Local Lyapunov Exponents: Sublimiting Growth Rates of Linear by Wolfgang Siegert

By Wolfgang Siegert

Establishing a brand new inspiration of neighborhood Lyapunov exponents the writer brings jointly separate theories, particularly Lyapunov exponents and the speculation of enormous deviations.

Specifically, a linear differential procedure is taken into account that's managed by means of a stochastic technique that in an appropriate noise-intensity-dependent time is trapped close to one among its so-called metastable states. The neighborhood Lyapunov exponent is then brought because the exponential development price of the linear approach in this time scale. not like classical Lyapunov exponents, which contain a restrict as time raises to infinity in a hard and fast approach, the following the process itself adjustments because the noise depth converges, too.

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Extra resources for Local Lyapunov Exponents: Sublimiting Growth Rates of Linear Random Differential Equations (Lecture Notes in Mathematics)

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E2 ............. r z s |z | sin α α | cos α | √ =: v r e Fig. 1 Geometrical interpretation of the quantity vt := (cos αt )2 where αt denotes the angle of zt as measured canonically with respect to the e1 -coordinate axis. 1 illustrates, vt = cos2 αt quantifies the distance between |zztt | and e2 and hence induces a metric on the projective space S 1 . Another popular choice to obtain a metric on S 1 is to work with | sin αt | which measures the distance between the projective lines |zztt | and e1 ; the latter metric is commonly used in the proof of the Multiplicative Ergodic Theorem; see Arnold [Ar 98, Prop.

The invariant set { x ∈ R2 : |x| = 1 } violates the stability criterion (K2) and hence constitutes the saddle for the radial motion. 1 Preliminaries and assumptions 59 −x2 + x1 η − (x21 + x22 ) x1 + x2 η − (x21 + x22 ) bη (x) := which represents the normal form of the Hopf-bifurcation, where η ∈ R denotes the bifurcation parameter; cf. ], Andronov et al. 158;52,226]; a slight generalization of this system is considered by Leng et al. 3]. Also in this case the drift decomposes orthogonally as bη = −∇Uη + L , where Uη (x) := 1 4 η |x| − |x|2 4 2 and L(x) := −x2 x1 ; in polar coordinates this ODE can be rewritten as ˙=− α˙ = 1 3 + η ≡ − Uη ( ) = (η − 2 ) ; again Uη (x) = Uη ( ) only depends on the absolute value of the position, and bη satisfies (K) with the only stable set K(η) := {0} , η≤0 √ 2 { x ∈ R : |x| = η } , η > 0 .

G. Markus [Mar 56] and Strauss and Yorke [Stra-Yor 67]. 5 Sample systems In this section several examples are presented which shall illustrate where linear, real noise driven stochastic systems (1) appear; these sample systems can be regarded as toy models for the different situations described by the definitions and assumptions in chapter 4. 1 (Linearized SDEs with constant noise coefficient σ). Let X ε be the diffusion given by the SDE √ dXtε = b (Xtε ) dt + ε σ dWt , where b ∈ C ∞ (Rd , Rd ), ε ≥ 0, and the Brownian motion W are as before and where σ ∈ GL(d, R) is now supposed to be a constant (invertible) matrix.

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