERATIONS OF DEPENDENT RANDOM MAPS AND EXOGENEITY IN NONLINEAR DYNAMICS

We discuss the existence and uniqueness of stationary and ergodic nonlinear autoregressive processes when exogenous regressors are incorporated into the dynamic. To this end, we consider the convergence of the backward iterations of dependent random maps. In particular, we give a new result when the classical condition of contraction on average is replaced with a contraction in conditional expectation. Under some conditions, we also discuss the dependence properties of these processes using the functional dependence measure of Wu (2005, Proceedings of the National Academy of Sciences 102, 14150–14154) that delivers a central limit theorem giving a wide range of applications. Our results are illustrated with conditional heteroscedastic autoregressive nonlinear models, Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) processes, count time series, binary choice models, and categorical time series for which we provide many extensions of existing results.

Path planning for unmanned wheeled robot based on improved ant colony optimization

This research presents a simple and novel improved ant colony optimization for path planning of unmanned wheeled robot. Our main concern is to avoid the random deadlock situation and to reach at the destination using the shortest path, to decrease lost ants and improve the efficiency of solutions. The aforementioned reasons, we design an adaptive heuristic function by adopting the Euclidean distance between the ant and the target destination, in order to avoid the initial blindness and later singleness of ant path searching. The historical best path when appropriate to retain the previous effort would supersede the current worst path. Simulation results under random maps show that the improved ant colony optimization considerably increases the number of effective ants. During the searching process, the probability to find the optimal path increases, as well as the search speed. Moreover, we also compare the improved ant colony optimization performance with the simple ant colony optimization.

A Linear Spline Markov Approximation Method for Random Maps with Position Dependent Probabilities

We present a rigorous convergence analysis of a linear spline Markov finite approximation method for computing stationary densities of random maps with position dependent probabilities, which consist of several chaotic maps. The whole analysis is based on a new Lasota–Yorke-type inequality for the Markov operator associated with the random map, which is better than the previous one in the literature and much simpler to obtain. We also present numerical results to support our theoretical analysis.