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For a fourth-order autonomous nonlinear electric circuit, we present two evolution patterns to complexity associated with the three coexisting equilibrium points. In the first pattern, stable periodic movement with symmetric structure can be observed by Hopf bifurcation from the unstable equilibrium point, which may lead to chaos via cascading of period-doubling bifurcations. All the attractors, including the chaos, keep the symmetric property. While in the second evolution pattern, two limit cycles symmetric to each other may occur via Hopf bifurcations from the other two stable equilibrium points, which may also lead to two chaotic attractors, respectively. Comparing with the two evolution procedures associated with the two stable equilibrium points, not only the bifurcations keep the same pace, but also the attractors including the two final chaotic attractors are still symmetric to each other. With further variation of the parameters, the two chaotic attractors may interact with each other to form another enlarged chaotic attractor, which is qualitatively equivalent to the chaos in the first evolution pattern.
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Keywords:
- nonlinear electric circuit /
- transition boundary /
- bifurcation /
- chaos
[1] ［1］Liu F C, Zang X F, Song J Q 2009 Acta Phys. Sin. 58 3765 (in Chinese) ［刘福才、臧秀凤、宋佳秋 2009 物理学报 58 3765］
[2] ［2］Liu M H, Feng J C 2009 Acta Phys. Sin. 58 4457 (in Chinese) ［刘明华、冯久超 2009 物理学报 58 4457］
[3] ［3］Li Y N, Chen L, Cai Z S, Zhao X Z 2004 Chaos, Solitons & Fractals 22 767
[4] ［4］Yu P 1997 ASME J. Appl. Mech 64 957
[5] ［5］Bi Q S 2007 Phys. Lett. A 369 418
[6] ［6］Cang S J, Chen Z Q, Yuan Z Z 2008 Acta Phys. Sin. 57 1493 (in Chinese) ［仓诗建、陈增强、袁著祉 2008 物理学报 57 1493］
[7] ［7］Zhang H, Ma X K, Xue B L, Liu W Z 2005 Chaos, Solitons & Fractals 23 431
[8] ［8］Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Analysis: Real World Appl. 9 1158
[9] ［9］Bi Q S 2004 Int. J. Non-linear Mech. 39 33
[10] ］Koliopanos C L, Kyprianidis I M, Stouboulos I N, Anagnostopoulos A N, Magafas L 2003 Chaos, Solitons & Fractals 16 173
[11] ］Stouboulos I N, Miliou A N, Valaristos A P, Kyprianidis I M, Anagnostopoulos A N 2007 Chaos, Solitons & Fractals 33 1256
[12] ］Karagiannopoulos C G 2007 Journal of Electrostatics 65 535
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[1] ［1］Liu F C, Zang X F, Song J Q 2009 Acta Phys. Sin. 58 3765 (in Chinese) ［刘福才、臧秀凤、宋佳秋 2009 物理学报 58 3765］
[2] ［2］Liu M H, Feng J C 2009 Acta Phys. Sin. 58 4457 (in Chinese) ［刘明华、冯久超 2009 物理学报 58 4457］
[3] ［3］Li Y N, Chen L, Cai Z S, Zhao X Z 2004 Chaos, Solitons & Fractals 22 767
[4] ［4］Yu P 1997 ASME J. Appl. Mech 64 957
[5] ［5］Bi Q S 2007 Phys. Lett. A 369 418
[6] ［6］Cang S J, Chen Z Q, Yuan Z Z 2008 Acta Phys. Sin. 57 1493 (in Chinese) ［仓诗建、陈增强、袁著祉 2008 物理学报 57 1493］
[7] ［7］Zhang H, Ma X K, Xue B L, Liu W Z 2005 Chaos, Solitons & Fractals 23 431
[8] ［8］Chen Z Y, Zhang X F, Bi Q S 2008 Nonlin. Analysis: Real World Appl. 9 1158
[9] ［9］Bi Q S 2004 Int. J. Non-linear Mech. 39 33
[10] ］Koliopanos C L, Kyprianidis I M, Stouboulos I N, Anagnostopoulos A N, Magafas L 2003 Chaos, Solitons & Fractals 16 173
[11] ］Stouboulos I N, Miliou A N, Valaristos A P, Kyprianidis I M, Anagnostopoulos A N 2007 Chaos, Solitons & Fractals 33 1256
[12] ］Karagiannopoulos C G 2007 Journal of Electrostatics 65 535
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