Parameter converting method for bifurcation analysis of nonlinear dynamical systems

Document Type : Article


Department of Mechanical Engineering, Babol Noshirvani University of Technology, Babol, P.O. Box 484, Postal Code 47148-71167, Mazandaran, Iran.


For detecting behavior of a dynamical system, bifurcation analysis is necessary with respect to change in parameters of system. In this work, based on the solution of ordinary differential equations from initial value and parameters, a simple method is presented, which can efficiently reveal different bifurcations of system. In addition to its simplicity, this method does not required to have deep physical and mathematical understanding of the problem, and because of its high precision and the speed of solutions, does not need to reduce the order of models in many complex problems or problems with high degrees of freedom. This method is named parameter converting method (PCM), which has two steps. In the first step the parameter is varied as a function of time and in the second step, time is expressed as inverse of this assumed function. With this method bifurcation and amplitude-frequency diagrams and hidden attractors of some complex dynamics will be analyzed and the sensitivity of the multi potential well systems to initial conditions is studied. With this algorithm, a simple way to find the domain of high-energy orbit in bistable systems is obtained.


Main Subjects

[1]. ALI HASAN NAYFEH, Introduction to Perturbation Techniques, Wiley Classics Library Edition Published (1993).
[2] M. Pakdemirli, M. M. F. Karahan and H. Boyac, “A new perturbation algorithm with better convergence properties: Multiple Scales Lindstedt Poincare Method”, Mathematical and Computational Applications, 14, pp. 31-44 (2009).
[3] M. Pakdemirli, “Review of the new Perturbation-Iteration method”, Mathematical and Computational Applications, 18, pp. 139-151 (2013).
[4] N. Damil and M. Potier-Ferry, “A new method to compute perturbed bifurcations: application to the buckling of imperfect elastic structures”, International Journal of Engineering Science, 28, pp. 943-957 (1990).
[5] P. Vannucci, B. Cochelin, N. Damil, M. Potier-Ferry, “An Asymptotic-Numerical method to compute bifurcating branches”, International Journal of Numerical Methods in Engineering, 41, pp. 1365-1389 (1998).
[6] E. H. Boutyour, H. Zahrouni, M. Potier-Ferry, M. Boudi, “Bifurcation points and bifurcated branches by an asymptotic numerical method and Padé approximants”, International Journal of Numerical Methods in Engineering, 60, pp. 1987-2012 (2004).
[7] B. Cochelin, N. Damil, M. Potier-Ferry, “Asymptotic Numerical Method and Padé Approximants for Nonlinear Elastic Structures”, International Journal for Numerical Methods in Engineering, 37 pp. 1187-1213 (1994).
[8] A. Hamdaoui, R.Hihi, B. Braikat, N. Tounsi, N. Damil, “A new class of vector padé approximants in the asymptotic numerical method: Application in nonlinear 2D elasticity”, World Journal of Mechanics, 4, 44-53 (2014).
[9] A. Elhage-Hussein, M. Potier-Ferry, N. Damil: “A numerical continuation method based on Pade approximants”, International Journal of Solids and Structures, 37, pp. 6981-7001 (2000).
[10] B. Cochelin, “A Path-Following technique via an Asymptotic-Numerical method”, Computers & Structures, 53, 1181-192 (1994).
[11] G. Kerschen, M. Peeters, J.C. Golinval, A.F. Vakakis, “Nonlinear normal modes, Part I: A useful framework for the structural dynamicis”, Mechanical Systems and Signal Processing, 23, pp. 170–194 (2009).
[12] M. Peeters, R. Viguie´, G. Se´randour, G. Kerschen, J.-C. Golinval, “Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques”, Mechanical Systems and Signal Processing, 23 pp. 195–216 (2009).
[13] D. A W Barton, B. Krauskopf, R. E Wilson, “Homoclinic bifurcations in a neutral delay model of a transmission line oscillator”, Nonlinearity, 20, pp. 809–829 (2007).
[14] W. J. Beyn, “The numerical computation of connecting orbits in dynamical systems”, IMA Journal of Numerical Analysis, 10, pp. 379–405 (1990).
[15]W. J. Beyn, Numerical methods for dynamical systems, Light W (ed) Advances in numerical analysis, Lancaster, Clarendon, Oxford, Vol. I (1990) 175–236.
[16]H.B. Keller, Lectures on Numerical Methods in Bifurcation Problems, Tata Institute Of Fundamental Research, (1986).
[17] Eugene L. Allgower, Kurt Georg, Introduction to Numerical Continuation Methods, Springer Series in Computational Mathematics (1990)
