This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

This study discusses the results of a Recurrence quantification analysis (RQA) of the Rössler system with a fractional order ($$q_1$$ q 1 ) of the derivative in the first equation. The fractional order $$q_1$$ q 1 changes slightly in the range $$q_1 \in \langle 0.9,1.0\rangle$$ q 1 ∈ ⟨ 0.9 , 1.0 ⟩ . Even with such relatively small changes in the $$q_1$$ q 1 derivative, significant changes in the dynamics of the system are observed between the bifurcation diagrams determined for the bifurcation parameter a. Nevertheless, as $$q_1$$ q 1 decreases one can notice the preservation of some structures of the bifurcation diagram, in particular the main periodic windows of the integer-order Rössler system. The RQA shows clear differences between various regular windows of the integer system and only slight changes in these windows are caused by an increase in the system’s fractionality. Nonetheless, by selecting appropriate recurrence variables it is possible to expose the changes occurring in the regular windows under the influence of the fractionality of the system. This approach allows for the detection of the fractional character of the system through a recurrence analysis of the time series taken from periodic regions.

In this paper, we consider two functionals of the Fekete–Szegö type Θ f ( μ ) = a 4 − μ a 2 a 3 and Φ f ( μ ) = a 2 a 4 − μ a 3 2 for a real number μ and for an analytic function f ( z ) = z + a 2 z 2 + a 3 z 3 + … , | z | < 1 . This type of research was initiated by Hayami and Owa in 2010. They obtained results for functions satisfying one of the conditions Re f ( z ) / z > α or Re f ′ ( z ) > α , α ∈ [ 0 , 1 ) . Similar estimates were also derived for univalent starlike functions and for univalent convex functions. We discuss Θ f ( μ ) and Φ f ( μ ) for close-to-convex functions such that f ′ ( z ) = h ( z ) / ( 1 − z ) 2 , where h is an analytic function with a positive real part. Many coefficient problems, among others estimating of Θ f ( μ ) , Φ f ( μ ) or the Hankel determinants for close-to-convex functions or univalent functions, are not solved yet. Our results broaden the scope of theoretical results connected with these functionals defined for different subclasses of analytic univalent functions.

In this article we take over methods for determination of Koebe set based on extremal sets for a given class of functions.

Abstract. In this paper we consider a class of univalent orientation-preserving harmonic functions defined on the exterior of the unit disk which satisfy the condition ∞ n=1 n p (|an| + |bn|) ≤ 1. We are interested in finding radius of univalence and convexity for such class and we find extremal functions. Convolution, convex combination, and explicit quasiconformal extension for this class are also determined.