Rate-induced Tipping in Discrete-time Dynamical Systems
Claire Kiers
math.DS


 Fitting In and Breaking Up: A Nonlinear Version of Coevolving Voter Models
    Mason A. Porter
    December 30, 2023
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We develop a definition of rate-induced tipping (R-tipping) in discrete-time dynamical systems (maps) and prove results giving conditions under which R-tipping will or will not happen. Specifically, we study (possibly non-invertible) maps with a time-varying parameter subject to a parameter shift. We show that each stable path has a unique associated solution (a local pullback attractor) which stays near the path for all negative time. When the parameter changes slowly, this local pullback attractor stays near the path for all time, but if the parameter changes quickly, the local pullback attractor may move away from the path in positive time; this is the phenomenon of R-tipping. We demonstrate that forward basin stability is an insufficient condition to prevent R-tipping in maps of any dimension but that forward inflowing stability is sufficient. Furthermore, we show that R-tipping will happen when there is a certain kind of forward basin instability, and we prove precisely what happens to the local pullback attractor as the rate of the parameter change approaches infinity. We then highlight the differences between discrete- and continuous-time systems by showing that when a map is obtained by discretizing a flow, the pullback attractors for the map and flow can have dramatically different behavior; there may be R-tipping in one system but not in the other. We finish by applying our results to demonstrate R-tipping in the 2-dimensional Ikeda map.




§ INTRODUCTION


In this paper, we consider what it means for there to be rate-induced tipping (R-tipping) in a discrete-time dynamical system (map or difference equation) with time-varying parameters. Rate-induced tipping has already been studied extensively in continuous-time dynamical systems, or flows (<cit.>, <cit.>, <cit.>), but one cannot simply deduce results about R-tipping for maps from flows. To begin with, not all maps come from flows; flows are invertible while many maps are non-invertible. Furthermore, even if a map is obtained from a flow (say, by a taking a Poincaré section or by evaluating the flow at evenly-spaced discrete time steps), the parameter change affects the flow and the map differently. In the flow the parameter changes constantly, while in the map the parameter is fixed for each evaluation of the map. As a result, solutions to the corresponding map and flow can look very different. Since some physical processes are better modelled with maps than with flows (such as the Ikeda map of <ref>), we endeavor here to establish the basic theory of R-tipping in maps.

Rate-induced tipping is, roughly, a drastic change in the behavior of a system due to quickly-changing parameters. Imagine trying to pull a tablecloth out from under a set of dishes on a table. If the tablecloth is pulled slowly, it will carry all the dishes with it, but (theoretically) if the tablecloth is pulled quickly enough the dishes will be left behind on the table. Here we get two distinct outcomes: dishes come with the tablecloth vs. dishes get left behind. The deciding factor between these outcomes is not how far the tablecloth moves, but how fast. Likewise, with R-tipping, the end behavior of solutions is determined not by how much the parameters change, but how quickly.

R-tipping was introduced in <cit.> and compared against other kinds of tipping (bifurcation and noise) in <cit.>. Bifurcation-induced tipping happens due to a bifurcation in the system (say, the annihilation of a stable fixed point) and is the result of parameters changing too much. With rate-induced tipping, there is no such bifurcation in the system to explain the sudden change in behavior. Noise-induced tipping happens as a result of noise in the system, which rate-induced tipping does not require, although there can be interplay between the two phenomena, as studied in <cit.>.

Most of the literature discusses rate-induced tipping in the context of flows. In <cit.>, R-tipping is studied in flows where parameters change according to a parameter shift (essentially, a smooth transition from one constant value to another). Conditions are given there and further developed in <cit.> for when one might expect to see R-tipping in such a system. The goal of this paper is to explore the possibility of R-tipping in maps and to propose conditions for tipping in this kind of system as an analog to <cit.> and <cit.>. Some initial work was done on this in Chapter 5 of <cit.>, and we hope to expand on their results.

Suppose we have a map of the form

    x_n+1 = f(x_n,λ),

where x_i ∈^ℓ, λ∈^m, and f: ^ℓ×^m →^ℓ is C^1. Note that f( · , λ) need not be invertible. We want to allow the parameter λ to vary in a continuous way from one value to another over time, so we replace it with a C^1 function Λ: →^m satisfying

    lim_s →±∞Λ(s)     = λ_±
    lim_s →±∞Λ'(s)     = 0

for some λ_±∈^n. (In <cit.>, such a function is called a parameter shift.) To allow the parameter change to happen at different rates, we introduce the rate r > 0 and obtain the map

    x_n+1 = f(x_n,Λ(rn)).

