Nonlinear stochastic differential equations and 1/f noise

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Point processes Example: equation dx = σ 2 x 4 dt + σx 5/2 dW leads to τ k+1 = τ k + σε k We obtained a simple random walk of inter-event time One of possible origins of 1/f noise Brownian motion in time axis leads to 1/f noise Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 25 / 32
Point processes General case τ k+1 = τ k + γτ 2µ−1 k where µ = 5/2 − η, γ = σ 2 (1 − η + ν/2). + στ µ k ε k Used for modeling of the internote interval sequences of the musical rhythms D. J. Levitin, P. Chordia, and V. Menon, Proc. Natl. Acad. Sci. U.S.A. 109, 3716 (2012). Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 26 / 32
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Nonlinear stochastic differential e
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Outline 1 Introduction: 1/f noise 2
Page 5 and 6:
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)
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1/f noise Many mathematical models:
Page 13 and 14: 1/f noise Many mathematical models:
Page 15 and 16: 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 and 38: Heuristic derivation of SDE To get
Page 39 and 40: Transformation property Introducing
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 55 and 56: Outline 1 Introduction: 1/f noise 2
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: Point processes Example: equation d
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
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Nonlinear stochastic differential equations and 1/f noise

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