November  2019, 24(11): 6025-6052. doi: 10.3934/dcdsb.2019119

Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems

1. 

School of Information Engineering, Southwestern University of Finance and Economics, Chengdu, Sichuan 610074, China

2. 

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Zhengdong Du

Received  December 2018 Published  November 2019 Early access  June 2019

Fund Project: The first author is supported by Humanities and Social Sciences Foundation of Ministry of Education of China under Grant Number 15YJAZH037. The second author is supported by NSFC (China) under Grant Number 11371264.

In the last few years, Battelli and Fečkan have developed a functional analytic method to rigorously prove the existence of chaotic behaviors in time-perturbed piecewise smooth systems whose unperturbed part has a piecewise continuous homoclinic solution. In this paper, by applying their method, we study the appearance of chaos in time-perturbed piecewise smooth systems with discontinuities on finitely many switching manifolds whose unperturbed part has a hyperbolic saddle in each subregion and a heteroclinic orbit connecting those saddles that crosses every switching manifold transversally exactly once. We obtain a set of Melnikov type functions whose zeros correspond to the occurrence of chaos of the system. Furthermore, the Melnikov functions for planar piecewise smooth systems are explicitly given. As an application, we present an example of quasiperiodically excited three-dimensional piecewise linear system with four zones.

Citation: Yurong Li, Zhengdong Du. Applying battelli-fečkan's method to transversal heteroclinic bifurcation in piecewise smooth systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6025-6052. doi: 10.3934/dcdsb.2019119
References:
[1]

J. Awrejcewicz, M. Fečkan and P. Olejnik, Bifurcations of planar sliding homoclinics, Mathematical Problems in Engineering, 2006 (2006), Art. ID 85349, 13 pp. doi: 10.1155/MPE/2006/85349.

[2]

J. Awrejcewicz and M. M. Holicke, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific, Singapore, 2007. doi: 10.1142/9789812709103.

[3]

M. BartuccelliP. L. ChristiansenN. F. Pedersen and M. P. Soerensen, Prediction of chaos in a Josephson junction by the Melnikov-function technique, Physical Review B, 33 (1986), 4686-4691. 

[4]

F. Battelli and C. Lazzari, Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations, 86 (1990), 342-366.  doi: 10.1016/0022-0396(90)90034-M.

[5]

F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems, J. Dynam. Differential Equations, 20 (2008), 337-376.  doi: 10.1007/s10884-007-9087-9.

[6]

F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics, J. Differential Equations, 248 (2010), 2227-2262.  doi: 10.1016/j.jde.2009.11.003.

[7]

F. Battelli and M. Fečkan, An example of chaotic behaviour in presence of a sliding homoclinic orbit, Ann. Mat. Pura Appl., 189 (2010), 615-642.  doi: 10.1007/s10231-010-0128-3.

[8]

F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems, J. Dynam. Differential Equations, 23 (2011), 495-540.  doi: 10.1007/s10884-010-9197-7.

[9]

F. Battelli and M. Fečkan, Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems, Physica D, 241 (2012), 1962-1975.  doi: 10.1016/j.physd.2011.05.018.

[10]

F. Battelli and M. Fečkan, Chaos in forced impact systems, Discrete and Continuous Dynamical Systems Series S, 6 (2013), 861-890.  doi: 10.3934/dcdss.2013.6.861.

[11]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008.

[12]

A. L. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 19 (1988), 1271-1294.  doi: 10.1137/0519093.

[13]

B. Bruhn and B. P. Koch, Heteroclinic bifurcations and invariant manifolds in rocking block dynamics, Z. Naturforsch. A, 46 (1991), 481-490.  doi: 10.1515/zna-1991-0603.

[14]

A. Calamai and M. Franca, Melnikov methods and homoclinic orbits in discontinuous systems, J. Dynam. Differential Equations, 25 (2013), 733-764.  doi: 10.1007/s10884-013-9307-4.

[15]

V. CarmonaS. Fernandez-GarciaE. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems, Physica D, 248 (2013), 44-54.  doi: 10.1016/j.physd.2013.01.002.

[16]

S. N. ChowJ. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373.  doi: 10.1016/0022-0396(80)90104-7.

[17]

S. N. Chow and S. W. Shaw, Bifurcations of subharmonics, J. Differential Equations, 65 (1986), 304-320.  doi: 10.1016/0022-0396(86)90022-7.

