信息系统的非线性动力学控制研究
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摘要
信息系统为复杂系统。复杂性科学被称为是科学史上继相对论和量子力学之后的又一次革命,它的出现极大地促进了科学向纵深发展,标志着人类的认识水平步入了一个崭新的阶段。非线性是复杂系统的一个重要特征,因此采用非线性动力学理论解决复杂系统问题成为一条主要而又有效的途径。本文将从非线性动力学角度出发,研究信息系统中,包括神经信息系统和通信信息系统,一些难以用传统方法解决的控制问题。
首先,对神经信息系统的研究表明,生理参数变化引发生物系统分岔可能是某些神经系统方面疾病的诱因。因此对分岔的稳定性控制研究在神经科学中占有十分重要的地位。本文以Hodgkin-Huxley神经元模型为研究对象,提出了一种利用washout滤波器控制函数的线性项来稳定Hopf分岔的新方法。通过应用Routh-Hurwitz稳定性判据,推导出使Hopf分岔稳定的控制系数范围,并且经过仿真证实了该方法的有效性。
其次,从非线性动力学角度来看,神经信息系统中的短期记忆工作在双稳态动力学区域,该区域的起点为双重极限环分岔。所以对双重极限环分岔点的控制研究将为与记忆有关疾病的治疗提供有效方法和理论依据。本文提出一种利用washout滤波器控制函数的立方项来改变FitzHugh-Nagumo神经元模型双重极限环分岔点的新方法。该方法应用中心流形定理和规范形理论,推导出加入控制后闭环系统控制系数的取值范围。改变控制系数的大小就能够将该分岔点前移或置后。
最后,基于对前两个具有明确数学表达形式系统的研究,提出了一种用于通信信息系统中抑制Turbo迭代译码算法的非线性动力学行为的新控制方法。研究表明,Turbo迭代译码中存在分岔、混沌等动力学行为,严重影响了译码性能。本文提出了一种改进的Mexican hat小波函数控制方法。该方法利用小波函数具有收敛性的特点,使迭代译码算法避开了低信噪比区域出现的动力学行为,快速收敛到一个不动点上。仿真结果表明所提出的方法明显降低了译码复杂度,使Turbo迭代译码算法在动力学区域内的译码性能平均改善了0.3dB,其结果优于已有报道的延迟反馈方法。
The information system is a complex system. Science of complexity is called the third scientific revolution following on the heels of relativity and quantum physics. The emergence of science of complexity greatly promotes the development of science in depth and indicates the new phase of awareness of human. Since nonlinearity is an important characteristic of complex system, it is the main and effective method to solve questions of complex system with theory of nonlinear dynamics. In this dissertation, control questions unsolvable with traditional method for information system, including neural information system and communicational information system, are studied from the point of view of dynamics
First, research on neural information system show, bifurcations of biological systems caused by variation of physiological parameters may be the reason for some neural system diseases. Hence study on stability control of bifurcation holds on an important position in neural science. In this dissertation, a new control method via washout filter controller with linear term is given to realize stability control of Hopf bifurcation in Hodgkin-Huxley neural model. By use of Routh-Hurwitz stability criterion, the range of control coefficient is derived. Further, simulation results demonstrate the effectiveness of the proposed method.
Second,from the point of view of dynamics, short-term memory of neural information system works during bistable dynamic regime. And double cycle bifurcation is the onset of bistable dynamic regime. Therefore research on control of location of double cycle bifurcation may provide therapeutic approaches and theory basis for memory problems. A new control method via washout filter controller with cubic term is presented to change location of double cycle bifurcation in this dissertation. By applying of center manifold and normal form theory, we can derive the range of control coefficient of closed system. Consequently, double cycle bifurcation poiont is advanced or delayed along with control coefficient.
Finally, based on the former study of two systems with definite mathematic expressions, a new control method is proposed to suppresse nonlinear dynamic behavior in Turbo iterative decoding algorithm of communicational information system. Research show there exist dynamic behaviors in Turbo iterative decoding algorithm such as bifurcation and chaos. Thus decoding performance is seriously affected. In this dissertation, control method with modified Mexican hat wavelet function is developed. By exploiting convergence property of wavelet function, decoding algorithm can avoid the dynamic behavior under low signal-to-noise condition and converges quickly to a fixed point. Simulation results show that the proposed method makes decoding complexity decreased greatly. Moreover, on average, turbo decoding algorithm presents a gain of about 0.3 dB in dynamic region, which is superior to that of the delay feedback method reported.
首先,对神经信息系统的研究表明,生理参数变化引发生物系统分岔可能是某些神经系统方面疾病的诱因。因此对分岔的稳定性控制研究在神经科学中占有十分重要的地位。本文以Hodgkin-Huxley神经元模型为研究对象,提出了一种利用washout滤波器控制函数的线性项来稳定Hopf分岔的新方法。通过应用Routh-Hurwitz稳定性判据,推导出使Hopf分岔稳定的控制系数范围,并且经过仿真证实了该方法的有效性。
其次,从非线性动力学角度来看,神经信息系统中的短期记忆工作在双稳态动力学区域,该区域的起点为双重极限环分岔。所以对双重极限环分岔点的控制研究将为与记忆有关疾病的治疗提供有效方法和理论依据。本文提出一种利用washout滤波器控制函数的立方项来改变FitzHugh-Nagumo神经元模型双重极限环分岔点的新方法。该方法应用中心流形定理和规范形理论,推导出加入控制后闭环系统控制系数的取值范围。改变控制系数的大小就能够将该分岔点前移或置后。
最后,基于对前两个具有明确数学表达形式系统的研究,提出了一种用于通信信息系统中抑制Turbo迭代译码算法的非线性动力学行为的新控制方法。研究表明,Turbo迭代译码中存在分岔、混沌等动力学行为,严重影响了译码性能。本文提出了一种改进的Mexican hat小波函数控制方法。该方法利用小波函数具有收敛性的特点,使迭代译码算法避开了低信噪比区域出现的动力学行为,快速收敛到一个不动点上。仿真结果表明所提出的方法明显降低了译码复杂度,使Turbo迭代译码算法在动力学区域内的译码性能平均改善了0.3dB,其结果优于已有报道的延迟反馈方法。
The information system is a complex system. Science of complexity is called the third scientific revolution following on the heels of relativity and quantum physics. The emergence of science of complexity greatly promotes the development of science in depth and indicates the new phase of awareness of human. Since nonlinearity is an important characteristic of complex system, it is the main and effective method to solve questions of complex system with theory of nonlinear dynamics. In this dissertation, control questions unsolvable with traditional method for information system, including neural information system and communicational information system, are studied from the point of view of dynamics
First, research on neural information system show, bifurcations of biological systems caused by variation of physiological parameters may be the reason for some neural system diseases. Hence study on stability control of bifurcation holds on an important position in neural science. In this dissertation, a new control method via washout filter controller with linear term is given to realize stability control of Hopf bifurcation in Hodgkin-Huxley neural model. By use of Routh-Hurwitz stability criterion, the range of control coefficient is derived. Further, simulation results demonstrate the effectiveness of the proposed method.
Second,from the point of view of dynamics, short-term memory of neural information system works during bistable dynamic regime. And double cycle bifurcation is the onset of bistable dynamic regime. Therefore research on control of location of double cycle bifurcation may provide therapeutic approaches and theory basis for memory problems. A new control method via washout filter controller with cubic term is presented to change location of double cycle bifurcation in this dissertation. By applying of center manifold and normal form theory, we can derive the range of control coefficient of closed system. Consequently, double cycle bifurcation poiont is advanced or delayed along with control coefficient.
Finally, based on the former study of two systems with definite mathematic expressions, a new control method is proposed to suppresse nonlinear dynamic behavior in Turbo iterative decoding algorithm of communicational information system. Research show there exist dynamic behaviors in Turbo iterative decoding algorithm such as bifurcation and chaos. Thus decoding performance is seriously affected. In this dissertation, control method with modified Mexican hat wavelet function is developed. By exploiting convergence property of wavelet function, decoding algorithm can avoid the dynamic behavior under low signal-to-noise condition and converges quickly to a fixed point. Simulation results show that the proposed method makes decoding complexity decreased greatly. Moreover, on average, turbo decoding algorithm presents a gain of about 0.3 dB in dynamic region, which is superior to that of the delay feedback method reported.
引文
[1] C. Flood, Dealing with complexity, New York: Plenum Press, 1993, 1~332
[2]钱学森,于景元,戴汝为,一个科学新邻域——开放复杂巨系统及其方法论,自然杂志,1990,13(1):3~10
[3] E. Richard, An overview of intentional change from a complexity perspective, Journal of Management Development, 2006, 25(7): 607~623
[4] G. Nicolis, C. Nicolis, Foundations of complex systems, European Review, 2009, 17(2): 237~248
[5] J. H. Kim, J. Stringer, Applied chaos, New York: John Wiley and Sons, 1992
[6]盛昭瀚,马军海,非线性动力学引论,北京:科学出版社, 2001
[7]刘式达,刘式适,非线性动力学和复杂现象,北京:气象出版社, 1989
[8]黄润生,黄浩,混沌及其应用(第二版, .武汉:武汉大学出版社, 2005
[9] E. N. Lorenz, The essence of chaos, Washington: University of Washington Press, 1993
[10] J. M. T. Thompson, H. B. Stewart, Nonlinear dynamics and chaos, New York: John Wiley and Sons, 2002
[11] A. T. Winfree, The Geometry of Biological Time., New York, Springer, 2000
[12] J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology, New York, Oxford University Press, 1994
[13] J. Keener, J. Sneyd, Mathematical Physiology, New York: Springer-Verlag, 1998
[14] L. Glass, M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton: Princeton University Press, 1988
[15] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, London: The MIT Press, 2005
[16] C. C. Canavier, Sodium dynamics underlying burst firing and putative mechanisms for the regulation of the firing pattern in midbrain dopamine neurons: a computational approach, Journal of Computational Neuroscience, 1999, 6(1): 49~69
[17] M. I. Rabinovich, D. I. Abarbanel, The role of chaos in neural systems, Neuroscience, 1998, 87(1): 5~14
[18] K. Aihara, G. Matsumoto, Chaotic oscillations and bifurcations in squid giant axons, Princeton, Manchester University Press, 1986, pp: 257~269
[19] G. Matsumoto, S. Aihara, Y. Hanyu, N. Takahashi, S. Yoshizawa, J. Nagumo, Chaos and phase locking in normal squid axons, Physics Letters A, 1987, 123(4): 162~166
[20] K. Aihara, G. Matsumoto, M. Ichikawa, An alternating periodic-chaotic sequence observed in neural oscillators, Physical Review Letters, 1985, 111(5): 251~255
[21] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane and its application to conduction and excitation in nerve, Journal of Physiology, 1952, 117: 500~544
[22] A. L. Hodgkin, A. F. Huxley, Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo, Journal of Physiology, 1952, 116: 449~472.
[23] A. L. Hodgkin, A. F. Huxley, The components of membrane conductance in the giant axon of Loligo, Journal of Physiology, 1952, 116: 473~496.
[24] A. L. Hodgkin, A. F. Huxley, The dual effect of membrane potential on sodium conductance in the giant axon of Loligo, Journal of Physiology, 1952, 116: 497~506
[25] R. Fitzhugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophysical Journal, 1961, 1(6): 445~456
[26] R. FitzHugh, Mathematical models of excitation and propagation in nerve, In: H. P. Schwan, Editor, Biological engineer, New York: McGraw-Hill, 1969
[27] R. E. Plant, Bifurcation and resonance in a model for bursting nerve cells, J. Math. Biol., 1981, 11(1): 15~32
[28] C. C. Canavier, J. W. Clark, J. H. Byrne, Simulation of the bursting activity of neuron R15 in Aplysia: role of ionic currents, calcium balance, and modulatory transmitters, Journal of Neurophysiology, 1991, 66(6): 2107~2124
[29] T. R. Chay, Bursting, spiking, chaos, fractals, and universality in biological rhythms, International Journal of Bifurcation and Chaos, 1995, 5(3): 595~635
[30] T. R. Chay, Electrical bursting and luminal calcium oscillation in excitable cell models, Biol. Cybern., 1996, 75(5): 419~431
[31] T. R. Chay, Chaos in a three-variable model of an excitable cell, Physica D, 1985, 16(2): 233~242
[32] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal, 1981, 35(1): 193~213
[33] J. Rinzel, Discussion: Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and an update, Bulletin of Mathematical Biology, 1990, 52(1-2): 5~23
[34] W. C. Troy, The bifurcation of periodic solutions in the Hodgkin-Huxley equations, Appl. Math. Q., 1978, 36: 73~83
[35] C. H. Tuckwell, J. Jost, Moment analysis of the Hodgkin–Huxley system with additive noise, Physica A: Statistical Mechanics and its Applications, 2009, 388(19): 4115~4125
