Bifurcations of Random Dynamical Systems

Bifurcations of random dynamical systems

or to be more speciﬁc: Additive noise does not destroy a pitchfork bifurcation

Martin Rasmussen

Department of Mathematics Imperial College London United Kingdom supported by an EPSRC Career Acceleration Fellowship (joint work with Mark Callaway, Doan Thai Son, Jeroen Lamb and Christian Rodrigues)

ICDEA 2013 Sultan Qaboos University, Muscat, Oman 26–30 May 2013

29 May 2013 Deterministic pitchfork bifurcation

α We consider for α ∈ [0, ∞) the diﬀe- x rence equation 1 1 x = α arctan(x ) + x , n+1 2 n 2 n which is a prototype for a deterministic pitchfork bifurcation.

• For α ≤ 1, there is only one equilibrium (x = 0), which is attractive. • For α > 1, x = 0 becomes repulsive, and there are two additional equilibria, which are attractive. Two possible answers NO: H. Crauel and F. Flandoli Additive noise destroys a pitchfork bifurcation, Journal of Dynamics and Diﬀerential Equations 10 (1998), 259–274.

YES: M. Callaway, T.S. Doan, J.S.W. Lamb, and M. Rasmussen The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, submitted.

Main question

Main question Does this bifurcation persist under bounded additive noise, i.e. for the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ , n+1 2 n 2 n n

where σ > 0, and (ξn)n∈Z is a sequence with values in [−1, 1]. Main question

Main question Does this bifurcation persist under bounded additive noise, i.e. for the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ , n+1 2 n 2 n n

where σ > 0, and (ξn)n∈Z is a sequence with values in [−1, 1].

Two possible answers NO: H. Crauel and F. Flandoli Additive noise destroys a pitchfork bifurcation, Journal of Dynamics and Diﬀerential Equations 10 (1998), 259–274.

YES: M. Callaway, T.S. Doan, J.S.W. Lamb, and M. Rasmussen The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, submitted. P-bifurcation: drawbacks (i) Problematic deﬁnition. (ii) Static concept, i.e. not directly linked to stability properties. (iii) Applies only to Markovian noise.

Random bifurcation theory: historical remarks

Consider a Markov semigroup on Rd with transition probabilities T (x, B). A probability measure ρ on Bd is called a stationary measure if Z ρ(B) = T (x, B) dρ(x) for all B ∈ Bd . R

P-bifurcation (phenomenological bifurcation): 1980s A P-bifurcation occurs if stationary probability densities change qualitatively, for instance, from unimodal to bimodal distributions. Main contributors: Arnold, Boxler, Namachchivaya, Schenk-Hopp´e. Random bifurcation theory: historical remarks

Consider a Markov semigroup on Rd with transition probabilities T (x, B). A probability measure ρ on Bd is called a stationary measure if Z ρ(B) = T (x, B) dρ(x) for all B ∈ Bd . R

P-bifurcation: drawbacks (i) Problematic deﬁnition. (ii) Static concept, i.e. not directly linked to stability properties. (iii) Applies only to Markovian noise. D-bifurcation: drawback The only known examples of D-bifurcations have multiplicative noise.

Random bifurcation theory: historical remarks

D-bifurcations concern the bifurcation of invariant measures of random dynamical systems. Invariant measures are canonical (random) generalisations of ﬁxed points and periodic orbits.

D-bifurcation (dynamic bifurcation): 1990s A D-bifurcation occurs if from an invariant reference measure, another invariant measure bifurcates in the sense of weak convergence. Main contributors: Arnold, Crauel, Flandoli, Imkeller, Schenk-Hopp´e. Random bifurcation theory: historical remarks

D-bifurcations concern the bifurcation of invariant measures of random dynamical systems. Invariant measures are canonical (random) generalisations of ﬁxed points and periodic orbits.

