Estimating the Hausdorff–Besicovitch dimension of boundary of basin of attraction in helicopter trim Saipraneeth Gouravaraju, Ranjan Ganguli ⇑ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560 012, India article info Keywords: Basin of attraction Chaos Fractal dimension Helicopter trim Nonlinear equations Statistical self-similarity abstract Helicopter trim involves solution of nonlinear force equilibrium equations. As in many nonlinear dynamic systems, helicopter trim problem can show chaotic behavior. This chaotic behavior is found in the basin of attraction of the nonlinear trim equations which have to be solved to determine the main rotor control inputs given by the pilot. This study focuses on the boundary of the basin of attraction obtained for a set of control inputs. We analyze the boundary by considering it at different magnification levels. The magnified views reveal intricate geometries. It is also found that the basin boundary exhibits the characteristic of statistical self-similarity, which is an essential property of fractal geome- tries. These results led the authors to investigate the fractal dimension of the basin bound- ary. It is found that this dimension is indeed greater than the topological dimension. From all the observations, it is evident that the boundary of the basin of attraction for helicopter trim problem is fractal in nature. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Fractals are an integral part of the natural world since most of the naturally formed geometries are fractals. For example clouds, coastlines, lightning, snow flakes, cauliflower and even the system of our blood vessels are all fractal in nature [1–4]. In fact, anything which is fragmented or irregular can be a fractal. Fractals are irregular and they cannot be described by Euclidean geometries. They are also characterized by self-similarity which essentially means that the each part of the geom- etry is a reduced scale image of the whole [1]. Even when the geometry is not identical at all the magnification levels, it can be fractal if there is any property of the geometry that repeats itself as it is magnified. This type of self-similarity is called statistical self-similarity [5]. The theory of fractals aids in analyzing the response of practical systems besides examining the complex geometries of nature. Fractal theory has a wide variety of applications ranging from economy [7], environmental sciences [8], mathematics [9,10], neurology [11,12] and thermal sciences [13,14]. An interesting study done by Mikiten et al. [15] proposes a fractal theory of the human mind to explain one aspect of how we interact with our environment. They propose that the mind establishes a connection with the environment by processing information, which is an important theme seen during the evo- lution of the brain. They also propose that analogies are developed for storing ideas and information within a fractal scheme. Researchers have found that many nonlinear dynamical systems exhibit chaotic behavior in their response. This is a very important issue as a small perturbation or uncertainty in the inputs may make the response erratic and impossible to ana- lyze. Alasty and Shabani [16] investigated the chaotic behavior of a spring pendulum which is a typical of many nonlinear systems. They showed that variation in damping coefficients results in chaotic motion. They also found that some regions of 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.104 ⇑ Corresponding author. E-mail address: ganguli@aero.iisc.ernet.in (R. Ganguli). Applied Mathematics and Computation 218 (2012) 10435–10442 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc