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To study the characteristics of the chaotic systems and their applications, an electronic circuit of simplified Lorenz chaotic system with one parameter is designed and experimented with discrete components. The system parameters correspond to the circuit element parameters. By regulating the variable resistor in the circuit, dynamic behaviors including limit cycle, pitchfork bifurcation, period-doubling bifurcation, chaos, and route to chaos by period-doubling bifurcation, are observed. The necessary condition for the existence of chaos in the fractional-order simplified Lorenz system is deduced. The lowest order of the fractional-order simplified Lorenz system and the variation law of the lowest order with system parameters are determined. Circuit simulations and experiments show that the simplified Lorenz system has rich dynamic characteristics, and that theoretical analysis and circuit experiment are accordant with each other.
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Keywords:
- chaos /
- fractional-order calculus /
- Lorenz system /
- circuit design
[1] Lorenz E N 1963 J. Atmos. Sci. 20 113
[2] Chen G R, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[3] Lü J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[4] Lü J H, Chen G R, Cheng D Z, elikovsk S 2002 Int. J. Bifur. Chaos 12 2917
[5] Li Y X, Tang W K S, Chen G R 2006 Proceedings International Conference on Communications Circuits and Systems (Vol.4) (Guilin: Guilin University of Electronic Technology) p2569
[6] Wang G Y, Qiu S S, Chen H, Cui J D 2008 J. Circ. Syst. 13 58 (in Chinese) [王光义、 丘水生、 陈 辉、 崔佳冬 2008 电路与系统学报 13 58]
[7] Tang L R, Li J, Fan B 2009 Acta Phys. Sin. 58 1446 (in Chinese) [唐良瑞、 李 静、 樊 冰 2009 物理学报 58 1446]
[8] Ahmad W, Sprott J C 2003 Chaos Solitons Fract. 16 339
[9] Arena P, Caponetto R, Fortuna L, Porto D 1997 Proceedings of the European Conference on Circuit Theory and Design (Budapest: Budapest University of Technology) p1259
[10] Li C G, Chen G R 2004 Chaos Solitons Fract. 22 549
[11] Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922 (in Chinese) [王发强、刘崇新 2006 物理学报 55 3922]
[12] Chen X R, Liu C X, Wang F Q 2008 Chin. Phys. 17 1664
[13] Lu J J, Liu C X 2007 Chin. Phys. 16 1586
[14] Sun K H, Sprott J C 2009 Int. J. Bifur. Chaos 19 1357
[15] Vaně ek A, elikovsk S 1996 Control Systems: From Linear Analysis to Synthesis of Chaos (London: Prentice-Hall)
[16] MoMmmad S T, Mohammed H 2007 Phys. Lett. A 367 102
[17] Li C G, Chen G R 2004 Physica A 341 55
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[1] Lorenz E N 1963 J. Atmos. Sci. 20 113
[2] Chen G R, Ueta T 1999 Int. J. Bifur. Chaos 9 1465
[3] Lü J H, Chen G R 2002 Int. J. Bifur. Chaos 12 659
[4] Lü J H, Chen G R, Cheng D Z, elikovsk S 2002 Int. J. Bifur. Chaos 12 2917
[5] Li Y X, Tang W K S, Chen G R 2006 Proceedings International Conference on Communications Circuits and Systems (Vol.4) (Guilin: Guilin University of Electronic Technology) p2569
[6] Wang G Y, Qiu S S, Chen H, Cui J D 2008 J. Circ. Syst. 13 58 (in Chinese) [王光义、 丘水生、 陈 辉、 崔佳冬 2008 电路与系统学报 13 58]
[7] Tang L R, Li J, Fan B 2009 Acta Phys. Sin. 58 1446 (in Chinese) [唐良瑞、 李 静、 樊 冰 2009 物理学报 58 1446]
[8] Ahmad W, Sprott J C 2003 Chaos Solitons Fract. 16 339
[9] Arena P, Caponetto R, Fortuna L, Porto D 1997 Proceedings of the European Conference on Circuit Theory and Design (Budapest: Budapest University of Technology) p1259
[10] Li C G, Chen G R 2004 Chaos Solitons Fract. 22 549
[11] Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922 (in Chinese) [王发强、刘崇新 2006 物理学报 55 3922]
[12] Chen X R, Liu C X, Wang F Q 2008 Chin. Phys. 17 1664
[13] Lu J J, Liu C X 2007 Chin. Phys. 16 1586
[14] Sun K H, Sprott J C 2009 Int. J. Bifur. Chaos 19 1357
[15] Vaně ek A, elikovsk S 1996 Control Systems: From Linear Analysis to Synthesis of Chaos (London: Prentice-Hall)
[16] MoMmmad S T, Mohammed H 2007 Phys. Lett. A 367 102
[17] Li C G, Chen G R 2004 Physica A 341 55
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