[18]

A. ColomboM. di BernardoS. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems, Physica D, 241 (2012), 1845-1860.  doi: 10.1016/j.physd.2011.09.017.

[19]

Z. DuY. LiJ. Shen and W. Zhang, Impact oscillators with homoclinic orbit tangent to the wall, Physica D, 245 (2013), 19-33.  doi: 10.1016/j.physd.2012.11.007.

[20]

Z. Du and W. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50 (2005), 445-458.  doi: 10.1016/j.camwa.2005.03.007.

[21]

M. Fečkan, Topological Degree Approach to Bifurcation Problems, Springer, Dordrecht, 2008.

[22] M. Fečkan, Bifurcation and Chaos in Discontinuous and Continuous Systems, Higher Education Press, Beijing, 2011. 
[23]

J. Gao and Z. Du, Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum, Nonlinear Dynamics, 79 (2015), 1061-1074.  doi: 10.1007/s11071-014-1723-4.

[24]

A. GranadosS. J. Hogan and T. M. Seara, The Melnikov method and subharmonic orbits in a piecewise-smooth system, SIAM J. Applied Dynamical Systems, 11 (2012), 801-830.  doi: 10.1137/110850359.

[25]

A. GranadosS. J. Hogan and T. M. Seara, The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks, Physica D, 269 (2014), 1-20.  doi: 10.1016/j.physd.2013.11.008.

[26]

J. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations, 122 (1995), 1-26.  doi: 10.1006/jdeq.1995.1136.

[27]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[28]

S. J. Hogan, Heteroclinic bifurcations in damped rigid block motion, Roy. Soc. London Ser. A, 439 (1992), 155-162.  doi: 10.1098/rspa.1992.0140.

[29]

P. Kukučka, Melnikov method for discontinous planar systems, Nonlinear Anal. Ser. A, 66 (2007), 2698-2719.  doi: 10.1016/j.na.2006.04.001.

[30]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. doi: 10.1007/BFb0103843.

[31]

M. Kunze and T. Küpper, Non-smooth dynamical systems: An overview, in Ergodic theory, analysis, and efficient simulation of dynamical systems (ed. B. Fiedler), Springer, Berlin, 2001,431–452. doi: 10.1007/978-3-642-56589-2.

[32]

T. Küpper, H. Hosham and D. Weiss, Bifurcation for non-smooth dynamical systems via reduction methods, in Recent trends in dynamical systems (eds. A. Johann, H.-P. Kruse, F. Rupp and S. Schmitz), Springer Proc. Math. Stat., 35, Springer, Basel, 2013, 79–105. doi: 10.1007/978-3-0348-0451-6.

[33]

S. Lenci and G. Rega, Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks, Internat. J. Bifur. Chaos, 15 (2005), 1901-1918.  doi: 10.1142/S0218127405013046.

[34]

S. LiS. ChaoW. Zhang and Y. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application, Nonlinear Dynamics, 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[35]

S. LiX. GongW. Zhang and Y. Hao, The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold, Nonlinear Dynamics, 89 (2017), 939-953.  doi: 10.1007/s11071-017-3493-2.

[36]

O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002.

[37]

V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math Soc., 12 (1963), 1-57. 

[38]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.

[39]

S. W. Shaw and R. H. Rand, The transition to chaos in a simple mechanical system, Internat. J. Non-Linear Mech., 24 (1989), 41-56.  doi: 10.1016/0020-7462(89)90010-3.

[40]

J. Shen and Z. Du, Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones, Z. Angew. Math. Phys., 67 (2016), Art. 42, 17 pages. doi: 10.1007/s00033-016-0642-2.

[41]

L. ShiY. Zou and T. Küpper, Melnikov method and detection of chaos for non-smooth systems, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 881-896.  doi: 10.1007/s10255-013-0265-8.

[42]

D. J. W. Simpson and J. D. Meiss, Aspects of bifurcation theory for piecewise-smooth, continuous systems, Physica D, 241 (2012), 1861-1868.  doi: 10.1016/j.physd.2011.05.002.

[43]

S. Wiggins, Global Bifurcations and Chaos - Analytical Methods, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1042-9.

[44]

J. X. XuR. Yan and W. Zhang, An algorithm for Melnikov functions and application to a chaotic rotor, SIAM J. Sci. Comput., 26 (2005), 1525-1546.  doi: 10.1137/S1064827503420726.

show all references

References:
[1]

J. Awrejcewicz, M. Fečkan and P. Olejnik, Bifurcations of planar sliding homoclinics, Mathematical Problems in Engineering, 2006 (2006), Art. ID 85349, 13 pp. doi: 10.1155/MPE/2006/85349.

