Nonlinear stochastic differential equations and 1/f noise

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Heuristic derivation of SDE To get the required scaling of transition probability: SDE should contain only powers of x The diffusion coefficient should be of the form x 2η The drift term is fixed by the requirement that the steady-state PDF should be x −ν Proposed SDE dx = σ 2 (η − ν/2)x 2η−1 dt + σx η dW t B. Kaulakys and J. Ruseckas, Phys. Rev. E 70, 020101(R) (2004). B. Kaulakys and J. Ruseckas, V. Gontis, and M. Alaburda, Physica A 365, 217 (2006). Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 13 / 32
Transformation property Introducing z = x α we get the equation of the same type dz = σ ′2 (η ′ − ν ′ /2)z 2η′ −1 dt + σ ′ z η′ dW t only with different parameters σ ′ = ασ , η ′ = (η − 1)/α + 1 , ν ′ = (ν − 1)/α + 1 Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 14 / 32
Page 1 and 2: Nonlinear stochastic differential e
Page 3 and 4: Outline 1 Introduction: 1/f noise 2
Page 7 and 8: What is 1/f noise? 1/f noise a type
Page 9 and 10: 1/f noise First observed (in 1925)
Page 11 and 12: 1/f noise Many mathematical models:
Page 17 and 18: 1/f noise Goal 1/f noise is interme
Page 19 and 20: 1/f noise Goal 1/f noise is interme
Page 21 and 22: Notes about 1/f noise Often 1/f noi
Page 23 and 24: Heuristic derivation of SDE If S(f
Page 25 and 26: Heuristic derivation of SDE If S(f
Page 27 and 28: Heuristic derivation of SDE Autocor
Page 29 and 30: Heuristic derivation of SDE Autocor
Page 31 and 32: Heuristic derivation of SDE Let us
Page 33 and 34: Heuristic derivation of SDE Let us
Page 35 and 36: Heuristic derivation of SDE To get
Page 37: Heuristic derivation of SDE To get
Page 41 and 42: Restriction of diffusion Because of
Page 43 and 44: Restriction of diffusion Possible f
Page 45 and 46: Restriction of diffusion q-exponent
Page 47 and 48: Connection with other equations For
Page 49 and 50: Connection with other equations For
Page 51 and 52: Connection with other equations SDE
Page 53 and 54: Intermittent behavior of solutions
Page 57 and 58: Point processes The signal of the m
Page 59 and 60: Point processes Let us assume that
Page 61 and 62: Point processes Let us assume that
Page 63 and 64: Point processes Example: equation d
Page 65 and 66: Point processes General case τ k+1
Page 67 and 68: Herding model Simple model describi
Page 69 and 70: Herding model Simple model describi
Page 71 and 72: Herding model Transition probabilit
Page 77 and 78: Herding model Ratio of the number o
Page 79 and 80: Herding model Ratio of the number o
Page 81 and 82: Herding model 10 2 10 -1 10 0 10 1
Page 83 and 84: Summary We obtain a class of nonlin
Page 85 and 86: Summary We obtain a class of nonlin
Page 87: Thank you for your attention! Juliu

Nonlinear stochastic differential equations and 1/f noise

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