• G. c. LayekEmail author


It is well known that some important properties of nonlinear equations can be determined through qualitative analysis. The general theory and solution methods for linear equations are highly developed in mathematics, whereas a very little is known about nonlinear equations. Linearization of a nonlinear system does not provide the actual solution behaviors of the original nonlinear system. Nonlinear systems have interesting solution features. It is a general curiosity to know in what conditions an equation has periodic or bounded solutions. Systems may have solutions in which the neighboring trajectories are closed, known as limit cycles. What are the conditions for the existence of such limiting solutions? In what conditions does a system have unique limit cycle? These were some questions both in theoretical and engineering interest at the beginning of twentieth century. This chapter deals with oscillatory solutions in linear and nonlinear equations, their mathematical foundations, properties, and some applications.


Limit cycleLimit Cycle Neighboring Trajectories Unique Stable Limit Cycle Close Path Successive Zeros 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Tricomi, F.: Differential Equations. Blackie and Son, London (1961)Google Scholar
  2. 2.
    Mickens. R.E.: Oscillations in Planar Dynamic Systems. World Scientific (1996)Google Scholar
  3. 3.
    Lakshmanan, M., Rajasekar, S.: Nonlinear Dynamics: Integrability. Springer, Chaos and Patterns (2003)Google Scholar
  4. 4.
    Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer, New York (2001)Google Scholar
  5. 5.
    Gluckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer (1983)Google Scholar
  6. 6.
    Grimshaw, R.: Nonlinear Ordinary Differential Equations. CRC Press (1993)Google Scholar
  7. 7.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 2nd edn. Springer (1996)Google Scholar
  8. 8.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos with Application to Physics, Biology, Chemistry and Engineering. Perseus Books, L.L.C, Massachusetts (1994)Google Scholar
  9. 9.
    Ya Adrianova, L.: Introduction to Linear Systems of Differential Equations. American Mathematical Society (1995)Google Scholar
  10. 10.
    Krasnov, M.L.: Ordinary Differential Equations. MIR Publication, Moscow (1987). (English translation)Google Scholar
  11. 11.
    Demidovick, B.P.: Lectures on Mathematical Theory of Stability. Nauka, Moscow (1967)Google Scholar

Copyright information

© Springer India 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe University of BurdwanBardhamanIndia

Personalised recommendations