Delay-induced switched states: A balancing act

Figure 6.4.: Numerical computation of the driftd^up =δ^up/εof up (blue) andd^down =δ^down/ε of down (green) transitions layers for fixedε= 0.01,f₂(x) = 2x/(1 + (0.8´x)⁴), and initial function x(t) = x0(t), t P [´1,0] with two sign changes (specified in the text). Panel (a)-(d) show the spatiotemporal plot of the solution to (6.19) for forT = 1 + 0.755εand different values ofy(0). Panel (e) showsd^up andd^down with varyingy(0)P[0,1].

6.5. Delay-induced switched states: A balancing act

According to Sec. 6.1, all solutions of (6.1)–(6.2) forηą0that are not converging to or identically zero are oscillating, i.e. its first component possesses infinitely many sign changes for t ą 0 (Props. 6.1 and 6.2). Consider the following definition motivated by the spatiotemporal plot in Fig. 6.4(c).

Definition (delay induced switched state). A solution(x, y)PC([´1,8),R²)of (6.1)(6.2) as a delay-induced switched state, if there exist1ďT ď2and 0ďsď1, such that x(t)has a constant even number of sign changes on each interval [nT +s,(n+ 1)T +s]for all ně0.

Additionally, refer to δ = T ´1, as the drift of a delay induced switched state [GMZY12, GMZY13, YG17]. It is easy to show thatδÑ0asεÑ0. In a neighborhood of the balance point βof system (6.19), it is possible to obtain delay-induced switched states for (6.1)-(6.2). This can be explained as follows: Ify(t)ăβ, the coarsening of the solution leads to an increase of the average value _T¹ ş0

´T x(t+s)dsofx(t)and therefore, to an increase ofy(t) =y(t´T) +ηş0

´T x(t+s)ds towards the valueβ. Analogously, fory(t)ąβ, the average _T¹ ş0

´T x(t+s)dsdecreases as well as y(t). This allows for the “dynamic stabilization” of the delay-induced switched states.

Figure 6.5 shows numerical study of the up- and down-drifts for thex-component of (6.1)-(6.2) for different values ofηandy(0). As the spatiotemporal plots in Figs. 6.5(a,b,c1,c2,c3,d) show, the coarsening may still occur for certain parameters as in Figs. 6.5(a,b,d), while switched states are observed for the other parameters as in Figs. 6.5(c1,c2,c3). The conditions, where switched states

exist, i.e. where the up- and down-drifts coincideδ⁺ = δ^´ = δ, are shown as the gray region in Figs. 6.5(e,f). The existence region has the form of a cone attached to a balance pointy(0) = β atη = 0. That is, the switched state forη = 0exists only in the balance point, and any deviation of the initial conditiony(0)fromβ would lead to the coarsening. With increasingη, the allowed range ofy(0), where switched states exist, increases. This indicates that such states become stable forη ą0, and in order to reach them, one should initiatey(0) in the neighborhood of the balance pointβ, while the size of this neighborhood grows withη.

The rigorous analysis in the following section shows that the solution of (6.1)-(6.2) can be ob-tained as a perturbation of (6.19) (Lemma 6.6) and that this perturbation contracts the distance of subsequent sign changes to a critical distance (implied by Lemma 6.7). The “largeness” of this correcting effect strongly depends onη. Too far from the balance point, it is too weak to prevent the coarsening of a phase, see Fig. 6.5(a). Similarly, the time scale of this process is of impor-tance. Closer to the balance point the influence ofηis strong enough to prevent the coarsening of one phase, possibly leading to the coarsening towards the other, see Fig. 6.5(b). Supported by the numerical evidence in Fig. 6.5 one arrives at the following conjecture.

Conjecture 6.5. Let f P C¹(R) satisfy (H1)(H3) with balance point β. Then, for all ε ą0 and η ą 0 sufficiently small, all initial conditions with y(0) sufficiently close to β, and x(θ) P [χ^´, χ⁺], for ´1 ď θ ď 0, with at least one sign changes lead to a delay-induced switched state.

