Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to the main page for the first course APMA0330
Return to the main page for the second course APMA0340
Return to Part III of the course APMA0340
Introduction to Linear Algebra with Mathematica
Glossary
Periodic Solutions
In this section, we consider systems of ordinary differential equations
It is well known that such non-autonomous systems may have a wide variety of periodic solutions (often called vibrations or oscillations). The period of these solutions may be T, i.e., the same as the period of the input function, or may be any integer m multiple of T, or any rational or even, in some exceptional cases, irrational number times T. The family of all the periodic solutions is, however, far from being arbitrary. Determination and analysis of periodic solutions for non-autonomous systems of equations is very difficult and interesting topic that is the area of active research. In this section, we mostly demonstrate ability of computational tools to help understand the behavior of periodic solutions, if any. The literature on the subject of periodic solutions is enormous, and we will restrict ourselves to describing and presenting a sequence of famous examples.
Theorem: Suppose that a one dimensional differential equation \( \dot{x} = f(x,t) \) has a bounded solution for any t greater than a certain finite value t_{0}. Then there exists a harmonic vibration.
Theorem: Suppose that all solutions to a two dimensional differential equation \( \dot{\bf x} = {\bf f}({\bf x},t) \) exist for any t greater than a certain finite value t_{0}. If one of them is bounded, then there exists a harmonic vibration.
When the dimension of the system \( \dot{\bf x} = {\bf f}({\bf x},t) \) exceeds 2, then there are known examples that the system has no periodic solutions or solutions are periodic with the period not compatible with the driven term.
Theorem: Let
Example: ■
Example: ■
Example: ■
Example: ■
Poincaré maps
A flow in three dimensions is cut locally by planes or other two-dimensional smooth surfaces that are chosen such that they do not contain any integral curves.
- Antosiewicz, H.A., Forced Periodic Solutions of Systems of Differential Equations, Annals of Mathematics, 1953, Vol. 57, No. 2, pp. 314--317; doi: 10.2307/1969861 https://www.jstor.org/stable/1969861
- Chen, H. and Li, Y., Bifurcation and stability of periodic solutions of Duffing equations, Nonlinearity, 2008, vol. 21, no. 11, pp. 2485--2503
- Fay, T.H., The forced hard spring equation, International Journal of Mathematical Education in Science and Technology, 2006, Vol. 37, No. 2, pp. 187--200. https://doi.org/10.1080/00207390500285735
- Fay, T.H. and Lott, P.A., Using the homotopy method to find periodic solutions of forced nonlinear differential equations, International Journal of Mathematical Education in Science and Technology, 2002, Vol. 33, No. 5, pp. 701--714. https://doi.org/10.1080/
- Ganji, S.S., Ganji, D.D., Ganji, Z.Z., Karimpour, S., Periodic solution for strongly nonlinear vibration systems by He’s Energy Balance Method, Acta Applied Mathematics, 2008, 106(1)79–92.
- Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 2002; https://doi.org/10.1007/978-1-4612-1140-2
- Lazer, A.C., and McKenna, P.J., On the existence of stable periodic solutions of differential equations of Duffing type, Procedings of the American Mathematical Society, 1990, Vol. 110, No. 1, pp. 125
- Li, W. and Shen, Z., Constructive proof on the existence of periodic solutions of Duffing equation, Chinese Science Bulletin, 1997, Vol. 42, pp. 1870--1874
- Li, Z. and Wang, M., On periodic solutions of Duffing's equation with damping, Chinese Science Bulletin, 1981, Vol. 26, No. 7, pp. 577--580
- Loud, W.S., On periodic solutions of Duffing's equation with damping, Studies in Applied Mathematics, 1955, Vol. 34, Issue 1-4, pp. 173--178. https://doi.org/10.1002/sapm1955341173 MathSciNet.
- Massera, J.L., The existence of periodic solutions of systems of differential equations, Duke Mathematical Journal, 1950, Vol. 17, No. 4, pp. 457--475. doi:10.1215/S0012-7094-50-01741-8. https://projecteuclid.org/euclid.dmj/1077476236
- Raghunandan, C.R. and Anand, G.V., Subharmonic vibrations of order 1/3 in stretched strings, The Journal of the Acoustical Society of America, 1978, Vol. 64, pp. 232 https://doi.org/10.1121/1.381966
- Schmidt, G. and Selsl, M., Subharmonic vibrations and chaos in forced nonlinear oscillators, Journal of Applied Mathematics and Mechanics, 1993, Vol. 73, Issue 2, pp. 93--107. ttps://doi.org/10.1002/zamm.19930730209
- Yin, Z. and Chen, H., On stability of periodic solutions of Lienard type equations, Discrete Dynamics in Nature and Society, 2017, Volume 2017, Article ID 1805760, 9 pages https://doi.org/10.1155/2017/1805760
Return to Mathematica page
Return to the main page (APMA0340)
Return to the Part 1 Matrix Algebra
Return to the Part 2 Linear Systems of Ordinary Differential Equations
Return to the Part 3 Non-linear Systems of Ordinary Differential Equations
Return to the Part 4 Numerical Methods
Return to the Part 5 Fourier Series
Return to the Part 6 Partial Differential Equations
Return to the Part 7 Special Functions