Nonlinear Dynamics

, Volume 91, Issue 4, pp 2429–2434 | Cite as

Shadowing in hidden attractors

  • N. K. Kamal
  • V. Varshney
  • M. D. ShrimaliEmail author
  • A. Prasad
  • N. V. Kuznetsov
  • G. A. Leonov
Original Paper


Hidden attractors found in physical systems are different from self-exited attractors and may have a small basin of attraction. The issue of shadowing in these attractors using dynamical noise is discussed. We have particularly considered two classes of dynamical systems which have hidden attractors in their state space. In one of the systems, there is no fixed point but only a hidden attractor in the state space, while in the other, the system has one unstable fixed point along with a hidden attractor in the state space. The effect of dynamical noise on these dynamical systems is studied by using the Hausdorff distance between the noisy and deterministic attractors. It appears that, up to some threshold value of noise, the noisy trajectory completely shadows the noiseless trajectory in these attractors which is quite different from the results of self-exited attractors. We compare the results of hidden chaotic attractors with the self-exited chaotic attractors.


Dynamical system Hidden attractor Shadowing 



Authors acknowledge the support from DST-RFBR for joint Indo–Russian collaborative research project (INT/RUS/RFBR/P-230 and 16-51-45002).


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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • N. K. Kamal
    • 1
  • V. Varshney
    • 2
  • M. D. Shrimali
    • 1
    Email author
  • A. Prasad
    • 2
  • N. V. Kuznetsov
    • 3
    • 4
  • G. A. Leonov
    • 3
  1. 1.Department of PhysicsCentral University of RajasthanAjmerIndia
  2. 2.Department of Physics and AstrophysicsDelhi UniversityDelhiIndia
  3. 3.Department of Applied CyberneticsSaint Petersburg State UniversitySaint PetersburgRussia
  4. 4.Department of Mathematical Information TechnologyUniversity of JyväskyläJyväskyläFinland

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