Initial conditions ’close’ the balance point lead to delay-induced switched states, whereas those too far from the balance point, may appear as delay-induced switched states at first, yet coarsen after some possibly long transient time, and as a result no delay-induced switching can be observed anymore. This can be explained by a competition of the fastε´dynamics and slowη-dynamics. In order to investigate these processes independently, one can ’split off’ theη-part of the dynamics, providing a conceptional mechanism for the stabilization of delay-induced switched states; the more involved arguments being given as self-contained lemmas. Recall from Sec. 6.1 that the solution to (6.1)–(6.2) can be written in integral form

x(t) =( x(t0))

e^´(t´t0)/ε+ żt

t₀

e^´(t´s)/ε ε

(f(x(s´1))´y(t0))

ds (6.20)

´η żt

t0

e^´(t´s)/ε ε

(żs t0

x(˜s)d˜s )

ds, y¹(t) =y(t0) +η

żt t0

x(s)ds, (6.21)

which can be readily checked using the variation of constants formula. The motivation is simple, find a transformation such that the resulting solution is independent of η. A closer look at

sys-6.5. DELAY-INDUCED SWITCHED STATES: A BALANCING ACT

Figure 6.5.: Up-driftd^up=δ^up/ε (blue) and down-driftd^down =δ^down/ε (green) of solutions for fixed ε = 0.01, f₂(x) = 2x/(1 + (0.8´ x)⁴), T = 1 + 0.514ε and initial function x(t) = x₀(t), t P [´1,0] with two sign changes (specified in the text).

Panels (a),(b),(c1),(d) show spatiotemporal plots of the solution for η = 0.001 and (a) y(0) = 0.2, (b) y(0) = 0.33, (c1) y(0) = 0.514 and (d) y(0) = 0.8.

Panels (c2),(c3) show spatiotemporal plots of the solution for y(0) = 0.514 and (c2) η = 0.0003 and (c3) y(0) = 0.0001, (c1) y(0) = 0.514 and (d) y(0) = 0.8.

Values of d^up and d^down varying the feedback parameter ηP [0,0.01] and initial conditiony(0)P[0,1]. Panel (e): numerically computed parameter regions (and individual points), where d^up and d^down coincide are shown in gray. Panel (f):

projection (from above) of Panel (e) onto the (y(0), η)-plane. Colors encode:

d^up ăd^down (blue),d^up =d^down (gray),d^up ąd^down (green).

tem (6.20)-(6.21) suggests to subtract the respective last terms´ηşt t0

e^´(t´s)/ε ε

(şs

t0x(˜s)d˜s) dsand ηşt

t0x(s)ds in (6.20)-(6.21). If one assumes delay-induced switched state ψ with length T, then this transformation takes the form of a linear operatorL^ε,η_T :C_T¹ ÑC_T¹, ψ ÞÑL^ε,η_T ψ,with

[L^ε,η_T φ](θ) = φ(θ) +η





1 ε

şθ

´T e^´(θ´s)ε (şs

´T φ¹(˜s)d˜s) ds

´şθ

´T φ¹(s)ds



. (6.22)

The next Lemma collects several properties ofL^ε,η_T . It is convenient to write Eqs. (6.1)–(6.2) in vec-tor form, such that for a given an initial functionφ P C(

[´1,0],R²)

,z(t, φ) = (x(t, φ), y(t, φ)), tě0is the unique solution of the initial value problem

dz

dt(t) =A^ε,ηz(t) +F (

z(t´1))

, z0 =φ, (6.23)

withA^ε,η PR^2ˆ2andF P C¹(R²)such that

A^ε,η =

( ´¹ε ´¹ε

η 0

)

, F(ζ) =

( f(ζ) 0

) .

Consider the solution operatorM_T^ε,ηthat maps segments of delay-induced switched states of length T to the next. More specifically, consider the solution segment φn(θ) := z(nT +θ) P C_T¹ :=

C¹([´T,0],R²), nPNandθ P[´T,0],1ďT ă2. Then, (6.23) is equivalent to d

dθφn+1(θ) = A^ε,ηφn+1(θ) +

$&

% F (

φn(θ+δ))

, ´T ăθă ´δ, F (

φn+1(θ´1))

, ´δ ăθ ă0, (6.24) φ_n+1(´T) = φn(0) for alln PN, φ₀|[´1,0] =z₀ PC([´1,0],Rⁿ).