[36] S. B. Kuang, J. F. Wang, T. Zeng, A. Y. Cao, Thermal impact on spiking properties in Hodgkin-Huxley neuron with synaptic stimulus, Pramana, 2008, 70(1): 183~190
[37] K. Aihara, G. Matsumoto, Y. Ikegaya, Periodic and Non-periodic response of a periodically forced Hodgkin-Huxley oscillator, J. Theor. Biol., 1984, 109: 249~269
[38] K. Aihara, G. Matsumoto, M. Ichikawa, An alternating periodic-chaotic sequence observed in neural oscillators, Physics Letters, 1985, 11(5): 251~255
[39] H. Fukai, S. Doi, T .Nomura, S. Sato, Hopf Bifurcations In Multiple Parameter Space of the Hodgkin-Huxley Equations, I. Global Organization Of Bistable Periodic Solutions, Biol. Cybern., 2000, 82(3): 215~222
[40] H. Fukai, T. Nomura, S. Doi, S. Sato, Hopf Bifurcations in Multiple-Parameter Space of the Hodgkin-Huxley Equations, II. Singularity Theoretic Approach and Highly Degenerate Bifurcations, Biol. Cybern., 2000, 82(3): 223~229
[41] Y. A. Bedrov, G. N. Akoev, O. E. Dick, Partition of the Hodgkin-Huxley type model parameter space in to the region of qualitatively different solutions, Biol. Cybern., 1992, 66(5): 413~418
[42] Y. A. Bedrov, O. E. Dick, etc., Method for constructing the boundary of the bursting oscillations region in the neuron model, Biol. Cybern., 2000, 82(6): 493~497
[43] Y. A. Bedrov, G. N. Akoev, O. E. Dick, On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values, Biol. Cybern., 1995, 73(2): 149~154
[44] M. C. Mackey, U. A. D. Heiden, Dynamic diseases and bifurcations in physiological control systems, Funk. Biol. Med., 1982, 1: 156~164
[45] L. Glass, M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, New Jersey: Princeton University Press, 1988
[46] J. Bélair, L. Glass, U. Heiden, J. Milton, Focus Issue: Dynamical Disease: Mathematical Analysis of Human Illness, Chaos, 1995, 5: 1~215
[47] P. E. Rapp, R. A. Latta, A. I. Mees, Parameter dependent transitions and the optimal control of dynamical diseases, Bull. Math. Biol., 1988, 50: 227~253
[48] J. Milton, P. Jung, Epilepsy as a dynamic disease, New York: Springer-Verlag, 2002
[49] U. A. D. Heiden, Schizophrenia as a dynamical disease, Pharmacopsychiatry 2006, 39(Suppl 1): S36~S42
[50] B. J. Gluckman, E. J. Neel, etc., Electric field suppression of epileptiform activity in hippocampal slices, Journal of Neurophysiology, 1996, 76(6): 4202~4205
[51] K. A. Richardson, B. J. Gluckman, etc., In Vivo Modulation of Hippocampal Epilepiform Activity with Radial Electric Fields, Epilepsia, 2003, 44: 768~777
[52] D. M. Durand, M. Bikson, Suppression and control of epileptiform activity by electrical stimulation: a review, Institute of Electrical and Electronics Engineers, 2001, 89(7): 1065~1082
[53] K. A. Richardson, S. J. Schiff, B. J. Gluckman, Control of Traveling Waves in Mammalian Cortex, Physics Review Letters, 2005, 94(2): 0281031
[54] J. Berzhanskaya, A. Gorchetchnikov, S. J. Schiff, Switching between gamma and theta: Dynamic network control using subthreshold electric fields, Neurocomputing, 2007, 70(10-12): 2091~2095
[55] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 1948, 27: 379~423
[56] C. Berrou, A. Glavieux, P. Thitimajshima, Shannon limit error-correction coding and decoding: Turbo codes, Proc. ICC'93, Geneva, Switzerland, 1993, pp. 1064~1070
[57] J. Hagenauer, E. Offer, L. Papke, Iterative decoding of binary block and convolutional codes, IEEE Trans. Inf. Theory, 1996, 42(2): 429~445
[58] 3GPP TS 25.212 V6.6.0, Multiplexing and channel coding (FDD), 2005.
[59] K. Sripimanwat, Turbo Code Applications: a Journey from a Paper to realization, Springer, 2005
[60] M. Aydinlik, M. Salehi, Performance Bounds for Unequal Error Protecting Turbo Codes,IEEE Trans. Commun., 2009, 57(5): 1215~1220
[61] F. M. Li, A. Y. Wu, On the New Stopping Criteria of Iterative Turbo Decoding by Using Decoding Threshold, IEEE Trans. Signal Process, 2007, 55(11): 5506~5516
[62] J. Skorupa, J. Slowack etc., Stopping criterions for turbo coding in a Wyner-Ziv video codec, Picture Coding Symposium, 2009, PCS 2009, Chicago, USA
[63] S. S. Shim, D. H. Jeong etc., A new stopping criterion for turbo codes, The 8th International Conference Advanced Communication Technology, 2006, ICACT, 2006, Phoenix Park, Korea
[64] Z. J. Wu, M. G. Peng, W. B. Wang, A new parity-check stopping criterion for turbo decoding, IEEE Commun. Lett., 2008, 12(4): 4304~4306
[65] M. Ferrari, S. Bellini, Rate variable, multi-binary turbo codes with controlled error-floor, IEEE Trans. Commun., 2009, 57(5):1209~1214
[66] E. Rosnes, O. Ytrehus, On lowering the error floor of bit-interleaved turbo-coded modulation, IEEE Int. Conf. on Commun., Paris, France, June, 2004, 2: 717~721
[67] A. Boronka, N. S. Muhammad, J. Speidel, Removing error floor for bit interleaved coded modulation MIMO transmission with iterative detection, IEEE Int.Conf.on Commun., Seoul, Korea, May, 2005, 4: 2392~2396
[68] T. Richradson, The Geometry of Turbo-Decoding Dyanamics, IEEE Trans. Inf. Theory, 2000, 46(1): 9~23
[69] L. Kocarev, Z. Tasev, A.Vardy, Improving turbo codes by control of transient chaos in turbo decoding algorithms, Electron. Lett., 2002, 38(20): 1184~1186
[70]张维,Turbo译码的分岔控制与时空混沌数字水印,硕士学位论文,北京:清华大学,2005
[71]张维,周淑华,任勇,山秀明,Turbo译码算法的分岔与控制,物理学报, 2006, 55(2): 622~627
[72] G. R. Chen, M. L. Moiola, H. O. Wang, Bifurcation control: Theories, methods, and applications, International Journal of Bifurcation and Chaos, 2000, 10(3): 515~548
[73] E. H. Abed, J. H. Fu, Local feedback stabilization and bifurcation control, I. Hopf bifurcation, System Control Letter, 1986, 7(1): 11~17
[74] E. H. Abed, J. H. Fu, Local feedback stabilization and bifurcation control, II. Stationary bifurcation, System Control Letter, 1987, 8(5): 467~473
