Nonlinear stochastic differential equations and 1/f noise

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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 2 / 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 2 / 32
Page 1: Nonlinear stochastic differential e
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 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
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Point processes Let us assume that
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Point processes Example: equation d
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Point processes General case τ k+1
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Herding model Simple model describi
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Herding model Simple model describi
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Herding model Transition probabilit
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Herding model Ratio of the number o
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Herding model Ratio of the number o
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Herding model 10 2 10 -1 10 0 10 1
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Summary We obtain a class of nonlin
Page 85 and 86:
Summary We obtain a class of nonlin
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Thank you for your attention! Juliu
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Nonlinear stochastic differential equations and 1/f noise

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