D-bifurcation: drawback The only known examples of D-bifurcations have multiplicative noise. Background on random dynamical systems Additive noise destroys a pitchfork bifurcation Additive noise does not destroy a pitchfork bifurcation (Part I) Additive noise does not destroy a pitchfork bifurcation (Part II) Contents

1 Background on random dynamical systems

2 Additive noise destroys a pitchfork bifurcation

3 Additive noise does not destroy a pitchfork bifurcation (Part I)

4 Additive noise does not destroy a pitchfork bifurcation (Part II)

Martin Rasmussen Bifurcations of random dynamical systems Background on random dynamical systems Deﬁnition Additive noise destroys a pitchfork bifurcation Generation Additive noise does not destroy a pitchfork bifurcation (Part I) Invariant measures Additive noise does not destroy a pitchfork bifurcation (Part II) Linear systems

Background on random dynamical systems

A random dynamical system is the combination of two systems (θ, ϕ):

model of noise model of dynamics

random inﬂuence

ergodic dynamical system θ cocycle over θ on Rd on a probability space (Ω, F, P)

The noise is modeled by an ergodic deterministic dynamical system. Model of noise Given a probability space (Ω, F, P), a (B ⊗ F, F)-measurable function θ : Z × Ω → Ω is called a ergodic dynamical system if the following four conditions are fulﬁlled.

(i) Initial value condition: θ0ω = ω.

(ii) Group property: θn+mω = θn(θmω). (iii) Invariance: P(θnA) = P(A). (iv) Ergodicity: θ1A = A =⇒ P(A) ∈ {0, 1}.

Model of dynamics A cocycle over the ergodic dynamical system θ : Z × Ω → Ω is d d (B ⊗ F ⊗ Bd , Bd )-measurable mapping ϕ : Z × Ω × R → R fulﬁlling (i) Initial value condition: ϕ(0, ω)x = x.

(ii) Cocycle property: ϕ(n + m, ω)x = ϕ(n, θmω)ϕ(m, ω)x.

ϕ(m,ω)x

x ϕ(n+m,ω)x =ϕ(n,θmω)ϕ(m,ω)x

θmω Ω ω θn+mω

Martin Rasmussen Bifurcations of random dynamical systems The cocycle property reads as

ϕ(n − 1, θ1ω)ϕ(1, ω)x = (hωn−1 ◦ · · · ◦ hω1 )(hω0 (x)) = ϕ(n, ω)x .

Background on random dynamical systems Deﬁnition Additive noise destroys a pitchfork bifurcation Generation Additive noise does not destroy a pitchfork bifurcation (Part I) Invariant measures Additive noise does not destroy a pitchfork bifurcation (Part II) Linear systems Generation of random dynamical systems

Consider a metric space X and two homeomorphisms h1, h2 : X → X . We want to study the random dynamics if h1 is used with probability p1 and h2 with probability p2. Generation of a discrete-time random dynamical systems (i) The ingredients for the ergodic dynamical system θ are: Ω := ω = (. . . , ω−2, ω−1, ω0, ω1, ω2,... ): ωi ∈ {1, 2} . Qn P(Ix1,...,xn ) := i=1 pxi (deﬁned on cylinder sets with xi ∈ {1, 2}). F = σ(cylinder sets). θ: left shift.

(ii) The cocycle ϕ is deﬁned by ϕ(n, ω)x := (hωn−1 ◦ · · · ◦ hω0 )(x) for n ∈ N.

(ii) The cocycle ϕ is deﬁned by ϕ(n, ω)x := (hωn−1 ◦ · · · ◦ hω0 )(x) for n ∈ N. The cocycle property reads as

ϕ(n − 1, θ1ω)ϕ(1, ω)x = (hωn−1 ◦ · · · ◦ hω1 )(hω0 (x)) = ϕ(n, ω)x .