[75] S. Behtash, S. S. Sastry, Stabilization of nonlinear systems with uncontrollable linearization, Systems and Control Letters, 1988, 33(6): 585~590
[76] C. H. Lee, E. H. Abed, Washout filters in the bifurcation control of high alpha flight dynamics, Proceeding of America of Control Conference, Boston, 1991
[77] H. O. Wang, E. H. Abed, Robust control of period doubling bifurcations and implications for control of chaos, Proceeding of the 33rd IEEE Conference Decision and Control, Orlando, 1994
[78] R. Genesio, A. Tesi, O. H. Wang, etc., Control of period doubling bifurcations using harmonic balance, Proceeding of the 32rd IEEE Conference Decision and Control, San Antonio, 1993
[79] A. Tesi, E. H. Abed, R. Genesio, etc., Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics, Automatica, 1996, 32(9): 1255~1271
[80] J. L. Moiola, D. W. Berns, G. R. Chen, Feedback control of limit cycle amplitudes, Proceeding of the 36rd IEEE Conference Decision and Control, San Diego CA, 1997
[81] D. Chen, H. O. Wang, L. E. Howle, Amplitude control of bifurcations and application to Rayleigh-Benard convection, Proceeding of the 37rd IEEE Conference Decision and Control, Tampa, Florida, 1998
[82] E. F. F. Angulo, M. diBernardo, G. Olivar, Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory, IEEE Trans. Circuits and Systs. I, 2005, 52(2): 366~378
[83] N. Khaehintung, P. Sirisuk, A. Kunakorn, Implementation of fuzzy logic controller with bifurcation control of a current-mode boost converter, International Conference on Power Electronics and Drive Systems, Bangkok, Thailand, 2007
[84] D. Ghosh, A. R. Chowdhury, P. Saha, Bifurcation continuation, chaos and chaos control in nonlinear Bloch system, Communications in Nonlinear Science and Numerical Simulation, 2008, 13(8): 1461~1471
[85]符文彬,唐驾时,强迫Duffing振动系统的主共鞍结分岔控制,振动工程学报,2004, 17(3): 365~368
[86]钱长照,唐驾时,一类非自治时滞反馈系统的分岔控制,物理学报,2006, 55(2): 617~621
[87]钱长照,符文彬,一类自治系统Hopf分岔及极限环幅值的时滞控制,动力学与控制学报,2005, 3(4): 7~11
[88] J. S. Tang, Z. L. Chen, Amplitude control of limit cycle in van der Pol system, International Journal Bifurcation and Chaos, 2006, 16(2): 487~495
[89] S. T. Guo, G. Feng, X. F. Liao, Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback, Chaos, 2008, 18, 043104
[90] E. Ott, C. Grebogi, J. A. Yorke, Controlling chaos, Physical Review E, 1990, 64(11): 1196~1199
[91] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 1992, 170(6): 421~426
[92] K. Pyragas, A. Tamasevicias, Experimental control of chaos by self-controlling feedback, Physics Letters A , 1993, 180(1): 97~102
[93] B. A. Hubermanand, E. Lumer, Dynamics of adaptive system, IEEE Trans. CAS, 1990, 37(4): 547~550
[94] S. Sinha, R. Ramaswamy, J. Rao, Adaptive control in nonlinear dynamics, Physica D, 1990, 43(1): 118~128
[95] J. J. Slotine, W. Li, Applied nonlinear control, China Machine Presss, 2004
[96] R. Meucci, W. Gadomski, M. Ciofini, F. T, Arecchi, Experimental control of chaos by means of weak parametric perturbations, Phys. Rev. E, 1994, 49, R2528
[97] Y. Brainam, I. Goldhirsch, Taming chaotic dynamics with weak periodic perturbations, Phys. Rev. Lett., 1991, 66: 2545~2548
[98] Z. L. Qu, G. Hu, G. J. Yang, G. R. Qin, Phase effect in taming nonautonomous chaos by weak harmonic perturbations, Phys. Rev. Lett., 1995, 74: 1736~1739
[99] E. A. Jackson, The entrainment and migration controls of multiple-attractor systems, Physics Letters A, 1990, A151: 478~484
[100] M. A. Matiasand, J. Guemez, Stabilization of chaos by proportional pulses in the system variables, Phys. Rev. Lett., 1994, 72: 1455~1458
[101] M. Ramesh, S. Narayanan, Chaos control by nonfeedback methods in the presence of noise, Chaos, Solitons & Fractals, 1999, 10(9): 1473~1484
[102] A. L. Fradkov, R. J. Evans, Control of chaos: Methods and applications in engineering, Annual Review in Control, 2005, 29(1): 33~56
[103] K. Pyragas, Analytical properties and optimization of time-delayed feedback control, Phys. Rev. E, 2002, 66, 026207
[104] S. Bowong, F. M. M. Kakmeni, J. L. Dimi, Chaos control in the uncertain Duffing oscillator, Journal of Sound and Vibrations, 2006, 292: 869~880
[105] A. Alasty, H. Salarieh, Nonlinear feedback control of chaotic pendulum in presence of saturation effect, Chaos, Solitons & Fractals, 2007, 31 (2): 292~304
[106] K. Konishi, M. Hirai, H. Kokame, Sliding mode control for a class of chaotic systems, Physics Letters A, 1998, 245: 511~517
[107] M. Bonakdar, M. Samadi, H. Salarieh, A. Alasty, Stabilizing periodic orbits of chaotic systems using fuzzy control of Poincarémap, Chaos, Solitons & Fractals, 2008, 36: 682~693
[108] H. Salarieh, M. Shahrokhi, Indirect adaptive control of discrete chaotic systems, Chaos, Solitons & Fractals, 2007, 34: 1188~1201
[109]郝柏林,从抛物线谈起——混沌动力学引论,上海:上海科技教育出版社,1992
[110]刘延柱,陈立群编著.非线性振动,北京:高等教育出版社,2001
[111]马知恩,常微分方程定性与稳定性理论,北京:科学出版社,2001
[112]陆启韶,分岔与奇异性,上海:上海科技教育出版社,1995
[113]姚妙新,陈芳启,非线性理论数学基础,天津:天津大学出版社,2005
[114] (美)詹姆斯·格雷克著,张淑誉译,混沌:开创新科学,北京:高等教育出版社,2004
[115]陆启韶,常微分方程的定性方法与分岔,北京:北京航空航天大学出版社,1989
[116]刘秉正,非线性动力学,北京:高等教育出版社,2004
[117]刘延柱,陈立群.非线性动力学[M],上海:上海交通大学出版社,2000
[118]王洪礼,张琪昌.非线性动力学理论及应用,天津:天津大学出版社,2002
[119]陈予恕,非线性振动系统的分岔和混沌理论,北京:高等教育出版社,1993
[120]李继彬,冯贝叶.稳定性、分支与混沌,昆明:云南科技出版社,1995
[121] S.Wiggins, Global Bifurcations and Chaos, New York: Springer-Verlag, 1998
[122] S. Wiggins, Intruction to Applied Nonlinear Dynamical Systems and Chaos, New York: Springer-Verlag, 1990
[123] Y. A.Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., New York: Springer-Verlag, 1990
[124] G. Iooss, D. D.Joseph, Elementary Stability and Bifurcation Theory, New York: Springer-Verlag, 1990
[125] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vertor Fields, New York: Springer-Verlag, 1983
