Exact solutions of the Liénard- and generalized Liénard-type ordinary nonlinear differential equations obtained by deforming the phase space coordinates of the linear harmonic oscillator

1. Liénard A (1928) Étude des oscillations entretenues. Revue générale de l’électricité 23:901–912, 946–954

2. Levinson N, Smith O (1942) A general equation for relaxation oscillations. Duke Math J 9:382–403

   Article  MathSciNet  MATH  Google Scholar 

3. Andronov AA, Leontovich EA, Gordon II, Maier AG (1973) Qualitative theory of second order dynamic systems. Wiley, New York

   MATH  Google Scholar 

4. Chandrasekar VK, Senthilvelan M, Kundu A, Lakshmanan M (2006) A nonlocal connection between certain linear and nonlinear ordinary differential equations/oscillators. J Phys A 39:9743–9754

   Article  ADS  MathSciNet  MATH  Google Scholar 

5. Liu X-G, Tang M-L, Martin RR (2008) Periodic solutions for a kind of Linard equation. J Comput Appl Math 219:263–275

   Article  ADS  MathSciNet  Google Scholar 

6. Zou L, Chen X, Zhang W (2008) Local bifurcations of critical periods for cubic Linard equations with cubic damping. J Comput Appl Math 222:404–410

   Article  ADS  MathSciNet  MATH  Google Scholar 

7. Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable cases of the Linard-type equation \(x{^{\prime \prime }}+f(x)x+g(x)\)=0: I. Equations having nonmaximal number of Lie point symmetries. J Math Phys 50:082702–082702-19

   Article  ADS  MathSciNet  MATH  Google Scholar 

8. Pandey SN, Bindu PS, Senthilvelan M, Lakshmanan M (2009) A group theoretical identification of integrable equations in the Linard-type equation \(x{^{\prime \prime }}+f(x)x+g(x)\)=0: II. Equations having maximal Lie point symmetries. J Math Phys 50:102701–102701-25

   Article  ADS  MathSciNet  MATH  Google Scholar 

9. Banerjee D, Bhattacharjee JK (2010) Renormalization group and Liénard systems of differential equations. J Phys A 43:062001

   Article  ADS  MathSciNet  MATH  Google Scholar 

10. Messias M, Alves G, Márcio R (2011) Time-periodic perturbation of a Liénard equation with an unbounded homoclinic loop. Physica D 240:1402–1409

    Article  ADS  MathSciNet  MATH  Google Scholar 

11. Mickens RE (2010) Truly nonlinear oscillations: harmonic balance, parameter expansions, iteration, and averaging methods. World Scientific, London

    Book  MATH  Google Scholar 

12. Nayfeh AH, Mook DT (1995) Nonlinear oscillations. Wiley, New York

    Book  MATH  Google Scholar 

13. DiBenedetto E (2011) Classical mechanics: theory and mathematical modeling. Springer, New York

    Book  MATH  Google Scholar 

14. Mak MK, Harko T (2012) New integrability case for the Riccati equation. Appl Math Comput 218:10974–10981

    MathSciNet  MATH  Google Scholar 

15. Mak MK, Harko T (2013) New further integrability cases for the Riccati equation. Appl Math Comput 219:7465–7471

    MathSciNet  MATH  Google Scholar 

16. Harko T, Lobo FSN, Mak MK (2014) Analytical solutions of the Riccati equation with coefficients satisfying integral or differential conditions with arbitrary functions. Univ J Appl Math 2:109–118

    Google Scholar 

17. Euler N (1997) Transformation properties of \(\frac{{\rm d}^2x}{{\rm d}t^2} +f_1 (t)\frac{{\rm d}x}{{\rm d}t} +f_2 (t) x+f_3 (t) x^n = 0\). J Nonlinear Math Phys 4:310–337

    Article  ADS  MathSciNet  MATH  Google Scholar 

18. Harko T, Lobo FSN, Mak MK (2013) Integrability cases for the anharmonic oscillator equation. J Pure Appl Math 10:115–129

    Google Scholar 

19. Harko T, Lobo FSN, Mak MK (2014) A class of exact solutions of the Linard type ordinary non-linear differential equation. J Eng Math 89:193–205

    Article  MathSciNet  Google Scholar 

20. Harko T, Lobo FSN, Mak MK (2013) A Chiellini type integrability condition for the generalized first kind Abel differential equation. Univ J Appl Math 1:101–104

    Google Scholar 

21. Mancas SC, Rosu HC (2013) Integrable dissipative nonlinear second order differential equations via factorizations and Abel equations. Phys Lett A 377:1234–1238

    Article  MathSciNet  MATH  Google Scholar 

22. Rosu HC, Cornejo-Perez O, Chen P (2012) Nonsingular parametric oscillators Darboux-related to the classical harmonic oscillator. Europhys Lett 100:60006

    Article  ADS  Google Scholar 

23. Mancas SC, Rosu HC (2015) Integrable Abel equations and Vein’s Abel equation. arXiv:1505.03548

24. Chandrasekar VK, Senthilvelan M, Lakshmanan M (2005) Unusual Liénard-type nonlinear oscillator. Phys Rev E 72:066203

    Article  ADS  Google Scholar 

25. Chandrasekar VK, Senthilvelan M, Lakshmanan M (2007) On the general solution for the modified Emden-type equation \(\ddot{x} + \alpha x\dot{x}+ \beta x^3 = 0\). J Phys A 40:4717–4727

