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Example: bifurcation of period doubling in the nonlinear map

Detection of forecoming bifurcations in plasma on the basis of prebifurcation noise enhancement phenomenon (project 2.1) Research group: Prof. S. Berczynski, Szczecin University of Technology Prof.. Yu.A.Kravtsov, Maritime University of Szczecin

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Example: bifurcation of period doubling in the nonlinear map

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  1. Detection of forecoming bifurcations in plasma on the basis of prebifurcation noise enhancement phenomenon (project 2.1) Research group: Prof. S. Berczynski, Szczecin University of Technology Prof.. Yu.A.Kravtsov, Maritime University of Szczecin Dr. habil. Elena Surovyatkina, Space Research Institute, Russ. Acad.Sci., Moscow Potential partners: Ph. Ghendrih and X. Garbet,, France, CEA and L. Villard, Switzerland.

  2. The concept of the NOISE PRECURSOR for forecomig bifurcations was introduced first by Wiesenfeld (1996). The linear theory of Wiesenfeld predicts the infinite growthof fluctuations near bifurcation threshold. Nonlinear theory was developed by the joint efforts of MUS, SUT and Space Research Institute (Moscow). Nonlinear theory implies SATURATION FOR FLUCTUATIONS GROWTH near the bifurcation threshold and thereby makes the problem of bifurcation prediction more realistic.

  3. Example: bifurcation of period doubling in the nonlinear map Map: Bifurcation Diagram T=1 T=2

  4. Linear teoria by Wiesenfel (unlimited growth of fluctuations) Nonlinear saturation Bifurcation diagram x Rysunek  Control parameter

  5. Bifurcation of the symmetry breaking in the nonlinear oscillator • Equation for nonlinear oscillator: B<0 B>0 albo

  6. Bifurcations, which take place in a hot plasma (information from Dr. L. Villard, Switzerland.) • ELM sequences, • L-H transitions, • NTM hysteresis, • Equilibrium bifurcations. Practical problem: lack of funds for trips to the Euratom Research Centers to initiate the joint studies of the experimental materials on bifurcations. (Fresh example: Invitation from Dr.Ph. Ghendrih, Cardache, to participate in „Pedestal” conference in France, 4-6 April 2006)

  7. Publications • 1. Kravtsov Yu.A., Belchinskaia S.G., Butkovskii O.Ya., Rychka I.A., Surovyatkina E.D. Prebifurcational noise rise in nonlinear systems. JETP, 2001, 93(6), 1323-1329. • 2. Yu.A. Kravtsov, E.D.Surovyatkina. Nonlinear saturation of prebifurcation noise amplification. Phys. Lett. A, 2003, 319(3-4), 348-351. • 3. E.D.Surovyatkina. Rise and saturation of the correlation time near bifurcation threshold. Phys.Lett. A, 2004, 323(2), 128-134. • 4. Yu.A.Kravtsov, S.G.Bilchinskaia, I.A.Rychka, E.D.Surovyatkina. Phenomenon of prebifurcation noise enhancement as diagnostic instrument for plasma systems, influenced by external noise. Plasma-2003, CBK PAN, Warszawa, 8-12 Sept. 2003 • 5. S.Berczyński, Yu.A.Kravtsov, E.D.Surovyatkina. Fluctuations enhancement ar a precursor of bifurcations in nonlinear systems. VIII School of Modal Analysis, AGH, Kraków, 1-2 Dec. 2003, pp. 7-12. • 6.Surovyatkina E.D., Kravtsov Yu.A., Kurths J. Fluctuation growth and saturation in nonlinear oscillator on the threshold of bifurcation. Phys. Rev. E, 2005, 72(4), 046125, pp.1-7.

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