Spatial chaos on surface and its associated bifurcation and Feigenbaum problem

详细信息    查看全文

• 作者：Shutang Liu ; Ping Liu ; Jian Liu ; Leyuan Wang
• 关键词：Spatial chaos ; Bifurcation ; Feigenbaum problem ; Surface polynomial
• 刊名：Nonlinear Dynamics
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：81
• 期：1-2
• 页码：283-298
• 全文大小：7,120 KB
• 参考文献：1.Winfree, A.T.: Scroll-shaped wave of chemical activity in three dimensions. Science 181, 937-39 (1973)View Article
  2.May, R.M.: Simple mathematical models with very complex dynamics. Science 186, 645-47 (1974)View Article
  3.Grebogi, C., Ott, E., Yorke, J.A.: Chaos, strange attractors, and fractal basin boundaries in nonlinear dynamics. Science 238, 632-38 (1987)MATH MathSciNet View Article
  4.Pool, R.: Putting chaos to work. Science 250, 626-28 (1990)View Article
  5.Ottino, J.M., Muzzio, F.J., Tjahjadi, M., Franjione, J.G., Jana, S.C., Kusch, H.A.: Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes. Science 257, 754-60 (1992)View Article
  6.Henson, S.M., Costantino, R.F., Cushing, J.M., Desharnais, R.A., Dennis, B., King, A.A.: Lattice effects observed in chaotic dynamics of experimental populations. Science 294, 602-05 (2001)View Article
  7.Otto, E.R., Klaus, W.: Chaos in the Zhabotinskii reaction. Nature 271, 89-0 (1978)View Article
  8.Juckes, M.N., Mclntyre, M.E.: A high-resolution one-layer model of breaking planetary waves in the stratosphere. Nature 328, 590-96 (1987)View Article
  9.Petrov, V., Gaspar, V., Masere, J., Showalter, K.: Controlling chaos in the Belousov–Zhabotinsky reaction. Nature 361, 240-46 (1993)View Article
  10.Schiff, S.J., Jerger, K., Duong, D.H., Chang, T., Spano, M.L., Ditto, W.L.: Controlling chaos in the brain. Nature 370, 615-21 (1994)View Article
  11.Paul, M.: Models for material failure and deformation. Science 252, 226-34 (1991)View Article
  12.Bertram, E.S., Chua, L.O.: Resistive grid image filtering: input/output analysis via the CNN Framework. IEEE Trans. Circuits Syst. I 39, 531-48 (1999)
  13.Li, X.P.: Partial difference equations used in the study of molecular orbits. Acta Chim. SINICA 40, 688-98 (1982)
  14.Liu, S.T., Wang, H.: Necessary and sufficient conditions for oscillations of a class of delay partial difference equations. Dyn. Syst. Appl. 7, 495-00 (1998)MATH
  15.Liu, S.T., Guan, X.P., Yang, J.: Nonexistence of positive solution of a class of nonlinear delay partial difference equation. J. Math. Anal. Appl. 234, 361-71 (1999)MATH MathSciNet View Article
  16.Liu, S.T., Liu, Y.Q.: Oscillation theorems for second-order nonlinear partial difference equations. J. Comput. Appl. Math. 132, 479-82 (2001)MATH MathSciNet View Article
  17.Liu, S.T., Liu, Y.Q.: Oscillation for nonlinear partial difference equation with positive and negative coefficients. Comput. Math. Appl. 43, 1219-230 (2002)MATH MathSciNet View Article
  18.Chen, G., Liu, S.T.: On spatial periodic orbits and spatial chaos. Int. J. Bifurc. Chaos 13, 935-41 (2003)MATH View Article
  19.Hao, B.L.: Symbolic dynamics and chaos in dissipative systems. World Scientific, Singapore (1989)MATH
  20.Kaneko, K.: Complex systems: chaos and beyond. Asakura Publishing Co., Ltd, Tokyo (1996)
  21.Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821-24 (1990)MATH MathSciNet View Article
  22.He, R., Vaidya, P.G.: Analysis and synthesis of synchronous periodic and chaotic systems. Phys. Rev. E 46, 7387-392 (1992)MathSciNet View Article
  23.Nikolai, F.R., Mikhai, M.S., Lev, S.T.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980-94 (1995)View Article
  24.Parlitz, U., Junge, L., Kocarev, L.: Subharmonic entrainment of unstable period orbits and generalized synchronization. Phys. Rev. Lett. 79, 3158-161 (1997)View Article
  25.Brown, R., Kocarev, L.: A unifying definition of synchronization for dynamical systems. Chaos 10, 344-49 (2000)MATH MathSciNet View Article
  26.Junge, L., Ulrich, P.: Synchronization using dynamic coupling. Phys. Rev. E 64, 2041-044 (2001)View Article
  27.Liu, S.T., Chen, G.: Asymptotic behavior of delay 2-D discrete logistic systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 49, 1677-682 (2002)View Article
  28.Liu, S.T., Chen, G.: Nonlinear feedback-controlled generalized synchronization of spatial chaos. Chaos Solitons Fractals 22, 35-6 (2002)MATH View Article
  29.Liu, S.T., Chen, G.: On spatial Lyapunov exponents and spatial chaos. Int. J. Bifurc. Chaos 13, 1163-181 (2003)MATH View Article
  30.Chen, G., Liu, S.T.: On generalized synchronization of spatial chaos. Chaos Solitons Fractals 15, 311-18 (2003)MATH MathSciNet View Article
  31.Devaney, R.L.: A first course in chaotic dynamical systems: theory and experiment, pp. 39-3. Westview Press, New York (1992)
  32.Liu, S.T., Wu, S.T.: Uniformity of spatial physical motion systems and spatial chaos behavior in the sense of Li–York. Int. J. Bifurc. Chaos Appl. Sci. Eng. 16, 2697-703 (2006)MATH View Article
  33.Liu, S.T.: Nuclear fission and spatial chaos. Chaos Solitons Fractals 30, 453-62 (2006)View Article
  34.Liu, S.T., Wu, S.T.: Spacial chaos behavior of molecular orbit. Chaos Solitons Fractals 31, 1
• 作者单位：Shutang Liu (1)
  Ping Liu (2) (3)
  Jian Liu (1) (4)
  Leyuan Wang (2)
  
  1. College of Control Science and Engineering, Shandong University, Jinan, 250061, China
  2. College of Mechanical and Electronic Engineering, Shandong Agricultural University, Taian, 271018, China
  3. College of Engineering, Peking University, Beijing, 100871, China
  4. School of Mathematical Sciences, University of Jinan, Jinan Shandong, 250022, China
  
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X

文摘

The qualitative theory of nonlinear spatial dynamical systems has attracted increasing attention recently. In particular, we have studied construction of spatial periodic orbits, dynamical behaviors of spatial chaos in the sense of Li–Yorke–Marotto, spatial Lyapunov exponents, control and generalized synchronization of spatial chaotic systems. In this paper, we apply a special mathematical transform to obtain spatial chaos on surface and its associated bifurcation and Feigenbaum problem. The 2D Logistic system is used for illustration. In addition, we also illustrate the difference in essence about chaos on surface as high-dimensional spatial system and dimension in phase space and we also analyze the parallel and different characters of theory system between 1D and 2D Logistic system.