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In this paper, we propose a joint estimation method of two parameters for phase space reconstruction in chaotic time series, based on radial basis function (RBF) neural networks. And we obtain the best estimation values, according to some objective standards. Furthermore, The single-step and multi-step RBF prediction model is used to estimate the best embedding dimension and delay time, and Lorenz system is selected as an example. Finally, the estimation values are tested in the original model. The simulations show that we can obtain the best estimation values through the method, and the prediction accuracy is significantly improved.
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Keywords:
- phase space reconstruction /
- radial basis function neural networks /
- parameter estimation /
- prediction
[1] Xu X, Zhang J, Small M 2008 Proc. Nat. Acad. Sci. USA 2008 19601
[2] Zhang J, Zhang K, Feng J F, Small M 2010 Plos Comput. Biol. 12 1001033
[3] Zhang J, Small M 2006 Phys. Rev. Lett. 96 238701
[4] Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510
[5] Chen D Y, Zhao W L, Ma X Y, Zhang R F 2011 Comput. Math. Appl. 61 3161
[6] Li H, Yang Z, Zhang Y M, Wen B C 2011 Acta Phys. Sin. 60 070512 (in Chinese) [李鹤, 杨周, 张义民, 闻邦椿 2011 物理学报 60 070512]
[7] Chen D Y, Shen T, Ma X Y 2011 Acta Phys. Sin. 60 050505 (in Chinese) [陈帝伊, 申滔, 马孝义 2011 物理学报 60 050505]
[8] Zhang C T, Ma Q L, Peng H 2010 Acta Phys. Sin. 59 7623 (in Chinese) [张春涛, 马千里, 彭宏 2010 物理学报 59 7623]
[9] Lü J H, Lu J A, Chen S H 2002 Analysis and Application of Chaos Time Series (Hubei: Wuhan University Press) p57 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用(湖北: 武汉大学出版社) 第57页]
[10] Prokhorov M D, Ponomarenko V I, Karavaev A S, Bezruchko B P 2005 Physica D: Nonlinear Phenomena 203 209
[11] Jin F B, Zhang K W, Zhong J X 2011 Chin. Phys. B 20 076701
[12] Wang J S, Yuan J, Li Q, Yuan R X 2011 Chin. Phys. B 20 050506
[13] Kostelich E J 1997 Sys. Control Lett. 31 313
[14] Lei M, Wang Z Z, Feng Z J 2002 Phys. Lett. A 303 179
[15] Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin. 59 1576 (in Chinese) [张淑清, 贾健, 高敏, 韩叙 2010 物理学报 59 1576]
[16] Maus A, Sprott J C Commun. Nonlinear Sci. Numer. Simul. 16 3294
[17] Small M, Tse C K 2004 Physica D: Nonlinear Phenomena 194 283
[18] Xie W J, Zhou W X 2011 Physica A 390 3592
[19] Yang S Q, Jia C Y 2002 Acta Phys. Sin. 51 2452 (in Chinese) [杨绍清, 贾传荧 2002 物理学报 51 2452]
[20] Yu D P, Zhao D Y 2009 J. Dalian Univ. Technol. 49 540
[21] You R Y, Chen Z, Xu S C, Wu B X 2004 Acta Phys. Sin. 53 2882 (in Chinese) [游荣义, 陈忠, 徐慎初, 吴伯僖 2004 物理学报 53 2882]
[22] Chen Z, Zeng Y C, Fu Z J 2008 Acta Phys. Sin. 57 46 (in Chinese) [陈争, 曾以成, 付志坚 2008 物理学报 57 46]
[23] Zhang C T, Ma Q L, Peng H 2011 Acta Phys. Sin. 60 7623 (in Chinese) [张春涛, 马千里, 彭宏 2011 物理学报 60 7623]
[24] John M, Chris D 1988 Neural Networks 1 202
[25] Takens F 1981 Lecture Notes Maths 898 366
[26] Jiang W J, Zhang L M, Wang P 2008 Appl. Math. Comput. 205 123
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[1] Xu X, Zhang J, Small M 2008 Proc. Nat. Acad. Sci. USA 2008 19601
[2] Zhang J, Zhang K, Feng J F, Small M 2010 Plos Comput. Biol. 12 1001033
[3] Zhang J, Small M 2006 Phys. Rev. Lett. 96 238701
[4] Bao B C, Liu Z, Xu J P 2010 Chin. Phys. B 19 030510
[5] Chen D Y, Zhao W L, Ma X Y, Zhang R F 2011 Comput. Math. Appl. 61 3161
[6] Li H, Yang Z, Zhang Y M, Wen B C 2011 Acta Phys. Sin. 60 070512 (in Chinese) [李鹤, 杨周, 张义民, 闻邦椿 2011 物理学报 60 070512]
[7] Chen D Y, Shen T, Ma X Y 2011 Acta Phys. Sin. 60 050505 (in Chinese) [陈帝伊, 申滔, 马孝义 2011 物理学报 60 050505]
[8] Zhang C T, Ma Q L, Peng H 2010 Acta Phys. Sin. 59 7623 (in Chinese) [张春涛, 马千里, 彭宏 2010 物理学报 59 7623]
[9] Lü J H, Lu J A, Chen S H 2002 Analysis and Application of Chaos Time Series (Hubei: Wuhan University Press) p57 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用(湖北: 武汉大学出版社) 第57页]
[10] Prokhorov M D, Ponomarenko V I, Karavaev A S, Bezruchko B P 2005 Physica D: Nonlinear Phenomena 203 209
[11] Jin F B, Zhang K W, Zhong J X 2011 Chin. Phys. B 20 076701
[12] Wang J S, Yuan J, Li Q, Yuan R X 2011 Chin. Phys. B 20 050506
[13] Kostelich E J 1997 Sys. Control Lett. 31 313
[14] Lei M, Wang Z Z, Feng Z J 2002 Phys. Lett. A 303 179
[15] Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin. 59 1576 (in Chinese) [张淑清, 贾健, 高敏, 韩叙 2010 物理学报 59 1576]
[16] Maus A, Sprott J C Commun. Nonlinear Sci. Numer. Simul. 16 3294
[17] Small M, Tse C K 2004 Physica D: Nonlinear Phenomena 194 283
[18] Xie W J, Zhou W X 2011 Physica A 390 3592
[19] Yang S Q, Jia C Y 2002 Acta Phys. Sin. 51 2452 (in Chinese) [杨绍清, 贾传荧 2002 物理学报 51 2452]
[20] Yu D P, Zhao D Y 2009 J. Dalian Univ. Technol. 49 540
[21] You R Y, Chen Z, Xu S C, Wu B X 2004 Acta Phys. Sin. 53 2882 (in Chinese) [游荣义, 陈忠, 徐慎初, 吴伯僖 2004 物理学报 53 2882]
[22] Chen Z, Zeng Y C, Fu Z J 2008 Acta Phys. Sin. 57 46 (in Chinese) [陈争, 曾以成, 付志坚 2008 物理学报 57 46]
[23] Zhang C T, Ma Q L, Peng H 2011 Acta Phys. Sin. 60 7623 (in Chinese) [张春涛, 马千里, 彭宏 2011 物理学报 60 7623]
[24] John M, Chris D 1988 Neural Networks 1 202
[25] Takens F 1981 Lecture Notes Maths 898 366
[26] Jiang W J, Zhang L M, Wang P 2008 Appl. Math. Comput. 205 123
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