Consider the random diﬀerence equation

xn+1 = f (xn) + σξn ,

where σ > 0, and (ξn)n∈Z is a sequence with values in [−1, 1]. We assume that ξn is distributed on [−1, 1] with the Lebesgue density ρ : [0, 1] → R+. Generation of a discrete-time random dynamical systems (i) The ingredients for the ergodic dynamical system θ are: Ω := ω = (. . . , ω−2, ω−1, ω0, ω1, ω2,... ): ωi ∈ [−1, 1] . Qn R yi (Ix ,y ,...,x ,y ) := ρ(r) dr. P 1 1 n n i=1 xi F = σ(cylinder sets). θ: left shift.

(ii) The cocycle ϕ is deﬁned by ϕ(1, ω)x := f (x) + σω0.

For a given random dynamical system (θ, ϕ), let Θ: Z × Ω × Rd → Ω × Rd denote the corresponding skew product ﬂow, given by Θ(n, ω, x) := (θnω, ϕ(n, ω)x) . This is a measurable dynamical system on the extended phase space Ω × X . Deﬁnition (Invariant measure of a random dynamical system) d A probability measure µ on (Ω × R , F ⊗ Bd ) is said to be an invariant measure of the random dynamical system (θ, ϕ) if

(i) µ(ΘnA) = µ(A) for all n ∈ Z and A ∈ F ⊗ Bd , (ii) πΩµ = P, where πΩµ denotes the marginal of µ on (Ω, F).

An invariant measure of a random dynamical system (θ, ϕ) on Ω × Rd can equivalently be represented by a family of measures on Rd . Theorem (Disintegration of invariant measures) An invariant measure µ of a random dynamical system (θ, ϕ) on Ω × Rd admits a P-almost surely unique disintegration, that is a family of probability measures (µω)ω∈Ω with Z Z µ(A) = 1A(ω, x) dµω(x) dP(ω) . Ω X

Remark This theorem extends to random dynamical systems with a Polish state space (i.e. a separable and complete metric space).

We cannot expect equilibria in the classical sense for random dynamical systems. The following deﬁnition, however, generalises this concept to random dynamical systems.

Definition (Random fixed point) A measurable mapping a :Ω → Rd is called a random fixed point for a random dynamical system (θ, ϕ) if

ϕ(n, ω)a(ω) = a(θnω) for all n ∈ Z and ω ∈ Ω .

Remark A random ﬁxed point a :Ω → Rd corresponds to an invariant measure of the random dynamical system the disintegration of which are families of Dirac measures concentrated on the values a:

µω = δa(ω) for all ω ∈ Ω .

A random dynamical system (θ, ϕ) is called linear if for given α, β ∈ R, we have ϕ(n, ω)(αx + βy) = αϕ(n, ω)x + βϕ(n, ω)y

for all n ∈ Z, ω ∈ Ω and x, y ∈ Rd . Given a linear random dynamical system (θ, ϕ), there exists a corresponding matrix-valued function Φ: Z × Ω → Rd×d with Φ(n, ω)x = ϕ(n, ω)x. Deﬁnition (Lyapunov exponent) Given a linear random dynamical system (θ, Φ) on Rd , a Lyapunov exponent is given by

1 kΦ(n, ω)xk d λ = lim ln for some ω ∈ Ω and 0 6= x ∈ R . n→∞ n kxk

Theorem (Multiplicative Ergodic Theorem) Consider a linear random dynamical system (θ, Φ) that fulﬁlls the integrability condition

+ 1 ln max kΦ(1, ·)k, kΦ(−1, ·)k ∈ L (Ω, P) ,

where ln+(x) := max{0, ln(x)}. Then almost surely, there exist at most d Lyapunov exponents λ1 < λ2 < ··· < λp and ﬁber-wise decomposition

d R = O1(ω) ⊕ O2(ω) ⊕ · · · ⊕ Op(ω) for almost all ω ∈ Ω

d into Oseledets subspaces Oi (ω) ⊂ R such that for all i ∈ {1,..., p} and almost all ω ∈ Ω, one has 1 kΦ(n, ω)xk lim ln = λi for all 0 6= x ∈ Oi (ω) . n→∞ n kxk

Martin Rasmussen Bifurcations of random dynamical systems Background on random dynamical systems Additive noise destroys a pitchfork bifurcation Additive noise does not destroy a pitchfork bifurcation (Part I) Additive noise does not destroy a pitchfork bifurcation (Part II)

Additive noise destroys a pitchfork bifurcation

Martin Rasmussen Bifurcations of random dynamical systems Random case (σ > 0)

There exists an α0 > 1 such that for all α ∈ [0, α0], there exists exactly one invariant Dirac measure, concentrated on a random ﬁxed point {aα(ω)}ω∈Ω.

