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The complicated dynamical evolution of a generalized BVP circuit system with piecewise linear characteristics is explored. The phase space is divided into different types of regions by the nonsmooth boundaries. In each region, the stabilities of the equilibrium points are investigated, from which the critical conditions related to simple bifurcations as well as Hopf bifurcations are obtained. By employing the analysis of the distribution of the eigenvalues of the generalized Jacobian matrix, the bifurcation behaviors related to the nonsmooth boundaries are explored in detail. It is pointed out that when pure imaginary eigenvalues associated with the generalized Jacobian matrix appear, the Hopf bifurcation may take place, leading the system to change from periodic motion into the quasi-periodic oscillation, while when zero eigenvalue occurs, it may lead the system to oscillate between different equilibrium points. Combined with the numerical simulations, two typical oscillation behaviors of the system verify the theoretical results.
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Keywords:
- generalized BVP oscillations /
- non-smooth bifurcation /
- generalized Jacobian matrix /
- Hopf bifurcation
[1] Hodgkin A L, Huxley A F 1952 J. Physiol. 116 449
[2] Hassard B 1978 J. Theor. Biol. 71 401
[3] Guchenhermer J, Oliva R A 2002 SIAM J. Appl. Dynamical Syst. 1 105
[4] Guido S,William C 2003 Physica D: Nonlinear Phenomena 177 1
[5] Zhang H, Holdem A V 1995 Chaos, Solitons and Fractals 10 303
[6] Rajasekar S 1996 Chaos, Solitons and Fractals 30 1799
[7] Ramesh M, Narayanan S 2001 Chaos, Solitons and Fractals 12 2395
[8] Ueta T, Kawakami H 2002 International Symposium on Circuts and Systems Toskushima Japan, 2002 May 26–29 II-544
[9] Wang J L, Feng G Q 2010 Int. J. Non-Linear Mech. 45 608
[10] Chimi EW, Fotsin H B,Woafo P 2008 Physica Scripta 77 045001
[11] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326 (in Chinese)[陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
[12] Sekikawa M, Inaba N, Yoshinaga T, Hikihara T 2010 Phys. Lett. A 374 3745
[13] Shimizu K, Sekikawa M, Inaba N 2011 Phys. Lett. A 375 1566
[14] Nishiuchi Y, Ueta T, Kawakami H 2006 Chaos, Solitons and Fractals 27 941
[15] Zhou G H, Xu J P, Bao B C 2010 Acta Phys. Sin. 59 2272 (in Chinese)[周国华, 许建平, 包伯成 2010 59 2272]
[16] Gonzalo M R, Jason A C 2010 Phys. Lett. A 375 143
[17] Avramov K V, Borysiuk O V 2008 J. Sound and Vibration 318 1197
[18] Zhusubaliyev T Z, Mosekilde E 2008 Physica D 237 930
[19] Makarenkov O, Nistri P 2008 J. Mathemat. Anal. Appl. 338 1401
[20] Santos B C, Savi M A 2009 Chaos, Solitons and Fractals 40 197
[21] Zhang G, Chen G, Chen T, Lin Y 2006 Chaos, Solitons and Fractals 30 1153
[22] Avrutin V, Schanz M 2004 Phys. Rev. E 70 026222
[23] Halse C, Homer M, Bernardo M D 2003 Chaos, Solitons and Fractals 18 953
[24] Zhusubaliyev Z H, Mosekilde E 2008 Phys. Lett. A 372 2237
[25] Liu M H, Yu S M 2006 Acta Phys. Sin. 55 5707 (in Chinese)[刘明华, 禹思敏 2006 55 5707]
[26] Gukenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Field (New York: Springer)
[27] Leine R I, Campen D H 2006 Eur. J. Mech. A Solids 25 595
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[1] Hodgkin A L, Huxley A F 1952 J. Physiol. 116 449
[2] Hassard B 1978 J. Theor. Biol. 71 401
[3] Guchenhermer J, Oliva R A 2002 SIAM J. Appl. Dynamical Syst. 1 105
[4] Guido S,William C 2003 Physica D: Nonlinear Phenomena 177 1
[5] Zhang H, Holdem A V 1995 Chaos, Solitons and Fractals 10 303
[6] Rajasekar S 1996 Chaos, Solitons and Fractals 30 1799
[7] Ramesh M, Narayanan S 2001 Chaos, Solitons and Fractals 12 2395
[8] Ueta T, Kawakami H 2002 International Symposium on Circuts and Systems Toskushima Japan, 2002 May 26–29 II-544
[9] Wang J L, Feng G Q 2010 Int. J. Non-Linear Mech. 45 608
[10] Chimi EW, Fotsin H B,Woafo P 2008 Physica Scripta 77 045001
[11] Chen Z Y, Zhang X F, Bi Q S 2010 Acta Phys. Sin. 59 2326 (in Chinese)[陈章耀, 张晓芳, 毕勤胜 2010 59 2326]
[12] Sekikawa M, Inaba N, Yoshinaga T, Hikihara T 2010 Phys. Lett. A 374 3745
[13] Shimizu K, Sekikawa M, Inaba N 2011 Phys. Lett. A 375 1566
[14] Nishiuchi Y, Ueta T, Kawakami H 2006 Chaos, Solitons and Fractals 27 941
[15] Zhou G H, Xu J P, Bao B C 2010 Acta Phys. Sin. 59 2272 (in Chinese)[周国华, 许建平, 包伯成 2010 59 2272]
[16] Gonzalo M R, Jason A C 2010 Phys. Lett. A 375 143
[17] Avramov K V, Borysiuk O V 2008 J. Sound and Vibration 318 1197
[18] Zhusubaliyev T Z, Mosekilde E 2008 Physica D 237 930
[19] Makarenkov O, Nistri P 2008 J. Mathemat. Anal. Appl. 338 1401
[20] Santos B C, Savi M A 2009 Chaos, Solitons and Fractals 40 197
[21] Zhang G, Chen G, Chen T, Lin Y 2006 Chaos, Solitons and Fractals 30 1153
[22] Avrutin V, Schanz M 2004 Phys. Rev. E 70 026222
[23] Halse C, Homer M, Bernardo M D 2003 Chaos, Solitons and Fractals 18 953
[24] Zhusubaliyev Z H, Mosekilde E 2008 Phys. Lett. A 372 2237
[25] Liu M H, Yu S M 2006 Acta Phys. Sin. 55 5707 (in Chinese)[刘明华, 禹思敏 2006 55 5707]
[26] Gukenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Field (New York: Springer)
[27] Leine R I, Campen D H 2006 Eur. J. Mech. A Solids 25 595
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