Estimating the Hausdorff-Besicovitch dimension of boundary of basin of attraction in helicopter trim. (English) Zbl 1246.37099
Summary: 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 geometries. These results led the authors to investigate the fractal dimension of the basin boundary. 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.
MSC:
| 37N35 | Dynamical systems in control |
Keywords:
basin of attraction; chaos; fractal dimension; helicopter trim; nonlinear equations; statistical self-similarityReferences:
| [1] | Mandelbrot, B. B., The Fractal Geometry of Nature (1977), W.H. Freeman and Company: W.H. Freeman and Company New York · Zbl 0504.28001 |
| [2] | Gabrys, E.; Rybaczuk, M.; Kedzia, A., Blood flow simulation through fractal models of circulatory system, Chaos Solitons Fract., 27, 1, 1-7 (2006) · Zbl 1140.76506 |
| [3] | Schneider, M.; Wohlke, T., Simplified model for fractal dimension of clouds, Physica A, 189, 1-2, 1-3 (1992) |
| [4] | Reiter, C. A., A local cellular model for snow crystal growth, Chaos Solitons Fract., 23, 4, 1111-1119 (2005) · Zbl 1102.82341 |
| [5] | Mandelbrot, B. B., How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156, 3775, 636-638 (1967) |
| [6] | Peterson, I., Islands of Truth: A Mathematical Mystery Cruise (1990), W.H. Freeman: W.H. Freeman New York |
| [7] | Mulligan, R. F.; Koppl, R., Monetary policy regimes in macroeconomic data: an application of fractal analysis, Quart. Rev. Econ. Finance, 51, 2, 201-211 (2011) |
| [8] | Yang, X. B.; Jin, X. Q.; Du, Z. M.; Zhu, Y. H., A novel model-based fault detection method for temperature sensor using fractal correlation dimension, Build. Environ., 46, 4, 970-979 (2011) |
| [9] | Chen, J.; Ngai, S. M., Eigenvalues and eigenfunctions of one-dimensional fractal Laplacians defined by iterated function systems with overlaps, J. Math. Anal. Appl., 364, 1, 222-241 (2010) · Zbl 1261.35095 |
| [10] | Pašić, M.; Tanaka, S., Fractal oscillations of self-adjoint and damped linear differential equations of second-order, J. Appl. Math. Comput., 218, 5, 2281-2293 (2011) · Zbl 1244.34052 |
| [11] | King, R. D.; Brown, B.; Hwang, M.; Jeon, T., A.T. George and the Alzheimer’s disease neuroimaging initiative, fractal dimension analysis of the cortical ribbon in mild Alzheimer’s disease, Neuroimage, 53, 2, 471-479 (2010) |
| [12] | Iftekharuddin, K. M.; Zheng, J.; Islam, Md. A.; Ogg, R. J., Fractal-based brain tumor detection in multimodal MRI, J. Appl. Math. Comput., 207, 1, 23-41 (2009) · Zbl 1155.92024 |
| [13] | Zhu, Q. Y.; Xie, M. H.; Yang, J.; Li, Y., A fractal model for the coupled heat and mass transfer in porous fibrous media, Int. J. Heat Mass Tran., 54, 7-8, 1400-1409 (2010) · Zbl 1211.80032 |
| [14] | Xiao, B.; Yu, B.; Wang, Z.; Chen, L., A fractal model for heat transfer of nanofluids by convection in a pool, Phys. Lett. A, 373, 45, 4178-4181 (2009) · Zbl 1236.76062 |
| [15] | Mikiten, T. M.; Salingaros, N. A.; Yu, H. S., Pavements as embodiments of meaning for a fractal mind, Nexus Netw. J., 2, 63-74 (2000) |
| [16] | Alasty, A.; Shabani, R., Chaotic motions and fractal basin boundaries in spring-pendulum system, Nonlinear Anal. Real World Appl., 7, 1, 81-95 (2006) · Zbl 1168.70319 |
| [17] | Larkin, J.; Goldburg, W.; Bandi, M. M., Time evolution of a fractal distribution: particle concentrations in free-surface turbulence, Physica D, 239, 14, 1264-1268 (2010) · Zbl 1193.37121 |
| [18] | Kou, J.; Wu, F.; Lu, H.; Xu, Y.; Song, F., The effective thermal conductivity of porous media based on statistical self-similarity, Phys. Lett. A, 374, 1, 62-65 (2009) · Zbl 1248.80005 |
| [19] | Carpinteri, A.; Lacidogna, G.; Niccolini, G., Fractal analysis of damage detected in concrete structural elements under loading, Chaos Solitons Fract., 42, 4, 2047-2056 (2009) · Zbl 1198.94022 |
| [20] | Chandrasekhar, D.; Ganguli, R., Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim, Comput. Math. Appl., 60, 10, 2834-2858 (2010) · Zbl 1207.65059 |
| [21] | Isomaki, H. M.; Boehm, J. V.; Raty, R., Fractal basin boundaries of an impacting particle, Phys. Lett. A, 126, 8-9, 484-490 (1988) |
| [22] | Berezowski, M., Fractal character of basin boundaries in a tubular chemical reactor with mass recycle, Chem. Eng. Sci., 61, 4, 1342-1345 (2006) |
| [23] | Silchenko, A. N.; Luchinsky, D. G.; McClintock, P. V.E., Noise-induced escape through a fractal basin boundary, Physica A, 327, 3-4, 371-377 (2003) · Zbl 1031.37035 |
| [24] | Yang, X. S., A proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems, Chaos Solitons Fract., 15, 4, 655-657 (2003) · Zbl 1070.37015 |
| [25] | Ardelean, G., A comparison between iterative methods by using the basin of attraction, J. Appl. Math. Comput., 218, 1, 88-95 (2011) · Zbl 1252.65086 |
| [26] | Scott, M.; Neta, B.; Chun, C., Basin attractors for various methods, J. Appl. Math. Comput., 218, 2584-2599 (2011) · Zbl 1478.65037 |
| [27] | Ganguli, R., Optimum design of a helicopter rotor for low vibration using aeroelastic analysis and response surface methods, J. Sound Vib., 258, 2, 327-344 (2002) |
| [28] | Ganguli, R.; Chopra, I.; Haas, D. J., Helicopter rotor system fault detection using physics based model and neural networks, AIAA J., 36, 6, 1078-1086 (1998) |
| [29] | Murugan, M. S.; Ganguli, R., Aeroelastic stability enhancement and vibration suppression in a composite helicopter rotor, J. Aircr. AIAA, 42, 4, 1013-1024 (2005) |
| [30] | Murugan, M. S.; Ganguli, R.; Harursampath, D., Aeroelastic response of composite helicopter rotor with random material properties, J. Aircr. AIAA, 45, 1, 306-322 (2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