When r is close to 0, Λ(rn) changes slowly as n increases, but when r is large, Λ(rn) approximates a step function from λ_- to λ_+. Solutions to <ref> also satisfy the map

    s_n+1   = s_n + r 
    
    x_n+1   = f(x_n,Λ(s_n))

where s_n = rn. However, <ref> and <ref> are not equivalent because a solution {x_n, s_n} to <ref> does not have to satisfy s_0 = 0. We will refer to <ref> as the autonomous map and (<ref>) as the nonautonomous map. Notice that if r = 0, then the nonautonomous map reduces to the autonomous map where λ = Λ(0). 

For a square matrix M, let ρ(M) = max{|λ|: λ is an eigenvalue of M} denote the spectral radius. Then we define a path as follows:

Suppose that for all s ∈, X(s) is an equilibrium of (<ref>) with λ = Λ(s) such that (s,X(s)) is a connected curve. Suppose also that this extends to the limits as s →±∞, so X_± = lim_s →±∞ X(s) are equilibria of (<ref>) with λ = λ_±, respectively. Then (s,X(s)) is


  * a stable path if ρ(D_x f(X(s),Λ(s))) < 1 for all s ∈∪{±∞};

  * an unstable path if ρ(D_x f(X(s),Λ(s))) > 1 for all s ∈∪{±∞}.



Paths are not solutions to the nonautonomous map <ref>; rather, they are a guideline against which we can compare solutions of the map. Note that paths are continuous curves, while solutions {x_n}_n ∈ of <ref> are sequences of discrete points. In this paper, it will be helpful to plot paths {(s,X(s))}_s ∈ and solutions {(rn,x_n)}_n ∈ together on the same axes; see <ref> as an illustration.




In <ref> we prove that for every stable path (s,X(s)) there is a unique solution {x_n^r} of (<ref>) that approaches X_- as n → -∞ and that this unique solution is a local pullback attractor. In <ref>, we use geometric singular perturbation theory to show that if r is sufficiently small, the pullback attractor to X_- will approach X_+ as n →∞, which we call endpoint tracking.

In <ref> we define rate-induced tipping for maps and give some conditions under which one can expect R-tipping to happen or not. Some of these conditions are different from conditions for R-tipping in flows, and we highlight these differences in <ref>, giving an example in which the continuous and discrete pullback attractors have drastically different behavior. Finally, in <ref> we look at an example of rate-induced tipping in the 2-dimensional Ikeda map to highlight the fact that the results given in this paper are not restricted to maps of one dimension.



§ EXISTENCE AND UNIQUENESS OF LOCAL PULLBACK ATTRACTORS BEGINNING ON STABLE PATHS



Assume that <ref> has a stable path (s,X(s)) and X_± = lim_s →±∞ X(s). The goal of this section is to prove that there is a unique solution of <ref> that limits to X_- as n → -∞ (<ref>) and that this solution is a local pullback attractor. The proof of <ref> relies on an important lemma, which we state and prove here first.


For every sufficiently small ϵ > 0 there exists an S > 0 such that if rN>S, then there is a unique solution {x_n} of (<ref>) that stays within an ϵ-neighborhood of X_- for all n ≤ -N.


Let ϵ > 0. For the sake of simplicity, suppose X_- = 0 and λ_- = 0. By definition of a stable path, all eigenvalues of A := D_xf(0,0) have norm less than 1. If we pick c ∈ (ρ(A),1), then by Lemma 5.6.10 of <cit.>, there is a matrix norm ||| · ||| such that |||A||| < c. Moreover, this is a matrix norm induced by a vector norm || · || on ^ℓ (see Example 5.6.4 of <cit.>). Therefore,

    ||Ax_0|| ≤ |||A|||  ||x_0|| < c ||x_0||

for all x_0 ∈^ℓ. Since f is C^1, for any x_0,y_0 ∈^ℓ we can write

    f(x_0,λ)-f(y_0,λ) = [∫_0^1 D_x f(tx_0 + (1-t)y_0,λ) dt] (x_0-y_0) =: g(x_0,y_0,λ)(x_0-y_0)

where g is continuous and g(0,0,0) = A. Therefore, by <ref> and <ref> we can make ϵ smaller if necessary and choose δ > 0 such that if x_0,y_0 ∈B_ϵ(0) and λ∈ B_δ(0), then

    || f(x_0,λ) - f(y_0,λ) ||= ||g(x_0,y_0,λ)(x_0-y_0)|| < c ||x_0-y_0||.