Eq. (6.24) defines a map given byM_T^ε,η :CT ÑCT, ψ ÞÑM_T^ηψ with [M_T^ε,ηψ]

(θ) = e^Aε,η(θ+T)ψ(´T) +

ż_mintθ,´δu

´T

e^{Aε,η(θ´s)}F (

ψ(s+δ))

ds (6.25)

+ żθ´1

mintθ´1,´Tu

e^{Aε,η(θ´1´s)}F (

e^Aε,η(s+T)ψ(´T) + żs

´T

e^{Aε,η(θ´˜s)}ψ(˜s+δ)d˜s )

ds,

which can be easily seen using the variations of constants formula.

Lemma 6.6. Let ε, η, T ą0, φPC_T¹ be fixed, and L^ε,η_T be defined as in (6.22). Then, (i) L^ε,η_T φ is a η-small perturbation of φ.

(ii) L^ε,η_T is one-to-one. K_T^ε,η = (L^ε,η_T )^´1 has the form (K_T^ε,ηφ)(θ) =φ(θ) +

żθ

´T

(

A^ε,ηe^{Aε,η(θ´s)}´e^{Aε,η(θ´s)}A^ε,0)

φ(s)ds. (6.26)

(iii) L^ε,η_T M_T^ε,η =M_T^ε,0, and M_T^ε,η =K_T^ε,ηM_T^ε,0.

Proof. Let φ P C_T¹ and denote φ¯ = L^ε,η_T φ. (i) One readily checks the estimates |φ¯²(θ)´ φ²(θ)|=η|şθ

´T φ¹(s)ds| ďηT|φ¹| ď2η}φ} and

|φ¯¹(θ)´φ¹(θ)|= η ε|

żθ

´T

e^´(θ´εs) (żs

´T

φ¹(˜s)d˜s )

ds| ď 2η ε }φ}

żθ

´T

e^´(θ´εs)ds ď2η}φ},

such that ››L^ε,η_T φ´φ›› ď 3η}φ} where 1ď T ă 2. Note that ( ¯φ¹)¹(θ) = (φ¹)¹(θ)´( ¯φ¹(θ)´ φ¹(θ) + ¯φ²(θ)´φ²(θ)),( ¯φ²)¹(θ) = (φ²)¹(θ) +ηφ¹(θ)such that››φ¯¹ ´φ¹››ă2››φ¯´φ››+ηT}φ} ă

6.5. DELAY-INDUCED SWITCHED STATES: A BALANCING ACT 8η}φ}. Thus, L^ε,η_T ´id is uniformly bounded by 8η. (ii) Taking the derivative of L^ε,η_T φ one observes thatφ¯PC_T¹, with

d

dθφ(θ) =¯ A^ε,0φ(θ) +¯ d

dθφ(θ)´A^ε,ηφ(θ),

and A^ε,η is defined as in Eq. (6.23). This is ODE that can be solved explicitly to recover Eq. (6.22). On the other hand, it can be rearranged as an initial value problem for φ, i.e.

d

dθφ(θ) = A^ε,ηφ(θ) + d

dθφ(θ)¯ ´A^ε,0φ(θ),¯ (6.27) φ(´T) = ¯φ(´T).

In this way φ is uniquely determined given φ¯P C_T¹ and the explicit form of K_T^ε,η is readily obtained by the variation of constants formula. (iii) Let φ₀ PC_T¹,and consider the sequence ( ¯φn)n where φ¯n =L^ε,η_T φn and φ_n+1 =M_T^ε,ηφn for all ně0. Then, it is easy to check thatφ¯n

satisfies

d

dθφ¯n(θ) = A^ε,0φ¯n(θ) +

$&

% F (

φ_n´1(θ+δ))

, ´T ăθ ă ´δ, F (φ¯n(θ´1))

, ´δăθ ă0, , φ¯n(´T) = φn´1(0),

such that L^ε,η_T M_T^ε,ηφ_n´1 = L^ε,η_T φn = M_T^ε,0φ_n´1. This is independent of φ_n´1 and therefore L^ε,η_T M_T^ε,η =M_T^ε,0. The second assertions follows immediately using (ii).

As a result, one obtains a decomposition of the solution operatorM_T^ε,η =K_T^ε,ηM_T^ε,0into two parts.