[126] B. D. Hassard, N. D. Kazarinoff, H. Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge, New York: Cambridge University Press, 1981
[127]唐焕文,神经信息学及其应用,北京:科学出版社,2007
[128]陈良泉,Hodgkin-Huxley模型的分岔分析与控制,硕士学位论文,天津:天津大学,2006
[129]车艳秋,外电场下神经元的分岔研究,博士学位论文,天津:天津大学,2008
[130]邓斌,神经元混沌、同步与控制,博士学位论文,天津:天津大学,2006
[131] Y. Xie, L. N. Chen, Y. M. Kang, K. Aihara, Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model, Physica. Rew. E, 2008, 77, 061921
[132] J. Wang, L. Q. Chen, X. Y. Fei, Bifurcation control of the Hodgkin-Huxley equations, Chaos, Solitons & Fractals, 2007, 33: 217~224
[133] J. Wang, K. M. Tsang, H. Zhang, Hopf bifurcation in the Hodgkin– Huxley model exposed to ELF electrical field, Chaos, Solitons & Fractals, 2004, 20: 759~764
[134] J. Wang, Y. Q. Che, X. Y. Fei, L. Li, Multi-parameter Hopf- bifurcation in Hodgkin–Huxley model exposed to ELF external electric field, Chaos, Solitons & Fractals, 2005, 26: 1221~1229
[135] W. M. Liu, Criterion of Hopf bifurcation without using eigenvalues, J. Math. Anal. Appl., 1994, 182: 250~255
[136] A. Hurwitz, On the conditions under which an equation has only roots with negative real parts, Math. Ann., 1895, 46: 273~284
[137] J. Rinzel, G. B. Ermentrout, Analysis of neural excitability and oscillations, in: C. Koch, I. Segev (Eds.), Methods in Neural Modeling, Cambridge: The MIT Press, 1989
[138] E. K. Kosmidis, K. Pakdaman, Stochastic chaos in a neuronal model, International Journal of Bifurcation and Chaos, 2006, 16(2): 395~410
[139]张琪昌,Hopf分叉范式理论及其在非线性振动系统中的应用,博士学位论文,天津:天津大学,1991
[140]张琪昌,分岔与混沌理论及应用,天津:天津大学出版社,2005
[141] B. D. Hassard, N. D. Kazarinoff, Y. H.Wan, Theory and application of Hopf bifurcation, London Mathematical Society Lecture Note Series, Cambridge: Cambridge Univ. Press, 1981, 41: 86-91
[142] J. Carr, Applications of centre manifold theory, New York: Spring-Verlag, 1981
[143] S. N. Chow, C. Li, D. Wang, Normal forms and bifurcation of Pplanar vector fields, Cambridge: Cambridge Univ. Press, 1994
[144] R. J. Mceliece, Theory of information and coding, 2nd edition, New York: Cambridge University Press, 2001
[145] J. M. Spinelli, Error correcting and detecting code, Encyclopedia of computer science, 4th edition, Chichester: John Wiley and Sons Ltd., 2003
[146] S. Benedetto, Implementation and performance of a Turbo/MAP decoder, International Journal of Satellite Communications, 1998, 15:23~46
[147] P. Robertson, E. Villebrun, P. Hoeher, A comparison of optimal and sub-optimal MAP decoding algorithms operating in the Log domain, IEEE Int. Commun. Conf., 1995, Washington, pp. 1009~1013
[148] D. Agrawal, and A. Vardy, The turbo decoding algorithm and its phase trajectories, IEEE Trans. Inf. Theory, 2001, 47(2): 699~722
[149] W. Zhang, S. H. Zhou, Y. Ren, X. M. Shan, Bifurcation analysis and control in Turbo decoding algorithm, Acta Phys. Sin., 2006, 55(2): 622~627
[150] L. Kocarev, F. Lehmann etc., Nonlinear Dynamics of Iterative Decoding Systems: Analysis and Applications, IEEE Trans. Inf. Theory, 2006, 52(4): 1366~1384
[151] I. Daubechies, Ten lectures on wavelets, Philadelphia: SIAM, 1992
[152] Y. Meyer, Wavelets: Algorithms and Applications, Philadelphia: SIAM, 1993
[153] S. J. Chapman, Matlab programming for engineers, Beijing: Science Press, 2003
[2]钱学森,于景元,戴汝为,一个科学新邻域——开放复杂巨系统及其方法论,自然杂志,1990,13(1):3~10
[3] E. Richard, An overview of intentional change from a complexity perspective, Journal of Management Development, 2006, 25(7): 607~623
[4] G. Nicolis, C. Nicolis, Foundations of complex systems, European Review, 2009, 17(2): 237~248
[5] J. H. Kim, J. Stringer, Applied chaos, New York: John Wiley and Sons, 1992
[6]盛昭瀚,马军海,非线性动力学引论,北京:科学出版社, 2001
[7]刘式达,刘式适,非线性动力学和复杂现象,北京:气象出版社, 1989
[8]黄润生,黄浩,混沌及其应用(第二版, .武汉:武汉大学出版社, 2005
[9] E. N. Lorenz, The essence of chaos, Washington: University of Washington Press, 1993
[10] J. M. T. Thompson, H. B. Stewart, Nonlinear dynamics and chaos, New York: John Wiley and Sons, 2002
[11] A. T. Winfree, The Geometry of Biological Time., New York, Springer, 2000
[12] J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology, New York, Oxford University Press, 1994
[13] J. Keener, J. Sneyd, Mathematical Physiology, New York: Springer-Verlag, 1998
[14] L. Glass, M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton: Princeton University Press, 1988
[15] E. M. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, London: The MIT Press, 2005
[16] C. C. Canavier, Sodium dynamics underlying burst firing and putative mechanisms for the regulation of the firing pattern in midbrain dopamine neurons: a computational approach, Journal of Computational Neuroscience, 1999, 6(1): 49~69
[17] M. I. Rabinovich, D. I. Abarbanel, The role of chaos in neural systems, Neuroscience, 1998, 87(1): 5~14
[18] K. Aihara, G. Matsumoto, Chaotic oscillations and bifurcations in squid giant axons, Princeton, Manchester University Press, 1986, pp: 257~269
[19] G. Matsumoto, S. Aihara, Y. Hanyu, N. Takahashi, S. Yoshizawa, J. Nagumo, Chaos and phase locking in normal squid axons, Physics Letters A, 1987, 123(4): 162~166
[20] K. Aihara, G. Matsumoto, M. Ichikawa, An alternating periodic-chaotic sequence observed in neural oscillators, Physical Review Letters, 1985, 111(5): 251~255
[21] A. L. Hodgkin, A. F. Huxley, A quantitative description of membrane and its application to conduction and excitation in nerve, Journal of Physiology, 1952, 117: 500~544
[22] A. L. Hodgkin, A. F. Huxley, Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo, Journal of Physiology, 1952, 116: 449~472.
[23] A. L. Hodgkin, A. F. Huxley, The components of membrane conductance in the giant axon of Loligo, Journal of Physiology, 1952, 116: 473~496.
