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Nonlinear Dynamics

, Volume 85, Issue 1, pp 439–452 | Cite as

Stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator

  • Pankaj Kumar
  • S. Narayanan
  • Sayan Gupta
Original Paper

Abstract

The stochastic bifurcations in a vibro-impact Duffing–Van der Pol oscillator, subjected to white noise excitations, are investigated. Bifurcations in noisy systems occur either due to topological changes in the phase space—known as D-bifurcations—or due to topological changes associated with the stochastic attractors—known as P-bifurcations. In either case, the singularities in the phase space near the grazing orbits due to impact lead to inherent difficulties in bifurcation analysis. Loss of dynamic stability—or D-bifurcations—is analyzed through computation of the largest Lyapunov exponent using the Nordmark–Poincare mapping that enables bypassing the problems associated with discontinuities. For P-bifurcation analysis, the steady-state solution of the Fokker–Planck equation is computed after applying suitable non-smooth coordinate transformations and mapping the problem into a continuous domain. A quantitative measure for P-bifurcations has been carried out using a newly developed measure based on Shannon entropy. A comparison of the stability domains obtained from P-bifurcation and D-bifurcation analyses is presented which reveals that these bifurcations need not occur in same regimes.

Keywords

Vibro-impact Duffing–Van der Pol oscillator Zhuravlev–Ivanov transformation Nordmark–Poincare mapping Shannon entropy Stochastic bifurcation 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Technology MadrasChennaiIndia
  2. 2.Indian Institute of Information Technology (Design and Manufacturing) KancheepuramChennaiIndia
  3. 3.Department of Applied MechanicsIndian Institute of Technology MadrasChennaiIndia

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