One can now use the qualitative information aboutM_T^ε,0, see Sec. 6.5. In particular, the number of sign changes of a given functionφP C_T¹ does not increase, and the amount of which sign changes drift can be read of numerically. In addition, one can show howK_T^ε,η effects the position of sign changes of a functionφ P CT along the interval[´T,0]via studying its inverseL^ε,η_T . Considerφ with the property

(SC_N) The first componentφ¹ ofφ has an even number of sign changes and no other zeros. More specifically, there existN P N, and a series of sign changes(θⁱ), 0 ď i ď 2N, ofφ¹ such thatφ¹(θⁱ) = 0,(φ¹)¹(θⁱ)‰0andφ¹has no other zeros.

The results are collected in the following Lemma; the proof of which is contained in the supple-mentary material.

Lemma 6.7 (dynamics of sign changes under L^ε,η_T ). Let ε, η, T ą 0 be fixed. Let φ P C_T¹ satisfy(SCN)and consider φ¯=L^ε,η_T φ. Then the following hold true.

(i) There existsη¯ą0, depending onφ, such that for all ηăη,¯ φ¯also satisfies(SCN). In particular, for each sign change θ of φ¹, there exists exactly one sign change θ¯of φ¯¹ with|θ´θ¯|=O(η)and φ¯¹ has no other zeros.

(ii) IfφP C_T², there existsη¯¹ ą0,depending onφ, such that for allη ăη¯¹, the sign changes θ¯of φ¯¹ are given by

θ¯=θ´ η ε(φ¹)¹(θ)

żθ

´T

e^´(θ´s)ε [żs

´T

φ¹(˜s)d˜s ]

ds+O(η²), (6.28) where θ is a sign change ofφ.

(iii) Assume as in (ii). There exist ε¯ą0such that if εăε, then for each subsequent pair¯ of sign changesθⁱ, θⁱ⁺¹, iď2N there exist unique (up to order η²) ϑⁱ ě0, depending on φ, such that if |θⁱ´θⁱ⁺¹| ăϑⁱ+O(η²) (respectively|θⁱ´θⁱ⁺¹| ąϑⁱ+O(η²)), then

|θ¯ⁱ´θ¯ⁱ⁺¹| ă |θⁱ´θⁱ⁺¹|(respectively |θ¯ⁱ´θ¯ⁱ⁺¹| ą |θⁱ´θⁱ⁺¹|). In particular, ϑⁱ ą0(up to order η²), if and only if

1 (φ¹)¹(θⁱ)



 żθⁱ

´T

e^´(θi´s)ε [żs

´T

φ¹(˜s)d˜s ]

ds



ă0.

(iv) assume as in (iii). If φ is (SC2) with ´T ă θ¹ ă θ² ă 0, ϑ¹ ą 0 (up to an error of orderη²), andϑ¹ is implicitly given (up to an error of orderη²) by

żθ¹+ϑ¹ θ¹

φ¹(s)ds=

((φ¹)¹(θ²) (φ¹)¹(θ¹) ´1

)żθ¹

´T

φ¹(s)ds.

Proof. The proof relies on Newton’s method to approximate the sign changes of φ¯for η ! 1. Note that φ(´T) = ¯φ(´T) such that without loss of generality, one may assume that φ¹(´T) ‰ 0, i.e. ´T is not a sign change of φ¹ or φ¯¹. Throughout the proof, restrict to φ¹(´T)ă0; the proof of the other case is completely analogous.

(i) For sufficiently smallη,φ¯also has property(SCN). From(SCN)ofφ, there existθⁱ,0ď iď 2N, φ¹(θⁱ) = 0 and intervals (αi, βi) Q θⁱ, such that (´1)^i´1φ¹(αi)ă 0ă (´1)^i´1φ¹(βi) and φ¹ˇˇ

[αi,βi] is strictly monotone. Let I =Ť

i[αi, βi]and denote c=infθP[´T,0]zI |φ¹(θ)|, c¹ = minθPI|( ¯φ¹)¹(θ)|. Using Lemma 6.6, φ¯ is a continuously differentiable, η-small perturbation of φ with C¹´uniform bound 8η}φ}. Thus, one may choose