[24] A. L. Hodgkin, A. F. Huxley, The dual effect of membrane potential on sodium conductance in the giant axon of Loligo, Journal of Physiology, 1952, 116: 497~506
[25] R. Fitzhugh, Impulses and Physiological States in Theoretical Models of Nerve Membrane, Biophysical Journal, 1961, 1(6): 445~456
[26] R. FitzHugh, Mathematical models of excitation and propagation in nerve, In: H. P. Schwan, Editor, Biological engineer, New York: McGraw-Hill, 1969
[27] R. E. Plant, Bifurcation and resonance in a model for bursting nerve cells, J. Math. Biol., 1981, 11(1): 15~32
[28] C. C. Canavier, J. W. Clark, J. H. Byrne, Simulation of the bursting activity of neuron R15 in Aplysia: role of ionic currents, calcium balance, and modulatory transmitters, Journal of Neurophysiology, 1991, 66(6): 2107~2124
[29] T. R. Chay, Bursting, spiking, chaos, fractals, and universality in biological rhythms, International Journal of Bifurcation and Chaos, 1995, 5(3): 595~635
[30] T. R. Chay, Electrical bursting and luminal calcium oscillation in excitable cell models, Biol. Cybern., 1996, 75(5): 419~431
[31] T. R. Chay, Chaos in a three-variable model of an excitable cell, Physica D, 1985, 16(2): 233~242
[32] C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber, Biophysical Journal, 1981, 35(1): 193~213
[33] J. Rinzel, Discussion: Electrical excitability of cells, theory and experiment: Review of the Hodgkin-Huxley foundation and an update, Bulletin of Mathematical Biology, 1990, 52(1-2): 5~23
[34] W. C. Troy, The bifurcation of periodic solutions in the Hodgkin-Huxley equations, Appl. Math. Q., 1978, 36: 73~83
[35] C. H. Tuckwell, J. Jost, Moment analysis of the Hodgkin–Huxley system with additive noise, Physica A: Statistical Mechanics and its Applications, 2009, 388(19): 4115~4125
[36] S. B. Kuang, J. F. Wang, T. Zeng, A. Y. Cao, Thermal impact on spiking properties in Hodgkin-Huxley neuron with synaptic stimulus, Pramana, 2008, 70(1): 183~190
[37] K. Aihara, G. Matsumoto, Y. Ikegaya, Periodic and Non-periodic response of a periodically forced Hodgkin-Huxley oscillator, J. Theor. Biol., 1984, 109: 249~269
[38] K. Aihara, G. Matsumoto, M. Ichikawa, An alternating periodic-chaotic sequence observed in neural oscillators, Physics Letters, 1985, 11(5): 251~255
[39] H. Fukai, S. Doi, T .Nomura, S. Sato, Hopf Bifurcations In Multiple Parameter Space of the Hodgkin-Huxley Equations, I. Global Organization Of Bistable Periodic Solutions, Biol. Cybern., 2000, 82(3): 215~222
[40] H. Fukai, T. Nomura, S. Doi, S. Sato, Hopf Bifurcations in Multiple-Parameter Space of the Hodgkin-Huxley Equations, II. Singularity Theoretic Approach and Highly Degenerate Bifurcations, Biol. Cybern., 2000, 82(3): 223~229
[41] Y. A. Bedrov, G. N. Akoev, O. E. Dick, Partition of the Hodgkin-Huxley type model parameter space in to the region of qualitatively different solutions, Biol. Cybern., 1992, 66(5): 413~418
[42] Y. A. Bedrov, O. E. Dick, etc., Method for constructing the boundary of the bursting oscillations region in the neuron model, Biol. Cybern., 2000, 82(6): 493~497
[43] Y. A. Bedrov, G. N. Akoev, O. E. Dick, On the relationship between the number of negative slope regions in the voltage-current curve of the Hodgkin-Huxley model and its parameter values, Biol. Cybern., 1995, 73(2): 149~154
[44] M. C. Mackey, U. A. D. Heiden, Dynamic diseases and bifurcations in physiological control systems, Funk. Biol. Med., 1982, 1: 156~164
[45] L. Glass, M. C. Mackey, From Clocks to Chaos: The Rhythms of Life, New Jersey: Princeton University Press, 1988
[46] J. Bélair, L. Glass, U. Heiden, J. Milton, Focus Issue: Dynamical Disease: Mathematical Analysis of Human Illness, Chaos, 1995, 5: 1~215
[47] P. E. Rapp, R. A. Latta, A. I. Mees, Parameter dependent transitions and the optimal control of dynamical diseases, Bull. Math. Biol., 1988, 50: 227~253
[48] J. Milton, P. Jung, Epilepsy as a dynamic disease, New York: Springer-Verlag, 2002
[49] U. A. D. Heiden, Schizophrenia as a dynamical disease, Pharmacopsychiatry 2006, 39(Suppl 1): S36~S42
[50] B. J. Gluckman, E. J. Neel, etc., Electric field suppression of epileptiform activity in hippocampal slices, Journal of Neurophysiology, 1996, 76(6): 4202~4205
[51] K. A. Richardson, B. J. Gluckman, etc., In Vivo Modulation of Hippocampal Epilepiform Activity with Radial Electric Fields, Epilepsia, 2003, 44: 768~777
[52] D. M. Durand, M. Bikson, Suppression and control of epileptiform activity by electrical stimulation: a review, Institute of Electrical and Electronics Engineers, 2001, 89(7): 1065~1082
[53] K. A. Richardson, S. J. Schiff, B. J. Gluckman, Control of Traveling Waves in Mammalian Cortex, Physics Review Letters, 2005, 94(2): 0281031
[54] J. Berzhanskaya, A. Gorchetchnikov, S. J. Schiff, Switching between gamma and theta: Dynamic network control using subthreshold electric fields, Neurocomputing, 2007, 70(10-12): 2091~2095
[55] C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 1948, 27: 379~423
[56] C. Berrou, A. Glavieux, P. Thitimajshima, Shannon limit error-correction coding and decoding: Turbo codes, Proc. ICC'93, Geneva, Switzerland, 1993, pp. 1064~1070
[57] J. Hagenauer, E. Offer, L. Papke, Iterative decoding of binary block and convolutional codes, IEEE Trans. Inf. Theory, 1996, 42(2): 429~445
[58] 3GPP TS 25.212 V6.6.0, Multiplexing and channel coding (FDD), 2005.
