Nonlinear stochastic differential equations and 1/f noise

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Summary We obtain a class of nonlinear SDEs giving the power-law behavior of the power spectral density in any desirably wide range of frequencies and power-law steady state distribution of the signal intensity. In special cases we obtain other well-known SDEs. One of the reasons for the appearance of the 1/f spectrum are scaling properties of the SDE. Proposed SDEs can be obtained from point processes with Brownian motion of inter-event time a simple agent model describing a herding behavior. Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 31 / 32
Summary We obtain a class of nonlinear SDEs giving the power-law behavior of the power spectral density in any desirably wide range of frequencies and power-law steady state distribution of the signal intensity. In special cases we obtain other well-known SDEs. One of the reasons for the appearance of the 1/f spectrum are scaling properties of the SDE. Proposed SDEs can be obtained from point processes with Brownian motion of inter-event time a simple agent model describing a herding behavior. Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 31 / 32
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Nonlinear stochastic differential e
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Outline 1 Introduction: 1/f noise 2
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Outline 1 Introduction: 1/f noise 2
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What is 1/f noise? 1/f noise a type
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1/f noise First observed (in 1925)
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1/f noise Many mathematical models:
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1/f noise Goal 1/f noise is interme
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1/f noise Goal 1/f noise is interme
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Notes about 1/f noise Often 1/f noi
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Heuristic derivation of SDE If S(f
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Heuristic derivation of SDE If S(f
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Heuristic derivation of SDE Autocor
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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 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: Herding model 10 2 10 -1 10 0 10 1
Page 85 and 86: Summary We obtain a class of nonlin
Page 87: Thank you for your attention! Juliu
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