¯

η=min"

c, c¹,minit|φ¹(αi)|u,min

i t|φ¹(βi)|u

*

/(8η}φ})ą0,

6.5. DELAY-INDUCED SWITCHED STATES: A BALANCING ACT such that (´1)^i´1φ¯¹(αi) ă 0 ă (´1)^i´1φ¯¹(βi), φ¯¹ˇˇ

[αi,βi] is strictly monotone with

¯

c¹ = minθPI|( ¯φ¹)¹(θ)| ą 0, and φ¯¹ is bounded away from zero outside of I for all η ă η.¯ Then, by the Intermediate Value Theorem, there exist unique (sinceφ¯¹ is strictly monotone there)θ¯ⁱ P[αi, βi] such that φ¯¹(¯θⁱ) = 0, ( ¯φ¹)¹(¯θⁱ)‰0, and there are no other zeros of φ¯¹.

Using the Mean Value Theorem for each i, there exist ξⁱ P [mintθⁱ,θ¯ⁱu,maxtθⁱ,θ¯ⁱu] Ă [αi, βi]such that ( ¯φ¹)¹(ξⁱ) =(φ¯¹(¯θⁱ)´φ¯¹(θⁱ))

/(θ¯ⁱ´θⁱ)

. Consequently,

|θ¯ⁱ´θⁱ| ď |φ¯¹(¯θⁱ)´φ¯¹(θⁱ)| minθPI|( ¯φ¹)¹(θ)| = 1

¯

c¹|φ¹(θⁱ)´φ¯¹(θⁱ)| ď bη for all i, whereb = 3T }φ}/¯c¹. Here, one uses φ¯¹(¯θⁱ) = 0 =φ¹(θⁱ) for all i.

(ii) Let ηăη. Fix a sign change¯ θ of φ¹ and the corresponding sign changeθ¯of φ¯¹ with

|θ¯´θ| = O(η). Under certain conditions to be specified θ¯is determined up to a bounded error of order η² by only one step of Newton’s method with the initial guess θ. Recall from (i) that there exists an interval [α, β] P tθ,θ¯u such that the derivatives of φ¹ and φ¯¹ are bounded away from 0 by a constant b¹. For example, b¹ =mintc¹,c¯¹u, where c¹,c¯¹ defined as in (i). Denote θ¹ the first iterate of the Newton approximation step θÞÑθ¹, then

θ¹ = θ´ φ¯¹(θ) ( ¯φ¹)¹(θ),

= θ´

η (φ¹)¹(θ)

1 ε

şθi

´T e^´(θiε´s) [şs

´Tφ¹(˜s)d˜s] ds 1´ _(φ^1η₎1(θ)

(

1 ε

şθi

´T e^´(θi´εs) [şs

´T φ¹(˜s)d˜s]

ds+şθ

´T φ¹(s)ds ).

Here, the definition of φ¯¹(θ), and the fact φ¹(θ) = 0 was used. Both the numerator and the denominator were multiplied by 1/(φ¹)¹(θ). For sufficiently small η ă b¹/(2T}φ}), one can expand the denominator into a geometric series such that θ¹ satisfies Eq. (6.28) (with θ¯ replaced by θ¹). φ¯ P C_T² satisfies the estimate ››φ¯²´φ²›› ă 10η}φ} such that ››φ¯²›› ă }φ²}+ 10η}φ} =: b². Then, the error |θ¯´θ¹| of the Newton approximation step θ ÞÑ θ¹ is bounded as

|θ¯´θ¹| ă b²

2b¹|θ¯´θ|² (

ăηb²b

2b¹|θ¯´θ| )

,

where for the last inequality, one uses |θ¯´θ| ă ηb from (i). It is a well known fact that if η^b_2b^2b₁ ă 1 here, Newton’s method converges, and quadratically so, such that |θ¯´θ¹| = O(|θ¯´θ|²) = O(η²). As a result, Eq. (6.28) holds for ηăη˜=min␣

¯

η, b¹/(2T }φ}),2b¹/b²b( . (iii) Consider two subsequent sign changes θⁱ, θⁱ⁺¹ as discussed in (i) and assume as in (ii).

One shows that applyingL^ε,η_T , for sufficiently smallε, the distance between the sign changes θⁱ⁺¹ ´θⁱ decreases (increases), if it is smaller (larger) than a given “critical” distance ϑⁱ, depending onφandθⁱ, such that|θ¯ⁱ⁺¹´θ¯ⁱ| ă |θⁱ⁺¹´θⁱ|(respectively|θ¯ⁱ⁺¹´θ¯ⁱ| ą |θⁱ⁺¹´θⁱ|).

