The generation of fractals using iterative procedures is the emerging and interesting field of research. The aim of this paper is to obtain some fixed point results in the generation of fractals using SP-iterative process with s-convexity for quadratic, cubic and higher degree polynomials. Our results extend and improve the corresponding recent results proved by other authors in the literature.

We present a new four-step feedback procedure to study the full dynamics of a nonlinear dynamical system, namely, the logistic map. We show that by using this procedure, the chaotic behavior of the logistic map can be controlled easily and rapidly or the system can be made stable for higher values of the population growth parameter. We utilize various dynamical techniques (orbit evolution, time series analysis, bifurcation diagrams, and Lyapunov exponents) to analyze the dynamics of the logistic map. Additionally, we adopt the switching strategy to control chaos or to increase the stability performance of the logistic map. Finally, we propose a modified traffic control model to enable rapid control of unexpected traffic on the road. The results of this model are supported by a physical interpretation. The model is found to be more efficient than existing models of Lo and Cho [J. Franklin Inst. 342, 839–851 (2005)] and Ashish et al. [Nonlinear Dyn. 94, 959–975 (2018)]. This work provides a novel feedback procedure that facilitates rapid control of chaotic behavior and increases the range of stability of dynamical systems.

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In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued G-contractions defined on multifractal space C ( U ) in the framework of a Hausdorff b-metric space, where U = U 1 × U 2 × ⋯ × U N , N being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff b-metric spaces which are more general spaces than Hausdorff metric spaces.

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