Describing Chaos of Continuous Time System Using Bounded Space Curve

Affiliation(s)

^{1}
Electrical Engineering School of Guangxi University, Nanning, China.

^{2}
Information Engineering School of Jimei University, Xiamen, China.

ABSTRACT

The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state; moreover the motion of phase point will not approach infinite at last; system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.

The qualitative solutions of dynamical system expressed with nonlinear differential equation can be divided into two categories. One is that the motion of phase point may approach infinite or stable equilibrium point eventually. Neither periodic excited source nor self-excited oscillation exists in such nonlinear dynamic circuits, so its solution cannot be treated as the synthesis of multiharmonic. And the other is that the endless vibration of phase point is limited within certain range, moreover possesses character of sustained oscillation, namely the bounded nonlinear oscillation. It can persistently and repeatedly vibration after dynamic variable entering into steady state; moreover the motion of phase point will not approach infinite at last; system has not stable equilibrium point. The motional trajectory can be described by a bounded space curve. So far, the curve cannot be represented by concretely explicit parametric form in math. It cannot be expressed analytically by human. The chaos is a most universally common form of bounded nonlinear oscillation. A number of chaotic systems, such as Lorenz equation, Chua’s circuit and lossless system in modern times are some examples among thousands of chaotic equations. In this work, basic properties related to the bounded space curve will be comprehensively summarized by analyzing these examples.

KEYWORDS

Nonlinear Oscillation, Lorenz Equation, Chaos, Chua’s Circuit, Lossless Circuit, Space Curve

Nonlinear Oscillation, Lorenz Equation, Chaos, Chua’s Circuit, Lossless Circuit, Space Curve

Cite this paper

Huang, B. , Wei, Y. , Huang, Y. and Liang, Y. (2014) Describing Chaos of Continuous Time System Using Bounded Space Curve.*Journal of Modern Physics*, **5**, 1489-1501. doi: 10.4236/jmp.2014.515151.

Huang, B. , Wei, Y. , Huang, Y. and Liang, Y. (2014) Describing Chaos of Continuous Time System Using Bounded Space Curve.

References

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http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[2] Feng, C.-W., Cai, L., Kang, Q. and Zhang, L.-S. (2011) A Novel Three-Dimensional Autonomuos Chaotic System. Acta Physica Sinica, 60, 030503/1-7.

[3] Tang, L.-r., Li, J. and Fan, B. (2009) A New Three-dimensional Chaotic System and Its Circuit Simulation. Acta Physica Sinica , 58, 785-793.

[4] Xe, Z., Liu, C.-X. and Yang, T. (2010) Study On A New Chaotic System With Analysis and Circuit Experiment. Acta Physica Sinica, 59, 131-139.

[5] Liu, Y.-Z., Lin, C.-S. and Wang, Z.-L. (2010) A New Switched Four-Scroll Hyperchaotic System and Its Circuit Implementation. Acta Physica Sinica, 59, 8407-8413.

[6] Yu, S.-M.,Qiu, S.-S. and Lin, Q.-H. (2003) New Results of Study on Generating Multiple-Scroll Chaotic Attractors. Science in China (Series F), 46, 104-115.

[7] Chua, L.O., Komuro, M. and Matsumoto, T. (1986) The Double Scroll Family. IEEE Transactions on Circuits and Systems, 33, 1073-1117.

[8] Kennedy, M.P. (1994) Chaos in the Colpitts Oscillator. IEEE Transactions on Circuits and Systems I, 41, 771-774.

[9] Maggio, G.M., De Feo, O. and Kennedy, M.P. (1999) Nonlinear Analysis of the Colpitts Oscillator and Applications to Design. IEEE Transactions on Circuits and Systems, 46, 1118-1130.

[10] Huang, B.-H., Niu, L.-R., Lin, L.-F. and Sun, C.-M. (2007) Fundamental Wave Analysis Based on Power Balance. Acta Electronica Sinica, 35, 1994-1998.

[11] Huang, B.H., Huang, X.M. and Wei, S.G. (2008) Journal on Communications, 29, 65-70.

[12] Huang, B.H., Li, G.M. and Wei, Y.F. (2012) Modern Physics, 2, 60-69.

[13] Huang, B.H., Li, G.M. and Liu, H.J. (2013) Modern Physics, 3, 1-8.

http://dx.doi.org/10.12677/MP.2013.31001

[14] Huang, B.H., Huang, X.M. and Li, H. (2011) Main Components of Harmonic Solutions of Nonlinear Oscillations. International Conference on Electric Information and Control Engineering, ICEICE, Wuhan, 15-17 April 2011, 2307-2310.