{aα(ω)}ω∈Ω has strictly negative Lyapunov exponent ⇒ no D-bifurcation

Background on random dynamical systems Additive noise destroys a pitchfork bifurcation Additive noise does not destroy a pitchfork bifurcation (Part I) Additive noise does not destroy a pitchfork bifurcation (Part II) Additive noise destroys a pitchfork bifurcation

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

Deterministic case (σ = 0) (i) For α ≤ 1, there exists exactly one invariant Dirac measure. (ii) For α > 1, there are exactly three invariant Dirac measures.

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

Deterministic case (σ = 0) (i) For α ≤ 1, there exists exactly one invariant Dirac measure. (ii) For α > 1, there are exactly three invariant Dirac measures.

Random case (σ > 0)

There exists an α0 > 1 such that for all α ∈ [0, α0], there exists exactly one invariant Dirac measure, concentrated on a random ﬁxed point {aα(ω)}ω∈Ω.

{aα(ω)}ω∈Ω has strictly negative Lyapunov exponent ⇒ no D-bifurcation

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

x aα(Ω) =support of stationary density

α = 1 α = α0

Martin Rasmussen Bifurcations of random dynamical systems Background on random dynamical systems Qualitative changes Additive noise destroys a pitchfork bifurcation Dichotomy spectrum Additive noise does not destroy a pitchfork bifurcation (Part I) Relationship to ﬁnite-time Lyapunov spectrum Additive noise does not destroy a pitchfork bifurcation (Part II)

Additive noise does not destroy a pitchfork bifurcation (Part I)

Martin Rasmussen Bifurcations of random dynamical systems In fact, there exists a λ < 0 such that

−λn |ϕ(n, ω)(aα(ω)+x)−aα(θt ω)| ≤ K(ω)e |x| for all n ∈ N0 and x ∈ R , where K(ω) ≤ K¯ < ∞ if and only if α < 1.

Background on random dynamical systems Qualitative changes Additive noise destroys a pitchfork bifurcation Dichotomy spectrum Additive noise does not destroy a pitchfork bifurcation (Part I) Relationship to ﬁnite-time Lyapunov spectrum Additive noise does not destroy a pitchfork bifurcation (Part II) Qualitative change in uniform local attractivity

The unique random attracting ﬁxed point {aα(ω)}ω∈Ω of 1 1 xn+1 = 2 α arctan(xn) + 2 xn + σξn is called locally uniformly attractive if there exists a δ > 0 such that

lim sup ess sup |ϕ(n, ω)(aα(ω) + x) − aα(θt ω)| = 0 . n→∞ x∈(−δ,δ) ω∈Ω

Theorem (Qualitative change in uniform local attractivity)

(i) For α < 1, the random ﬁxed point {aα(ω)}ω∈Ω is locally uniformly attractive (even globally uniformly exponential attractive).

(ii) For α > 1, the random ﬁxed point {aα(ω)}ω∈Ω is not locally uniformly attractive.

The unique random attracting ﬁxed point {aα(ω)}ω∈Ω of 1 1 xn+1 = 2 α arctan(xn) + 2 xn + σξn is called locally uniformly attractive if there exists a δ > 0 such that

lim sup ess sup |ϕ(n, ω)(aα(ω) + x) − aα(θt ω)| = 0 . n→∞ x∈(−δ,δ) ω∈Ω

Theorem (Qualitative change in uniform local attractivity)

(i) For α < 1, the random ﬁxed point {aα(ω)}ω∈Ω is locally uniformly attractive (even globally uniformly exponential attractive).