[18] H. Meijer, F. Dercole, B. Oldman, Numerical Bifurcation Analysis. In: Meyers R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY, (2012).
[19] E. J. Doedel, Nonlinear Numerics, Journal of the Franklin Institute, 334 (5–6), pp. 1049-1073 (1997).
[20] S. Karkar, B. Cochelin, Ch. Vergez, “A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities”, Journal of Sound and Vibration, 332, pp. 968–977 (2013).
[21]S. Karkar, B. Cochelin, Ch. Vergez, “A comparative study of the harmonic balance method and the orthogonal collocation method on stiff nonlinear systems”, Journal of Sound and Vibration, 333, pp. 2554–2567 (2014).
[22] G. Rega, H. Troger, “Dimension reduction of dynamical systems: methods, models, applications”, Nonlinear Dynamics, 41, pp. 1–15 (2005).
[23] A. Steindl, H. Troger, “Methods for dimension reduction and their application in nonlinear dynamics”, International Journal of Solids and Structures, 38, pp. 2131–2147 (2001).
[24] F. Terragni, J. M. Vega, “On the use of POD-based ROMs to analyze bifurcations in some dissipative systems”, Physica D, 241, pp. 1393–1405 (2012).
[25] M. Couplet, C. Basdevant, P. Sagaut, “Calibrated reduced-order POD- Galerkin system for fluid flow modeling”, Journal of Computational Physics, 207, pp. 192–220 (2005).
[26] S. Sirisup, G. E. Karniadakis, I. G. Kevrekidis, “Equations-free/Galerkin-free POD assisted computation of incompressible flows”, Journal of Computational Physics, 207, pp. 568–587 (2005).
[27] M. L. Rapun, J. M. Vega: “Reduced order models based on local POD plus Galerkin projection”, Journal of Computational Physics, 229, pp. 3046–3063 (2010).
[28] F. Terragni, E. Valero, J. M. Vega, Local POD plus Galerkin projection in the unsteady lid-driven cavity problem, Journal on Scientific Computing archive, 33, pp. 3538–3561 (2011).
[29] Eusebius Doedel Laurette, S. Tuckerman, Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, Springer, The IMA Volumes In Mathematics and its Applications 119 (1999).
[30] W. Govaerts, “Numerical bifurcation analysis for ODEs”, Journal of Computational and Applied Mathematics, 125, pp. 57-68 (2000).
[31] J. Heyman, G. Girault, Y. Guevel, C. Allery, A. Hamdouni, J. M. Cadou, “Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method”, Computers & Fluids, 76, pp. 73–85 (2013).
[32] Robert A. Meyers: Mathematics of Complexity And Dynamical Systems, Springer Science + Business Media, LLC (2012).
[33] H. A. Dijkstra, F. W. Wubs, A. K. Cliffe, E. Doedel, I. F. Dragomirescu, B. Eckhardt, A. Yu. Gelfgat, A. L. Hazel, V. Lucarini , A. G. Salinger, E. T. Phipps, J. Sanchez-Umbria, H. Schuttelaars ,L. S. Tuckerman, U. Thiele, “Numerical Bifurcation Methods and their Application to Fluid Dynamics: Analysis beyond Simulation”, Communications in Computational Physics, 15, pp. 1-45 (2014).
[34] G. Gai, S. Timme, “Nonlinear reduced-order modelling for limit-cycle oscillation analysis”, Nonlinear Dynamics, 84, pp. 991–1009 (2016).
[35] Y. Feng, W. Pan, “Hidden attractors without equilibrium and adaptive reduced-order function projective synchronization from hyper chaotic Rikitake system”, Pramana journal of physics, 88 :62 (2017).
[36] C. Chnafa, K. Valen-Sendstad, O. Brina, V.M. Pereira, D.A. Steinman, “Improved reduced-order modelling of cerebrovascular flow distribution by accounting for arterial bifurcation pressure drops”, Journal of Biomechanics, 51, pp. 83–88 (2017).
[37] Amabili M, Karazis K, Khorshidi K. “Nonlinear vibrations of rectangular laminated composite plates with different boundary conditions”, International Journal of Structural Stability and Dynamics. 11, pp. 673-95 (2011).
[38] Kurt E, Ciylan B, Taskan OO, Kurt HH. “Bifurcation analysis of a resistor-double inductor and double diode circuit and a comparison with a resistor-inductor-diode circuit in phase space and parametrical responses”, Scientia Iranica. Transaction D, Computer Science & Engineering, Electrical. 21, pp. 935-944 (2014).
[39] Sayyaadi H, Tadayon MA, Eftekharian AA. “Micro resonator nonlinear dynamics considering intrinsic properties”, Transaction B: Mechanical Engineering, 16, pp. 121- 129, (2009).
[44] E. N. Lorenz, “Deterministic nonperiodic flow”, Journal of the atmospheric sciences, 20, pp. 130-141 (1963).
[45] O. E. Rossler, “An Equation for Continuous Chaos”, Physics Letters, 57A, pp. 397-398 (1976).
[46] L. O. Chua, G. N. Lin, “Canonical Realization of Chua's Circuit Family”, IEEE Transactions on Circuits and Systems, 37, pp. 885-902 (1990).
[47] G. Chen, T. Ueta, “Yet another chaotic attractor”, International Journal of Bifurcation and Chaos, 9, pp. 1465-1466 (1999).