[2]

J. Awrejcewicz and M. M. Holicke, Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-type Methods, World Scientific, Singapore, 2007. doi: 10.1142/9789812709103.

[3]

M. BartuccelliP. L. ChristiansenN. F. Pedersen and M. P. Soerensen, Prediction of chaos in a Josephson junction by the Melnikov-function technique, Physical Review B, 33 (1986), 4686-4691. 

[4]

F. Battelli and C. Lazzari, Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations, 86 (1990), 342-366.  doi: 10.1016/0022-0396(90)90034-M.

[5]

F. Battelli and M. Fečkan, Homoclinic trajectories in discontinuous systems, J. Dynam. Differential Equations, 20 (2008), 337-376.  doi: 10.1007/s10884-007-9087-9.

[6]

F. Battelli and M. Fečkan, Bifurcation and chaos near sliding homoclinics, J. Differential Equations, 248 (2010), 2227-2262.  doi: 10.1016/j.jde.2009.11.003.

[7]

F. Battelli and M. Fečkan, An example of chaotic behaviour in presence of a sliding homoclinic orbit, Ann. Mat. Pura Appl., 189 (2010), 615-642.  doi: 10.1007/s10231-010-0128-3.

[8]

F. Battelli and M. Fečkan, On the chaotic behaviour of discontinuous systems, J. Dynam. Differential Equations, 23 (2011), 495-540.  doi: 10.1007/s10884-010-9197-7.

[9]

F. Battelli and M. Fečkan, Nonsmooth homoclinic orbits, Melnikov functions and chaos in discontinuous systems, Physica D, 241 (2012), 1962-1975.  doi: 10.1016/j.physd.2011.05.018.

[10]

F. Battelli and M. Fečkan, Chaos in forced impact systems, Discrete and Continuous Dynamical Systems Series S, 6 (2013), 861-890.  doi: 10.3934/dcdss.2013.6.861.

[11]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008.

[12]

A. L. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flows, SIAM J. Math. Anal., 19 (1988), 1271-1294.  doi: 10.1137/0519093.

[13]

B. Bruhn and B. P. Koch, Heteroclinic bifurcations and invariant manifolds in rocking block dynamics, Z. Naturforsch. A, 46 (1991), 481-490.  doi: 10.1515/zna-1991-0603.

[14]

A. Calamai and M. Franca, Melnikov methods and homoclinic orbits in discontinuous systems, J. Dynam. Differential Equations, 25 (2013), 733-764.  doi: 10.1007/s10884-013-9307-4.

[15]

V. CarmonaS. Fernandez-GarciaE. Freire and F. Torres, Melnikov theory for a class of planar hybrid systems, Physica D, 248 (2013), 44-54.  doi: 10.1016/j.physd.2013.01.002.

[16]

S. N. ChowJ. K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits, J. Differential Equations, 37 (1980), 351-373.  doi: 10.1016/0022-0396(80)90104-7.

[17]

S. N. Chow and S. W. Shaw, Bifurcations of subharmonics, J. Differential Equations, 65 (1986), 304-320.  doi: 10.1016/0022-0396(86)90022-7.

[18]

A. ColomboM. di BernardoS. J. Hogan and M. R. Jeffrey, Bifurcations of piecewise smooth flows: Perspectives, methodologies and open problems, Physica D, 241 (2012), 1845-1860.  doi: 10.1016/j.physd.2011.09.017.

[19]

Z. DuY. LiJ. Shen and W. Zhang, Impact oscillators with homoclinic orbit tangent to the wall, Physica D, 245 (2013), 19-33.  doi: 10.1016/j.physd.2012.11.007.

[20]

Z. Du and W. Zhang, Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Comput. Math. Appl., 50 (2005), 445-458.  doi: 10.1016/j.camwa.2005.03.007.

[21]

M. Fečkan, Topological Degree Approach to Bifurcation Problems, Springer, Dordrecht, 2008.

[22] M. Fečkan, Bifurcation and Chaos in Discontinuous and Continuous Systems, Higher Education Press, Beijing, 2011. 
[23]

J. Gao and Z. Du, Homoclinic bifurcation in a quasiperiodically excited impact inverted pendulum, Nonlinear Dynamics, 79 (2015), 1061-1074.  doi: 10.1007/s11071-014-1723-4.