(ii) For α > 1, the random ﬁxed point {aα(ω)}ω∈Ω is not locally uniformly attractive.

In fact, there exists a λ < 0 such that

−λn |ϕ(n, ω)(aα(ω)+x)−aα(θt ω)| ≤ K(ω)e |x| for all n ∈ N0 and x ∈ R , where K(ω) ≤ K¯ < ∞ if and only if α < 1.

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

α = 1 α = α0

Martin Rasmussen Bifurcations of random dynamical systems It implies that after the bifurcation, we observe a positive Lyapunov exponent with positive probability.

This loss of uniform attractivity has immediate practical consequences (here practical means that we ﬁx a ﬁnite time interval I = [0, T ]).

It implies that after the bifurcation, we observe a positive Lyapunov exponent with positive probability.

Let {aα(ω)}ω∈Ω, α ∈ (0, α0), denote the unique random attracting ﬁxed point of 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

Definition (Finite-time Lyapunov exponent) Given N ∈ N, the finite-time Lyapunov exponent associated to the invariant measure aα(ω) is defined by

N 1 ∂ϕα λ (ω) := ln (N, ω, aα(ω)) . α N ∂x

Note that this is a random variable! Clearly, the (classical) Lyapunov ∞ exponent λα associated with the random ﬁxed point aα(ω) is given by

∞ N λα = lim λα (ω) . N→∞

Martin Rasmussen Bifurcations of random dynamical systems This bifurcation does not manifest itself in a qualitative change of the Lyapunov spectrum.

Theorem (Qualitative change in finite-time attractivity) Consider the random difference equation 1 1 xn+1 = 2 α arctan(xn) + 2 xn + σξn on the finite time interval [0, N].

Before the bifurcation (α < 1) After the bifurcation (α > 1) The random fixed point The random fixed point {aα(ω)}ω∈Ω is finite-time {aα(ω)}ω∈Ω is not finite-time attractive, i.e. attractive, i.e.

N N λα (ω) < 0 for all ω ∈ Ω . P ω ∈ Ω: λα (ω) > 0 > 0 .

Theorem (Qualitative change in finite-time attractivity) Consider the random difference equation 1 1 xn+1 = 2 α arctan(xn) + 2 xn + σξn on the finite time interval [0, N].

N N λα (ω) < 0 for all ω ∈ Ω . P ω ∈ Ω: λα (ω) > 0 > 0 .

This bifurcation does not manifest itself in a qualitative change of the Lyapunov spectrum.

Dichotomy (or Sacker–Sell) spectrum: central spectral concept for nonautonomous dynamical systems

Dichotomy spectrum: historical remarks Compact base: R.J. Sacker & G.R. Sell (1978). Non-compact base: B. Aulbach & S. Siegmund and A. Ben Artzi & I. Gohberg (1990s). Alternative approach: M. Rasmussen (this afternoon).

We have combined these studies with ergodic properties of the base ﬂow to obtain the dichotomy spectrum for random dynamical systems.

Deﬁnition (Invariant projector) Let (θ, Φ) be a linear random dynamical system on Rd . A measurable function P :Ω → Rd×d is called an invariant projector if P2(ω) = P(ω) for all ω ∈ Ω ,

Φ(n, ω)P(ω) = P(θnω)Φ(n, ω) for all n ∈ N and ω ∈ Ω

Properties of invariant projectors

(i) The ranges {R(P(ω))}ω∈Ω and the null spaces {N (P(ω))}ω∈Ω of an invariant projector P are invariant in the following sense:

Φ(n, ω)R(P(ω)) = R(P(θnω)) and Φ(n, ω)N (P(ω)) = N (P(θnω)) .

(ii) rk P(ω) is almost surely constant.

The concept of an exponential dichotomy is a generalisation of hyperbolicity.

