The fractional derivative holds long-time memory effects or non-locality. It successfully depicts the dynamical systems with long-range interactions. However, it becomes challenging to investigate chaos in the deformed fractional discrete-time systems. This study turns to fractional quantum calculus on the time scale and reports chaos in fractional $q$ -deformed maps. The discrete memory kernels are used, and a weight function approach is proposed for fractional modeling. Rich $q$ -deformed dynamics are demonstrated, which shows the methodology’s efficiency.

Modelling is challenging topic and using parametric models is important stage to reach flexible function for modelling. Weibull distribution has shape and scale parameters which play the main role for modelling. Bimodality parameter is added and so bimodal Weibull distribution can capture real data set with bimodality which can be actually combination of two populations. The properties of the proposed distribution and estimation method are examined extensively to show its usability in modelling accurately and safely for practitioners. After examination as first stage in modelling issue, it is appropriate to use bimodal Weibull for modelling bimodality in real data sets if it exists. Two estimation methods including objective functions are used to estimate the parameters of shape, scale and bimodality parameters of function. The second stage in modelling is overcome by using heuristic algorithms for optimization of function according to parameters due to the fact that converging to global point of objective function is performed by heuristic algorithms from stochastic optimization. Real data sets are provided to show the modelling competence of objective functions from bimodal forms of Weibull and Gamma distributions having well defined shape, scale and bimodality parameters and potentially less parameters when compared with the existing distributions.

A bimodal extension of the generalized gamma distribution is proposed by using a mixing approach. Some distributional properties of the new distribution are investigated. The maximum likelihood (ML) estimators for the parameters of the new distribution are obtained. Real data examples are given to show the strength of the new distribution for modeling data.

The asymmetric bimodal exponential power (ABEP) distribution is an extension of the generalized gamma distribution to the real line via adding two parameters that fit the shape of peakedness in bimodality on the real line. The special values of peakedness parameters of the distribution are a combination of half Laplace and half normal distributions on the real line. The distribution has two parameters fitting the height of bimodality, so capacity of bimodality is enhanced by using these parameters. Adding a skewness parameter is considered to model asymmetry in data. The location-scale form of this distribution is proposed. The Fisher information matrix of these parameters in ABEP is obtained explicitly. Properties of ABEP are examined. Real data examples are given to illustrate the modelling capacity of ABEP. The replicated artificial data from maximum likelihood estimates of parameters of ABEP and other distributions having an algorithm for artificial data generation procedure are provided to test the similarity with real data. A brief simulation study is presented.