Nonlinear stochastic differential equations and 1/f noise

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Heuristic derivation of SDE If S(f ) ∼ f −β then power spectral density has a scaling property S(af ) = a −β S(f ) Wiener-Khintchine theorem C(t) = ∫ +∞ 0 S(f ) cos(2πft) df Autocorrelation function C(t) has scaling property C(at) ∼ a β−1 C(t) Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 10 / 32
Heuristic derivation of SDE If S(f ) ∼ f −β then power spectral density has a scaling property S(af ) = a −β S(f ) Wiener-Khintchine theorem C(t) = ∫ +∞ 0 S(f ) cos(2πft) df Autocorrelation function C(t) has scaling property C(at) ∼ a β−1 C(t) Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 10 / 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: 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 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 73 and 74: Herding model Transition probabilit
Page 75 and 76:
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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