Deﬁnition (Exponential dichotomy) Let (θ, Φ) be a linear random dynamical system, and let γ ∈ R and P :Ω → Rd×d be an invariant projector of (θ, ϕ). Then (θ, ϕ) is said to admit an exponential dichotomy with growth rate γ ∈ R, constants α > 0, K ≥ 1 if for almost all ω ∈ Ω, one has

kΦ(n, ω)P(ω)k ≤ Ke(γ−α)n for all n ≥ 0 , kΦ(n, ω)(1 − P(ω))k ≤ Ke(γ+α)n for all n ≤ 0 .

Deﬁnition (Dichotomy spectrum) The dichotomy spectrum of a linear random dynamical system is deﬁned by [ Σ := R \ {γ} ∪{−∞} ∪ {∞} . ED with growth rate γ | {z } possibly

Remark This deﬁnition does not require the integrability condition of the Multiplicative Ergodic Theorem. If this condition is fullﬁlled, then the dichotomy spectrum contains all Lyapunov exponents.

Theorem (Spectral Theorem) The dichotomy spectrum is given by

Σ = [a1, b1] ∪ · · · ∪ [an, bn] ,

where −∞ ≤ a1 ≤ b1 < a2 ≤ · · · < an ≤ bn ≤ ∞ and 1 ≤ n ≤ N, and for each spectral interval [ai , bi ], there exists a family of linear subspaces {Si (ω)}ω∈Ω with

d S1(ω) ⊕ · · · ⊕ Sn(ω) = R for all ω ∈ Ω .

Remark If the integrability condition of the Multiplicative Ergodic Theorem is fullﬁlled, then the spaces Si (ω) are direct sums of the Oseledet spaces.

Proposition (Boundedness of the dichotomy spectrum) Consider a linear random dynamical system (θ, Φ), let Σ denote the dichotomy spectrum of (θ, Φ), and deﬁne

Γ±(ω) := ln+ kΦ(1, ω)±1k .

Then Σ is bounded from above if and only if

ess sup Γ+(ω) < ∞ , ω∈Ω

and Σ is bounded from below if and only if

ess sup Γ−(ω) < ∞ . ω∈Ω

Let {aα(ω)}ω∈Ω, α ∈ (0, α0], denote the unique random attracting ﬁxed point of 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

Proposition (Lyapunov and dichotomy spectrum at the bifurcation point)

(i) The Lyapunov exponent associated with {aα(ω)}ω∈Ω is negative, i.e. λα < 0 for all α ∈ (0, α0) .

(ii) The dichotomy spectrum associated with {aα(ω)}ω∈Ω satisﬁes 1 1 sup Σ = α − for all α ∈ (0, α ) . α 2 2 0

Let (θ, Φ) be a linear random dynamical system on Rd with dichotomy spectrum Σ. Deﬁne the ﬁnite-time Lyapunov exponent by

1 kΦ(N, ω)xk λN (ω, x) := ln for all N ∈ , ω ∈ Ω and x ∈ d \{0} . N kxk N R

Proposition The limit relation

lim ess sup sup λN (ω, x) = sup Σ < ∞ N→∞ d ω∈Ω x∈R \{0} holds, provided that sup Σ < ∞. A similar equality holds for the lower bound of the spectrum if inf Σ > −∞.

Martin Rasmussen Bifurcations of random dynamical systems Background on random dynamical systems Set-valued dynamical systems Additive noise destroys a pitchfork bifurcation Topological bifurcations Additive noise does not destroy a pitchfork bifurcation (Part I) Period doubling in the random H´enonmap Additive noise does not destroy a pitchfork bifurcation (Part II)

Additive noise does not destroy a pitchfork bifurcation (Part II)

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

x ? α ?

α = 1 α = α0

α < 1 σ = 0

α = 1 σ = 0

α > 1 σ = 0

α > 1 σ > 0

There is no attracting random ﬁxed point near 0!

Martin Rasmussen Bifurcations of random dynamical systems Instead we look at set-valued dynamical systems, where we consider topological bifurcations.