Let(φ¹)¹(θⁱ) ą0, (φ¹)¹(θⁱ⁺¹)ă0 be fixed such that φ¹(θ) ą0 for allθ P(θⁱ, θⁱ⁺¹) and vary ϑ = θⁱ⁺¹ ´θⁱ ą 0 as an independent variable; the proof of the case with signs exchanged is completely analogous. To increase readability, one introduces new variables Y(´T, θ) = şθ

´Tφ¹(s)ds and X(´T, θ) = ¹_εşθ

´T e^´(θ´s)ε Y(´T, s)ds such that at θⁱ⁺¹, one can express the value of X(´T, θⁱ⁺¹) as

X(´T, θⁱ+ϑ) = X(θⁱ, θⁱ+ϑ) +Y(´T, θⁱ) +e^´ϑε(X(´T, θⁱ)´Y(´T, θⁱ)).

Note that |X(´T, θ)´Y(´T, θ)| ď εT }φ} for all θ P [´T,0]. Using the above expression and Eq. 6.28, the differenceθ¯ⁱ⁺¹´θ¯ⁱ can be written as

θ¯ⁱ⁺¹´θ¯ⁱ =ϑ+ aiη

(φ¹)¹(θⁱ)Zⁱ(ϑ) +O(η²), whereai =´(φ¹)¹(θⁱ)/(φ¹)¹(θⁱ⁺¹)ą0 and

Zⁱ(ϑ) = X(´T, θⁱ+ϑ) + 1 ai

X(´T, θⁱ),

= X(θⁱ, θⁱ+ϑ) +Y(´T, θⁱ) + 1 ai

X(´T, θⁱ) +e^´ϑε(X(´T, θⁱ)´Y(´T, θⁱ)).

Thus, up to an error of order η², θ¯ⁱ⁺¹ ´θ¯ⁱ is smaller (larger) than ϑ, if Zⁱ(ϑ) ă 0 (re-spectively Zⁱ(ϑ) ą 0). Clearly, Y(θⁱ, θⁱ +ϑ) = şθⁱ+ϑ

θⁱ φ¹(s)ds and therefore X(θⁱ, θⁱ +ϑ) =

1 ε

şθⁱ+ϑ

θⁱ e^{´(θi+ϑ´s)ε} Y(θⁱ, θⁱ+s)dsare monotonously increasing withY(θⁱ, θⁱ) = 0andX(θⁱ, θⁱ) = 0. Note that in the limit εÑ0, statement (iii) is trivial now: Consider

εÑ0limZⁱ(ϑ) =Y(θⁱ, θⁱ +ϑ) + (1 + 1 ai

)Y(´T, θⁱ),

and observe that the second term is constant with 1 + 1/ai ą 0. Thus, if Y(´T, θⁱ) ă 0, there exist a uniqueϑⁱ ą0 such that Zⁱ(ϑⁱ) = 0 and θ¯ⁱ⁺¹´θ¯ⁱ = ϑⁱ+O(η²). If Y(θⁱ)ě 0, then limεÑ0Zⁱ(ϑ)ą0 for all ϑą0 and one may choose ϑⁱ = 0.

For 0ăεăε¯=η/(T}φ}), one has that |X(´T, θ)´Y(´T, θ)| ă η and thus, Zⁱ(ϑ) = X(θⁱ, θⁱ+ϑ) + (1 + 1

ai

)X(´T, θⁱ) +O(η).

One can apply the same reasoning as before. The remaining terms of order η affect the difference θ¯ⁱ⁺¹´θ¯ⁱ only of order η² and can be neglected. In particular, ϑⁱ ą 0 (up to an error of orderη²), if and only X(´T, θⁱ)ă0.

(iv) Let X(´T, θ) and Y(´T, θ) be defined as in (iii). By assumption (φ¹)¹(θ1) ą 0 and φ¹(θ) ă 0 for all θ P [´T, θ1) such that X(´T, θ1) ă 0. Then, Z¹(ϑ¹) = 0 implies ϑ¹ ą 0.

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