    Article  ADS  MathSciNet  MATH  Google Scholar 

26. Chandrasekar VK, Sheeba JH, Pradeep G, Divyasreea RS, Lakshmanan M (2012) A class of solvable coupled nonlinear oscillators with amplitude independent frequencies. Phys Lett A 376:2188–2194

    Article  ADS  MATH  Google Scholar 

27. Calogero F (2008) Isochronous systems. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

28. Durga Devi A, Gladwin Pradeep R, Chandrasekar VK, Lakshmanan M (2013) Method of generating N-dimensional isochronous nonsingular Hamiltonian systems. J Nonlinear Math Phys 20:78–93

    Article  MathSciNet  Google Scholar 

29. Chithiika Ruby V, Senthilvelan M, Lakshmanan M (2012) Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator. J Phys A 45:382002

    Article  ADS  MathSciNet  MATH  Google Scholar 

30. Gubbiottia G, Nucci MC (2014) Noether symmetries and the quantization of a Liénard-type nonlinear oscillator. J Nonlinear Math Phys 21:248–264

    Article  MathSciNet  Google Scholar 

31. Herbst RT (1956) The equivalence of linear and non-linear equations. Proc Am Math Soc 7:95–97

    Article  MathSciNet  MATH  Google Scholar 

32. Dasarathy BV (1975) Equivalent linear models for non-linear non-autonomous systems. J Sound Vib 42:447–452

    Article  ADS  MATH  Google Scholar 

33. Finch MR (1989) The first integral of a class of non-linear second order ordinary differential equations. J Sound Vib 130:321–329

    Article  ADS  MathSciNet  MATH  Google Scholar 

34. Sawada K, Osawa T (1978) On exactly soluble nonlinear ordinary differential equations of the Liénard type. J Phys Soc Jpn 44:1730–1732

    Article  ADS  MathSciNet  MATH  Google Scholar 

35. Merzbacher E (1998) Quantum mechanics. Wiley, New York

    MATH  Google Scholar 

36. Reyes MA, Rosu HC, Gutierrez MR (2011) Self-adjoint oscillator operator from modifed factorization. Phys Lett A 375:2145–2148

    Article  ADS  MathSciNet  MATH  Google Scholar 

37. Arcos-Olalla R, Reyes MA, Rosu HC (2012) An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators. Phys Lett A 376:2860–2865

    Article  ADS  MathSciNet  Google Scholar 

38. Ibragimov NH (2009) A practical course in differential equations and mathematical modelling. World Scientific, London

    Book  Google Scholar 

39. Gilding BH, Kersner R (1996) The characterization of reaction–convection–diffusion processes by travelling waves. J Differ Equ 124:27–79

    Article  ADS  MathSciNet  MATH  Google Scholar 

40. Mei Z (2000) Numerical bifurcation analysis for reaction–diffusion equations. Springer, Berlin

    Book  MATH  Google Scholar 

41. Wilhelmsson H, Lazzaro E (2001) Reaction–diffusion problems in the physics of hot plasmas. Institute of Physics Publishing, Bristol

    Book  Google Scholar 

42. Gilding BH, Kersner R (2004) Travelling waves in nonlinear diffusion convection reaction. Birkhäuser, Basel

    Book  MATH  Google Scholar 

43. Harko T, Mak MK (2015) Travelling wave solutions of the reaction–diffusion mathematical model of glioblastoma growth: an Abel equation based approach. Math Biosci Eng 12:41–69

    Article  MathSciNet  MATH  Google Scholar 

44. Solovyova A, Schuck P, Costenaro L, Ebel C (2001) Non-ideality by sedimentation velocity of halophilic malate dehydrogenase in complex solvents. Biophys J 81:1868–1880

    Article  Google Scholar 

45. Follain N, Valleton J-M, Lebrun L, Alexandre B, Schaetzel P, Metayer M, Marais S (2010) Simulation of kinetic curves in mass transfer phenomena for a concentration-dependent diffusion coefficient in polymer membranes. J Membr Sci 349:195–207

    Article  Google Scholar 

46. Joannés S, Mazé L, Bunsell AR (2014) A concentration-dependent diffusion coefficient model for water sorption in composite. Compos Struct 108:111–118

    Article  Google Scholar 

47. Hamdan MN, Dado MHF (1997) Large aplitude free vibrations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. J Sound Vib 206:151–168

    Article  ADS  Google Scholar 

48. He JH (2005) Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 6:207–208

    MathSciNet  Google Scholar 

49. Yildirim A (2010) Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt–Poincaré method. Meccanica 45:1–6

    Article  MathSciNet  MATH  Google Scholar 

50. Ramos JI (2009) An artificial parameter Linstedt–Poincaré method for oscillators with smooth odd nonlinearities. Chaos Solitons Fractals 41:380–393

    Article  ADS  MATH  Google Scholar 

51. Herisanu N, Marinca V (2010) Explicit analytical approximation to large-amplitude non-linear oscillations of a uniform cantilever beam carrying an intermediate lumped mass and rotary inertia. Meccanica 45:847–855

    Article  MathSciNet  MATH  Google Scholar 

52. Akbarzade M, Khanb Y (2012) Dynamic model of large amplitude non-linear oscillations arising in the structural engineering: analytical solutions. Math Comput Model 55:480–489

    Article  MathSciNet  MATH  Google Scholar 

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