Background on random dynamical systems Set-valued dynamical systems Additive noise destroys a pitchfork bifurcation Topological bifurcations Additive noise does not destroy a pitchfork bifurcation (Part I) Period doubling in the random H´enonmap Additive noise does not destroy a pitchfork bifurcation (Part II) Topological bifurcations of set-valued dynamical systems

We do not need the theory of random dynamical systems to understand these bifurcations!

Instead we look at set-valued dynamical systems, where we consider topological bifurcations.

x f (x)

f Inﬂation

x f (x) Bσ(f (x))

d In addition to a diﬀerence equation xn+1 = f (xn) on R , we consider iterations of the set-valued mapping F : Rd → P(Rd ), where

d F (x) := Bσ(f (x)) for all x ∈ R . Minimal invariant sets A compact set M ⊂ Rd is called invariant if F (M) = M. An invariant set is called minimal if it does not contain a proper invariant set.

f Inﬂation

x f (x) Bσ(f (x))

d In addition to a diﬀerence equation xn+1 = f (xn) on R , we consider iterations of the set-valued mapping F : Rd → P(Rd ), where

Minimal invariant sets support stationary measures (H. Zmarrou & A.J. Homburg, 2007).

Martin Rasmussen Bifurcations of random dynamical systems In general, minimal attractors persist as minimal invariant sets under small perturbation.

Proposition

Let the mapping f of the diﬀerence equation xn+1 = f (xn) be a contraction, i.e. there exists an L ∈ (0, 1) with

kf (x) − f (y)k ≤ Lkx − yk for all x, y ∈ X .

Then for all σ > 0, the mapping F is also a contraction, and the uniquely determined ﬁxed point of F is the uniquely determined minimal invariant set.

Proposition

Let the mapping f of the diﬀerence equation xn+1 = f (xn) be a contraction, i.e. there exists an L ∈ (0, 1) with

kf (x) − f (y)k ≤ Lkx − yk for all x, y ∈ X .

Then for all σ > 0, the mapping F is also a contraction, and the uniquely determined ﬁxed point of F is the uniquely determined minimal invariant set.

In general, minimal attractors persist as minimal invariant sets under small perturbation.

α < 1 σ = 0

α = 1 σ = 0

α > 1 σ = 0

α > 1 σ > 0

Martin Rasmussen Bifurcations of random dynamical systems Theorem (Types of topological bifurcations) Topological bifurcations can only occur in one of the following ways: A minimal invariant set explodes lower semi-continuously, or a minimal invariant set disappears instantaneously.

Deﬁnition (Topological bifurcation) d d Consider the set-valued mappings Fα : R → P(R ), where α is a real parameter, and let Mα denote the union of minimal invariant sets of Fα.

We say that Fα∗ admits a topological bifurcation of minimal sets if for any neighbourhood V of λ∗, there does not exist a family of d d homeomorphisms (hλ)λ∈V , hλ : R → R , depending continuously on λ, with the property that

hλ(Mλ) = Mλ∗ for all λ ∈ V .

Deﬁnition (Topological bifurcation) d d Consider the set-valued mappings Fα : R → P(R ), where α is a real parameter, and let Mα denote the union of minimal invariant sets of Fα.

hλ(Mλ) = Mλ∗ for all λ ∈ V .

Theorem (Types of topological bifurcations) Topological bifurcations can only occur in one of the following ways: A minimal invariant set explodes lower semi-continuously, or a minimal invariant set disappears instantaneously.

α = 1 α = α0

? α

α = 1 α = α0

Does the trivial repeller x = 0 for α > α0 of the deterministic equation 1 1 x = α arctan(x ) + x n+1 2 n 2 n correspond to a minimal invariant set of its random version?

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

α > 1 σ > 0

There is no minimal invariant set containing 0!

Martin Rasmussen Bifurcations of random dynamical systems An invariant set of F ∗ is not invariant for F , but it can be characterised as a set in which it is possible to stay for a particular realisation of noise.