[59] K. Sripimanwat, Turbo Code Applications: a Journey from a Paper to realization, Springer, 2005
[60] M. Aydinlik, M. Salehi, Performance Bounds for Unequal Error Protecting Turbo Codes,IEEE Trans. Commun., 2009, 57(5): 1215~1220
[61] F. M. Li, A. Y. Wu, On the New Stopping Criteria of Iterative Turbo Decoding by Using Decoding Threshold, IEEE Trans. Signal Process, 2007, 55(11): 5506~5516
[62] J. Skorupa, J. Slowack etc., Stopping criterions for turbo coding in a Wyner-Ziv video codec, Picture Coding Symposium, 2009, PCS 2009, Chicago, USA
[63] S. S. Shim, D. H. Jeong etc., A new stopping criterion for turbo codes, The 8th International Conference Advanced Communication Technology, 2006, ICACT, 2006, Phoenix Park, Korea
[64] Z. J. Wu, M. G. Peng, W. B. Wang, A new parity-check stopping criterion for turbo decoding, IEEE Commun. Lett., 2008, 12(4): 4304~4306
[65] M. Ferrari, S. Bellini, Rate variable, multi-binary turbo codes with controlled error-floor, IEEE Trans. Commun., 2009, 57(5):1209~1214
[66] E. Rosnes, O. Ytrehus, On lowering the error floor of bit-interleaved turbo-coded modulation, IEEE Int. Conf. on Commun., Paris, France, June, 2004, 2: 717~721
[67] A. Boronka, N. S. Muhammad, J. Speidel, Removing error floor for bit interleaved coded modulation MIMO transmission with iterative detection, IEEE Int.Conf.on Commun., Seoul, Korea, May, 2005, 4: 2392~2396
[68] T. Richradson, The Geometry of Turbo-Decoding Dyanamics, IEEE Trans. Inf. Theory, 2000, 46(1): 9~23
[69] L. Kocarev, Z. Tasev, A.Vardy, Improving turbo codes by control of transient chaos in turbo decoding algorithms, Electron. Lett., 2002, 38(20): 1184~1186
[70]张维,Turbo译码的分岔控制与时空混沌数字水印,硕士学位论文,北京:清华大学,2005
[71]张维,周淑华,任勇,山秀明,Turbo译码算法的分岔与控制,物理学报, 2006, 55(2): 622~627
[72] G. R. Chen, M. L. Moiola, H. O. Wang, Bifurcation control: Theories, methods, and applications, International Journal of Bifurcation and Chaos, 2000, 10(3): 515~548
[73] E. H. Abed, J. H. Fu, Local feedback stabilization and bifurcation control, I. Hopf bifurcation, System Control Letter, 1986, 7(1): 11~17
[74] E. H. Abed, J. H. Fu, Local feedback stabilization and bifurcation control, II. Stationary bifurcation, System Control Letter, 1987, 8(5): 467~473
[75] S. Behtash, S. S. Sastry, Stabilization of nonlinear systems with uncontrollable linearization, Systems and Control Letters, 1988, 33(6): 585~590
[76] C. H. Lee, E. H. Abed, Washout filters in the bifurcation control of high alpha flight dynamics, Proceeding of America of Control Conference, Boston, 1991
[77] H. O. Wang, E. H. Abed, Robust control of period doubling bifurcations and implications for control of chaos, Proceeding of the 33rd IEEE Conference Decision and Control, Orlando, 1994
[78] R. Genesio, A. Tesi, O. H. Wang, etc., Control of period doubling bifurcations using harmonic balance, Proceeding of the 32rd IEEE Conference Decision and Control, San Antonio, 1993
[79] A. Tesi, E. H. Abed, R. Genesio, etc., Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics, Automatica, 1996, 32(9): 1255~1271
[80] J. L. Moiola, D. W. Berns, G. R. Chen, Feedback control of limit cycle amplitudes, Proceeding of the 36rd IEEE Conference Decision and Control, San Diego CA, 1997
[81] D. Chen, H. O. Wang, L. E. Howle, Amplitude control of bifurcations and application to Rayleigh-Benard convection, Proceeding of the 37rd IEEE Conference Decision and Control, Tampa, Florida, 1998
[82] E. F. F. Angulo, M. diBernardo, G. Olivar, Feedback control of limit cycles: a switching control strategy based on nonsmooth bifurcation theory, IEEE Trans. Circuits and Systs. I, 2005, 52(2): 366~378
[83] N. Khaehintung, P. Sirisuk, A. Kunakorn, Implementation of fuzzy logic controller with bifurcation control of a current-mode boost converter, International Conference on Power Electronics and Drive Systems, Bangkok, Thailand, 2007
[84] D. Ghosh, A. R. Chowdhury, P. Saha, Bifurcation continuation, chaos and chaos control in nonlinear Bloch system, Communications in Nonlinear Science and Numerical Simulation, 2008, 13(8): 1461~1471
[85]符文彬,唐驾时,强迫Duffing振动系统的主共鞍结分岔控制,振动工程学报,2004, 17(3): 365~368
[86]钱长照,唐驾时,一类非自治时滞反馈系统的分岔控制,物理学报,2006, 55(2): 617~621
[87]钱长照,符文彬,一类自治系统Hopf分岔及极限环幅值的时滞控制,动力学与控制学报,2005, 3(4): 7~11
[88] J. S. Tang, Z. L. Chen, Amplitude control of limit cycle in van der Pol system, International Journal Bifurcation and Chaos, 2006, 16(2): 487~495
[89] S. T. Guo, G. Feng, X. F. Liao, Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback, Chaos, 2008, 18, 043104
[90] E. Ott, C. Grebogi, J. A. Yorke, Controlling chaos, Physical Review E, 1990, 64(11): 1196~1199
[91] K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 1992, 170(6): 421~426
[92] K. Pyragas, A. Tamasevicias, Experimental control of chaos by self-controlling feedback, Physics Letters A , 1993, 180(1): 97~102
[93] B. A. Hubermanand, E. Lumer, Dynamics of adaptive system, IEEE Trans. CAS, 1990, 37(4): 547~550
[94] S. Sinha, R. Ramaswamy, J. Rao, Adaptive control in nonlinear dynamics, Physica D, 1990, 43(1): 118~128
[95] J. J. Slotine, W. Li, Applied nonlinear control, China Machine Presss, 2004
[96] R. Meucci, W. Gadomski, M. Ciofini, F. T, Arecchi, Experimental control of chaos by means of weak parametric perturbations, Phys. Rev. E, 1994, 49, R2528
[97] Y. Brainam, I. Goldhirsch, Taming chaotic dynamics with weak periodic perturbations, Phys. Rev. Lett., 1991, 66: 2545~2548
[98] Z. L. Qu, G. Hu, G. J. Yang, G. R. Qin, Phase effect in taming nonautonomous chaos by weak harmonic perturbations, Phys. Rev. Lett., 1995, 74: 1736~1739
