Applications of Mathematics

, Volume 61, Issue 1, pp 79–102 | Cite as

Response of a class of mechanical oscillators described by a novel system of differential-algebraic equations

  • Josef Málek
  • Kumbakonam R. Rajagopal
  • Petra Suková


We study the vibration of lumped parameter systems whose constituents are described through novel constitutive relations, namely implicit relations between the forces acting on the system and appropriate kinematical variables such as the displacement and velocity of the constituent. In the classical approach constitutive expressions are provided for the force in terms of appropriate kinematical variables, which when substituted into the balance of linear momentum leads to a single governing ordinary differential equation for the system as a whole. However, in the case considered we obtain a system of equations: the balance of linear momentum, and the implicit constitutive relation for each constituent, that has to be solved simultaneously. From the mathematical perspective, we have to deal with a differential-algebraic system. We study the vibration of several specific systems using standard techniques such as Poincaré’s surface of section, bifurcation diagrams, and Lyapunov exponents. We also perform recurrence analysis on the trajectories obtained.


chaos differential-algebraic system Poincaré’s sections recurrence analysis; bifurcation diagram implicit constitutive relations Duffing oscillator Bingham dashpot rigid-elastic spring 

MSC 2010

34C28 70K55 34A09 


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Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  • Josef Málek
    • 1
  • Kumbakonam R. Rajagopal
    • 3
  • Petra Suková
    • 2
  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic
  2. 2.Center for Theoretical PhysicsPolish Academy of SciencesWarsawPoland
  3. 3.Texas A&M UniversityCollege StationUSA

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