[48] D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, A. Prasad, “Hidden attractors in dynamical systems”, Physics Reports, 637, pp. 1-50 (2016).
[49] N. V. Kuznetsov, G. A. Leonov, V. I. Vagaitsev, "Analytical-numerical method for attractor localization of generalized Chua’s system", IFAC Proceedings Volumes 43, pp. 29-33 (2010).
[50] G. A. Leonov, V.I. Vagaitsev, N.V. Kuznetsov, “Algorithm for Localizing Chua Attractors Based on the Harmonic Linearization Method”, Doklady Mathematics, 433, pp. 323–326 (2010).
[51] G. A. Leonov, N. V. Kuznetsov, V. I. Vagaitsev, "Localization of hidden Chua’s attractors", Physics Letters A, 375, pp. 2230–2233(2011).
[52] G.A. Leonov, N.V. Kuznetsov, "Algorithms for searching for hidden oscillations in the Aizerman and Kalman Problems", Doklady Mathematics, 84 (1), 475–481 (2011).
[53] G. A. Leonov, N. V. Kuznetsov, "Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems", IFAC Proceedings Volumes, 44 (1), pp. 2494-2505 (2011).
[54] V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov, "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua’s Circuits", Journal of Computer and Systems Sciences International, 50 (4), pp. 511–543 (2011).
[55] N. Kuznetsov, O. Kuznetsova, G. Leonov, V. Vagaitsev, Analytical-numerical localization of hidden attractor in electrical Chua's circuit, Informatics in Control, Automation and Robotics. Lecture Notes in Electrical Engineering, Vol. 174. Springer, Berlin, Heidelberg
[56] I. Zelinka, "Evolutionary identification of hidden chaotic attractors", Engineering Applications of Artificial Intelligence, 50, pp. 159–167 (2016).
[57] V. G. Ivancevic, T. T. Ivancevic, High-Dimensional Chaotic and Attractor Systems: A Comprehensive Introduction, Intelligent systems, control and automation: Science and Engineering, 1st edition, Springer, 2007.
[58] Q. Li, H. Zeng, X.S. Yang, "On hidden twin attractors and bifurcation in the Chua’s circuit", Nonlinear Dynamics, 77 (1-2), pp. 255–266 (2014).
[59] D. F. Griffiths, D. J. Higham, Numerical methods for Ordinary Differential Equations: Initial Value Problems; Springer-Verlag London, 2010.
[60] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta formula", Journal of Computational and Applied Mathematics, 6, pp. 19–26 (1980).
[62] Ivana Kovacic, and Michael J. Brennan, The Duffing Equation: Nonlinear Oscillators and their Behaviour, WILEY, 2011.
[64] R. L. Harne, K. W. Wang, "A review of the recent research on vibration energy harvesting via bistable systems", Smart Materials and Structures, 22 (2), 023001 (12pp) (2013).
[65] Meghashyam Panyam Mohan Ram, Characterizing the effective bandwidth of nonlinear vibratory energy harvesters possessing multiple stable equilibria, PhD Dissertation, Clemson University, December 2015.
[66] R. Ramlan, M. J. Brennan, B. R. Mace, I. Kovacic, "Potential benefits of a non-linear stiffness in an energy harvesting device", Nonlinear Dynamics, 59 (4), pp. 545–558 (2010).
[67] A. Erturk, D. J. Inman, "Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling", Journal of Sound and Vibration, 330, pp. 2339–2353 (2011).
[68] M. A. Karami, D. J. Inman, "Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems", Journal of Sound and Vibration, 330, pp. 5583–5597 (2011).
[69] M. Panyamn, R. Masana, M. F. Daqaq, "On approximating the effective bandwidth of bi-stable energy harvesters", International Journal of Non-Linear Mechanics, 67, pp. 153–163 (2014).
[70] M. Panyam, M. F. Daqaq, "A comparative performance analysis of electrically optimized nonlinear energy harvesters", Journal of Intelligent Material Systems and Structures, 27 (4), pp. 537-548 (2016).
[70] W. Guang-Qing, L. Wei-Hsin, "A strategy for magnifying vibration in High-Energy orbits of a bistable oscillator at low excitation levels", Chinese Physics Letters, 32 (6), pp. 068503 (1-4) (2015).
[72] Sh. Zhou, J. Cao,D. J. Inman, Sh. Liu, W. Wang, J. Lin, "Impact-induced high-energy orbits of nonlinear energy harvesters", Applied Physics Letters, 106, pp. 093901 (1-4) (2015).
[73] G. Q.  Wang, W. H. Liao, "A bistable piezoelectric oscillator with an elastic magnifier for energy harvesting enhancement", Journal of Intelligent Material Systems and Structures, 28 (3), pp. 392-407 (2017).
[74] Sh. Zhou, J. Cao, D.J. Inman, J. Lin, D. Li, "Harmonic balance analysis of nonlinear tristable energy harvesters for performance enhancement", Journal of Sound and Vibration, 373 (7), pp. 223-235 (2016).