Nonlinear stochastic differential equations and 1/f noise

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Intermittent behavior of solutions 60 50 40 I(t) 30 20 10 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t Signals generated by proposed SDE exhibit intemittent behavior: there are bursts corresponding to large deviations, separated by laminar phases. Bursts are characterized by power-law distributions of burst size, burst duration, and interburst time. Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 21 / 32
Outline 1 Introduction: 1/f noise 2 Stochastic differential equations giving 1/f noise 3 Some models resulting in proposed SDE Point processes Simple model of herding behavior 4 Summary Julius Ruseckas (Lithuania) Nonlinear stochastic differential equations August 28, 2012 22 / 32
Page 1 and 2:
Nonlinear stochastic differential e
Page 3 and 4: 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)
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 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: 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
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Nonlinear stochastic differential equations and 1/f noise

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