[24]

A. GranadosS. J. Hogan and T. M. Seara, The Melnikov method and subharmonic orbits in a piecewise-smooth system, SIAM J. Applied Dynamical Systems, 11 (2012), 801-830.  doi: 10.1137/110850359.

[25]

A. GranadosS. J. Hogan and T. M. Seara, The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks, Physica D, 269 (2014), 1-20.  doi: 10.1016/j.physd.2013.11.008.

[26]

J. Gruendler, Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations, 122 (1995), 1-26.  doi: 10.1006/jdeq.1995.1136.

[27]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[28]

S. J. Hogan, Heteroclinic bifurcations in damped rigid block motion, Roy. Soc. London Ser. A, 439 (1992), 155-162.  doi: 10.1098/rspa.1992.0140.

[29]

P. Kukučka, Melnikov method for discontinous planar systems, Nonlinear Anal. Ser. A, 66 (2007), 2698-2719.  doi: 10.1016/j.na.2006.04.001.

[30]

M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin-Heidelberg, 2000. doi: 10.1007/BFb0103843.

[31]

M. Kunze and T. Küpper, Non-smooth dynamical systems: An overview, in Ergodic theory, analysis, and efficient simulation of dynamical systems (ed. B. Fiedler), Springer, Berlin, 2001,431–452. doi: 10.1007/978-3-642-56589-2.

[32]

T. Küpper, H. Hosham and D. Weiss, Bifurcation for non-smooth dynamical systems via reduction methods, in Recent trends in dynamical systems (eds. A. Johann, H.-P. Kruse, F. Rupp and S. Schmitz), Springer Proc. Math. Stat., 35, Springer, Basel, 2013, 79–105. doi: 10.1007/978-3-0348-0451-6.

[33]

S. Lenci and G. Rega, Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks, Internat. J. Bifur. Chaos, 15 (2005), 1901-1918.  doi: 10.1142/S0218127405013046.

[34]

S. LiS. ChaoW. Zhang and Y. Hao, The Melnikov method of heteroclinic orbits for a class of planar hybrid piecewise-smooth systems and application, Nonlinear Dynamics, 85 (2016), 1091-1104.  doi: 10.1007/s11071-016-2746-9.

[35]

S. LiX. GongW. Zhang and Y. Hao, The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold, Nonlinear Dynamics, 89 (2017), 939-953.  doi: 10.1007/s11071-017-3493-2.

[36]

O. Makarenkov and J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D, 241 (2012), 1826-1844.  doi: 10.1016/j.physd.2012.08.002.

[37]

V. K. Melnikov, On the stability of the center for time periodic perturbations, Trans. Moscow Math Soc., 12 (1963), 1-57. 

[38]

K. J. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984), 225-256.  doi: 10.1016/0022-0396(84)90082-2.

[39]

S. W. Shaw and R. H. Rand, The transition to chaos in a simple mechanical system, Internat. J. Non-Linear Mech., 24 (1989), 41-56.  doi: 10.1016/0020-7462(89)90010-3.

[40]

J. Shen and Z. Du, Heteroclinic bifurcation in a class of planar piecewise smooth systems with multiple zones, Z. Angew. Math. Phys., 67 (2016), Art. 42, 17 pages. doi: 10.1007/s00033-016-0642-2.

[41]

L. ShiY. Zou and T. Küpper, Melnikov method and detection of chaos for non-smooth systems, Acta Math. Appl. Sin. Engl. Ser., 29 (2013), 881-896.  doi: 10.1007/s10255-013-0265-8.

[42]

D. J. W. Simpson and J. D. Meiss, Aspects of bifurcation theory for piecewise-smooth, continuous systems, Physica D, 241 (2012), 1861-1868.  doi: 10.1016/j.physd.2011.05.002.

[43]

S. Wiggins, Global Bifurcations and Chaos - Analytical Methods, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1042-9.

[44]

J. X. XuR. Yan and W. Zhang, An algorithm for Melnikov functions and application to a chaotic rotor, SIAM J. Sci. Comput., 26 (2005), 1525-1546.  doi: 10.1137/S1064827503420726.

Figure 1.  A heteroclinic cycle $ \Gamma $ of the unperturbed system (2) with $ n = 2 $ and $ m = 4 $
Figure 2.  The heteroclinic cycle $ \Gamma $ of the unperturbed system of (24) with $ \lambda = 1.05 $ and $ \eta = 0.75 $
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