[15] Huang, B.H., Huang, X.M. and Li, H. (2011) Procedia Engineering, 16, 325-332.

http://dx.doi.org/10.1016/j.proeng.2011.08.1091

[16] Huang, B.-H., Yang, G.-S., Wei, Y.-F. and Huang, Y. (2013) Harmonic Analysis Method Based on Power Balance. Applied Mechanics and Materials, 325-326, Manufacturing Engineering and Process II, 1508-1514

[17] Huang, B.-H., Yang, G.-S., Wei, Y.-F. and Huang, Y. (2013) Harmonic Analysis Method Based on Power Balance. Applied Mechanics and Materials, 327, Advanced Research on Materials, Applied Mechanics and Design Science, 1508-1514

[18] Sixty-Six Plane Phase Portraits Are Drawn from Lorenz Equation.

[1] Lorenz, E.N. (1963) Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20, 130-141.

http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

[2] Feng, C.-W., Cai, L., Kang, Q. and Zhang, L.-S. (2011) A Novel Three-Dimensional Autonomuos Chaotic System. Acta Physica Sinica, 60, 030503/1-7.

[3] Tang, L.-r., Li, J. and Fan, B. (2009) A New Three-dimensional Chaotic System and Its Circuit Simulation. Acta Physica Sinica , 58, 785-793.

[4] Xe, Z., Liu, C.-X. and Yang, T. (2010) Study On A New Chaotic System With Analysis and Circuit Experiment. Acta Physica Sinica, 59, 131-139.

[5] Liu, Y.-Z., Lin, C.-S. and Wang, Z.-L. (2010) A New Switched Four-Scroll Hyperchaotic System and Its Circuit Implementation. Acta Physica Sinica, 59, 8407-8413.

[6] Yu, S.-M.,Qiu, S.-S. and Lin, Q.-H. (2003) New Results of Study on Generating Multiple-Scroll Chaotic Attractors. Science in China (Series F), 46, 104-115.

[7] Chua, L.O., Komuro, M. and Matsumoto, T. (1986) The Double Scroll Family. IEEE Transactions on Circuits and Systems, 33, 1073-1117.

[8] Kennedy, M.P. (1994) Chaos in the Colpitts Oscillator. IEEE Transactions on Circuits and Systems I, 41, 771-774.

[9] Maggio, G.M., De Feo, O. and Kennedy, M.P. (1999) Nonlinear Analysis of the Colpitts Oscillator and Applications to Design. IEEE Transactions on Circuits and Systems, 46, 1118-1130.

[10] Huang, B.-H., Niu, L.-R., Lin, L.-F. and Sun, C.-M. (2007) Fundamental Wave Analysis Based on Power Balance. Acta Electronica Sinica, 35, 1994-1998.

[11] Huang, B.H., Huang, X.M. and Wei, S.G. (2008) Journal on Communications, 29, 65-70.

[12] Huang, B.H., Li, G.M. and Wei, Y.F. (2012) Modern Physics, 2, 60-69.

[13] Huang, B.H., Li, G.M. and Liu, H.J. (2013) Modern Physics, 3, 1-8.

http://dx.doi.org/10.12677/MP.2013.31001

[14] Huang, B.H., Huang, X.M. and Li, H. (2011) Main Components of Harmonic Solutions of Nonlinear Oscillations. International Conference on Electric Information and Control Engineering, ICEICE, Wuhan, 15-17 April 2011, 2307-2310.

[15] Huang, B.H., Huang, X.M. and Li, H. (2011) Procedia Engineering, 16, 325-332.

http://dx.doi.org/10.1016/j.proeng.2011.08.1091

[16] Huang, B.-H., Yang, G.-S., Wei, Y.-F. and Huang, Y. (2013) Harmonic Analysis Method Based on Power Balance. Applied Mechanics and Materials, 325-326, Manufacturing Engineering and Process II, 1508-1514

[17] Huang, B.-H., Yang, G.-S., Wei, Y.-F. and Huang, Y. (2013) Harmonic Analysis Method Based on Power Balance. Applied Mechanics and Materials, 327, Advanced Research on Materials, Applied Mechanics and Design Science, 1508-1514

[18] Sixty-Six Plane Phase Portraits Are Drawn from Lorenz Equation.