Consider a set-valued mapping F : Rd → P(Rd ). Then the dual of F is given by the mapping F ∗ : Rd → P(Rd )

∗ d d F (x) := ξ ∈ R : x ∈ F (ξ) for all x ∈ R .

Consider a set-valued mapping F : Rd → P(Rd ). Then the dual of F is given by the mapping F ∗ : Rd → P(Rd )

∗ d d F (x) := ξ ∈ R : x ∈ F (ξ) for all x ∈ R .

An invariant set of F ∗ is not invariant for F , but it can be characterised as a set in which it is possible to stay for a particular realisation of noise.

α > 1 σ > 0

The collision of a minimal invariant set with an F ∗-invariant set is a necessary condition for a topological bifurcation.

Consider the random diﬀerence equation 1 1 x = α arctan(x ) + x + σξ . n+1 2 n 2 n n

α = 1 α = α0

Consider the H´enonmap

2 xn+1 = a + byn − xn ,

yn+1 = xn ,

1 2 and ﬁx b ∈ (0, 1). For a > − 4 (1 − b) , the ﬁxed points are given by 1 p 2 p 2 z1 = 2 −(1 − b) + (1 − b) + 4a, −(1 − b) + (1 − b) + 4a

and 1 p 2 p 2 z2 = 2 −(1 − b) − (1 − b) + 4a, −(1 − b) − (1 − b) + 4a ,

3 2 and at a = 4 (1 − b) , a period-doubling bifurcation occurs.

a = 2.03 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 2.02 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 2.01 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 2.00 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.99 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.98 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.97 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.96 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.95 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.94 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.93 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.92 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.91 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.90 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.89 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.88 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.87 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.86 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.85 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.84 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.83 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.82 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.81 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.80 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.79 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

a = 1.78 σ = 0.01 σ = 0.007 σ = 0.005

b = 0.5

We have seen that adding noise to a deterministic pitchfork bifurcation does not destroy the bifurcation, but doubles the number of bifurcations: (i) The first bifurcation is associated to a qualitative change in the uniform and finite-time attractivity of a random fixed point. (ii) The second bifurcation is associated to a topological bifurcation of the corresponding set-valued dynamical system. These observations are based on a case study, but we have developed a general theory around this study (e.g. dichotomy spectrum, nature and characterisation of topological bifurcations). Several open questions are urgent: (i) How can we compute the dichotomy spectrum? (i) What is a good indicator for an upcoming topological bifurcation? (ii) How can we efficiently approximate minimal invariant sets (e.g. by using recent results by J. Rieger)?

Martin Rasmussen Bifurcations of random dynamical systems Thanks for your attention!

References • L. Arnold, Random Dynamical Systems, Springer, 1998. • M. Callaway, T.S. Doan, J.S.W. Lamb, and M. Rasmussen, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, submitted. • H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, Journal of Dynamics and Differential Equations 10 (1998), 259–274. • J.S.W. Lamb, M. Rasmussen, and C.S. Rodrigues, Topological bifurcations of set-valued dynamical systems, submitted. • H. Zmarrou and A.J. Homburg, Bifurcations of stationary measures of random diffeomorphisms, Ergodic Theory and Dynamical Systems 27 (2007), 1651–1692. References • L. Arnold, Random Dynamical Systems, Springer, 1998. • M. Callaway, T.S. Doan, J.S.W. Lamb, and M. Rasmussen, The dichotomy spectrum for random dynamical systems and pitchfork bifurcations with additive noise, submitted. • H. Crauel and F. Flandoli, Additive noise destroys a pitchfork bifurcation, Journal of Dynamics and Differential Equations 10 (1998), 259–274. • J.S.W. Lamb, M. Rasmussen, and C.S. Rodrigues, Topological bifurcations of set-valued dynamical systems, submitted. • H. Zmarrou and A.J. Homburg, Bifurcations of stationary measures of random diffeomorphisms, Ergodic Theory and Dynamical Systems 27 (2007), 1651–1692. Thanks for your attention!