[99] E. A. Jackson, The entrainment and migration controls of multiple-attractor systems, Physics Letters A, 1990, A151: 478~484
[100] M. A. Matiasand, J. Guemez, Stabilization of chaos by proportional pulses in the system variables, Phys. Rev. Lett., 1994, 72: 1455~1458
[101] M. Ramesh, S. Narayanan, Chaos control by nonfeedback methods in the presence of noise, Chaos, Solitons & Fractals, 1999, 10(9): 1473~1484
[102] A. L. Fradkov, R. J. Evans, Control of chaos: Methods and applications in engineering, Annual Review in Control, 2005, 29(1): 33~56
[103] K. Pyragas, Analytical properties and optimization of time-delayed feedback control, Phys. Rev. E, 2002, 66, 026207
[104] S. Bowong, F. M. M. Kakmeni, J. L. Dimi, Chaos control in the uncertain Duffing oscillator, Journal of Sound and Vibrations, 2006, 292: 869~880
[105] A. Alasty, H. Salarieh, Nonlinear feedback control of chaotic pendulum in presence of saturation effect, Chaos, Solitons & Fractals, 2007, 31 (2): 292~304
[106] K. Konishi, M. Hirai, H. Kokame, Sliding mode control for a class of chaotic systems, Physics Letters A, 1998, 245: 511~517
[107] M. Bonakdar, M. Samadi, H. Salarieh, A. Alasty, Stabilizing periodic orbits of chaotic systems using fuzzy control of Poincarémap, Chaos, Solitons & Fractals, 2008, 36: 682~693
[108] H. Salarieh, M. Shahrokhi, Indirect adaptive control of discrete chaotic systems, Chaos, Solitons & Fractals, 2007, 34: 1188~1201
[109]郝柏林,从抛物线谈起——混沌动力学引论,上海:上海科技教育出版社,1992
[110]刘延柱,陈立群编著.非线性振动,北京:高等教育出版社,2001
[111]马知恩,常微分方程定性与稳定性理论,北京:科学出版社,2001
[112]陆启韶,分岔与奇异性,上海:上海科技教育出版社,1995
[113]姚妙新,陈芳启,非线性理论数学基础,天津:天津大学出版社,2005
[114] (美)詹姆斯·格雷克著,张淑誉译,混沌:开创新科学,北京:高等教育出版社,2004
[115]陆启韶,常微分方程的定性方法与分岔,北京:北京航空航天大学出版社,1989
[116]刘秉正,非线性动力学,北京:高等教育出版社,2004
[117]刘延柱,陈立群.非线性动力学[M],上海:上海交通大学出版社,2000
[118]王洪礼,张琪昌.非线性动力学理论及应用,天津:天津大学出版社,2002
[119]陈予恕,非线性振动系统的分岔和混沌理论,北京:高等教育出版社,1993
[120]李继彬,冯贝叶.稳定性、分支与混沌,昆明:云南科技出版社,1995
[121] S.Wiggins, Global Bifurcations and Chaos, New York: Springer-Verlag, 1998
[122] S. Wiggins, Intruction to Applied Nonlinear Dynamical Systems and Chaos, New York: Springer-Verlag, 1990
[123] Y. A.Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., New York: Springer-Verlag, 1990
[124] G. Iooss, D. D.Joseph, Elementary Stability and Bifurcation Theory, New York: Springer-Verlag, 1990
[125] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vertor Fields, New York: Springer-Verlag, 1983
[126] B. D. Hassard, N. D. Kazarinoff, H. Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge, New York: Cambridge University Press, 1981
[127]唐焕文,神经信息学及其应用,北京:科学出版社,2007
[128]陈良泉,Hodgkin-Huxley模型的分岔分析与控制,硕士学位论文,天津:天津大学,2006
[129]车艳秋,外电场下神经元的分岔研究,博士学位论文,天津:天津大学,2008
[130]邓斌,神经元混沌、同步与控制,博士学位论文,天津:天津大学,2006
[131] Y. Xie, L. N. Chen, Y. M. Kang, K. Aihara, Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model, Physica. Rew. E, 2008, 77, 061921
[132] J. Wang, L. Q. Chen, X. Y. Fei, Bifurcation control of the Hodgkin-Huxley equations, Chaos, Solitons & Fractals, 2007, 33: 217~224
[133] J. Wang, K. M. Tsang, H. Zhang, Hopf bifurcation in the Hodgkin– Huxley model exposed to ELF electrical field, Chaos, Solitons & Fractals, 2004, 20: 759~764
[134] J. Wang, Y. Q. Che, X. Y. Fei, L. Li, Multi-parameter Hopf- bifurcation in Hodgkin–Huxley model exposed to ELF external electric field, Chaos, Solitons & Fractals, 2005, 26: 1221~1229
[135] W. M. Liu, Criterion of Hopf bifurcation without using eigenvalues, J. Math. Anal. Appl., 1994, 182: 250~255
[136] A. Hurwitz, On the conditions under which an equation has only roots with negative real parts, Math. Ann., 1895, 46: 273~284
[137] J. Rinzel, G. B. Ermentrout, Analysis of neural excitability and oscillations, in: C. Koch, I. Segev (Eds.), Methods in Neural Modeling, Cambridge: The MIT Press, 1989
[138] E. K. Kosmidis, K. Pakdaman, Stochastic chaos in a neuronal model, International Journal of Bifurcation and Chaos, 2006, 16(2): 395~410
[139]张琪昌,Hopf分叉范式理论及其在非线性振动系统中的应用,博士学位论文,天津:天津大学,1991
[140]张琪昌,分岔与混沌理论及应用,天津:天津大学出版社,2005
[141] B. D. Hassard, N. D. Kazarinoff, Y. H.Wan, Theory and application of Hopf bifurcation, London Mathematical Society Lecture Note Series, Cambridge: Cambridge Univ. Press, 1981, 41: 86-91
[142] J. Carr, Applications of centre manifold theory, New York: Spring-Verlag, 1981
[143] S. N. Chow, C. Li, D. Wang, Normal forms and bifurcation of Pplanar vector fields, Cambridge: Cambridge Univ. Press, 1994
[144] R. J. Mceliece, Theory of information and coding, 2nd edition, New York: Cambridge University Press, 2001
[145] J. M. Spinelli, Error correcting and detecting code, Encyclopedia of computer science, 4th edition, Chichester: John Wiley and Sons Ltd., 2003
[146] S. Benedetto, Implementation and performance of a Turbo/MAP decoder, International Journal of Satellite Communications, 1998, 15:23~46
[147] P. Robertson, E. Villebrun, P. Hoeher, A comparison of optimal and sub-optimal MAP decoding algorithms operating in the Log domain, IEEE Int. Commun. Conf., 1995, Washington, pp. 1009~1013
[148] D. Agrawal, and A. Vardy, The turbo decoding algorithm and its phase trajectories, IEEE Trans. Inf. Theory, 2001, 47(2): 699~722
[149] W. Zhang, S. H. Zhou, Y. Ren, X. M. Shan, Bifurcation analysis and control in Turbo decoding algorithm, Acta Phys. Sin., 2006, 55(2): 622~627
[150] L. Kocarev, F. Lehmann etc., Nonlinear Dynamics of Iterative Decoding Systems: Analysis and Applications, IEEE Trans. Inf. Theory, 2006, 52(4): 1366~1384
[151] I. Daubechies, Ten lectures on wavelets, Philadelphia: SIAM, 1992
[152] Y. Meyer, Wavelets: Algorithms and Applications, Philadelphia: SIAM, 1993
[153] S. J. Chapman, Matlab programming for engineers, Beijing: Science Press, 2003