Now, let S>0 be sufficiently large so that Λ(rn) ∈ B_δ(0) and ||f(0,Λ(rn))|| ≤ϵ(1-c) for all rn < -S. Fix r,N>0 such that rN > S. Let 𝒫 denote the set of sequences {x_n}_n = -∞^-N such that x_n ∈B_ϵ(0) for all n ≤ -N. For {x_n}∈𝒫, define

    ||{x_n}|| := sup_n ≤ -N ||x_n||

using the same vector norm as above. We define the distance between two sequences in 𝒫 to be d({x_n},{y_n}) := ||{x_n - y_n}||. Then 𝒫 is a complete metric space under this distance function. Define ϕ: 𝒫→𝒫 by

    ϕ(…, x_-N-2, x_-N-1, x_-N) 
       = (…, f(x_-N-3,Λ(r(-N-3))), f(x_-N-2,Λ(r(-N-2))), f(x_-N-1,Λ(r(-N-1)))).

To verify that the image of 𝒫 under ϕ is a subset of 𝒫, pick {x_n}∈𝒫. For any fixed n, x_n ∈B_ϵ(0). Then

    ||f(x_n,Λ(rn))||    ≤ ||f(x_n,Λ(rn)) - f(0,Λ(rn))|| + ||f(0,Λ(rn))|| 
        < c||x_n|| + ϵ(1-c) , by <ref>
       ≤ϵ c + ϵ(1-c) 
        = ϵ,

so f(x_n,Λ(rn)) ∈B_ϵ(0). Thus, ϕ({x_n}) ∈𝒫. A sequence in 𝒫 is a fixed point under ϕ if and only if it is the beginning of a solution of <ref>. Furthermore, ϕ is a contraction mapping because if {x_n},{y_n}∈𝒫,

    d(ϕ({x_n}),ϕ({y_n}))     = ||ϕ({x_n}) - ϕ({y_n}) || 
        = sup_n ≤ -N ||f(x_n-1,Λ(r(n-1))) - f(y_n-1,Λ(r(n-1)))|| 
       ≤ c sup_n ≤ -N ||x_n-1 - y_n-1|| , by <ref>
       ≤ c sup_n ≤ -N ||x_n - y_n|| 
        = c  d({x_n},{y_n}).

The unique fixed point of this mapping gives the only solution {x_n} of <ref> that stays within an ϵ-neighborhood of 0 for all n ≤ -N.


Therefore, we can conclude


For each r > 0, there is a unique solution {x_n^r} of <ref> satisfying lim_n → -∞ x_n^r = X_-.


Fix r > 0. Pick ϵ > 0 sufficiently small for <ref>. Then there is an N ∈ and a unique solution {x_n^r} of <ref> such that ||x_n^r - X_-|| < ϵ for all n ≤ -N. Let δ∈ (0,ϵ). Then there is an N_1 ≥ N and a unique solution {y_n^r} of <ref> such that ||y_n^r - X_-|| < δ for all n ≤ -N_1. Since ||x_n^r - X_-|| , ||y_n^r - X_-|| < ϵ for all n ≤ -N_1, uniqueness implies x_n^r = y_n^r for all n ∈. In particular, ||x_n^r - X_-|| < δ for all n ≤ -N_1. Since our choice of δ∈ (0,ϵ) was arbitrary, lim_n → -∞ x_n^r = X_-.

Furthermore, there cannot be another {z_n^r} satisfying lim_n → -∞ z_n^r = X_- because that would violate uniqueness in <ref>.


Following the example of <cit.>, we would like to use the term local pullback attractor to describe this unique solution {x_n^r} from <ref>. To justify doing this, we will show that {x_n^r} is related to the pullback attractors defined in <cit.>, although it is not compact and it may not be a global attractor. We introduce the following notation to help state precisely what we mean by local pullback attractor:

 Let ϕ(n_1,n_0,y) denote y_n_1 where {y_n} is a solution to <ref> with initial condition y_n_0 = y and n_0 ≤ n_1.


Then we have


The solution {x_n^r} guaranteed by <ref> is a local pullback attractor, in the sense that there exists an open U ⊂^ℓ containing X_- such that for all y ∈ U,

    lim_n_0 → -∞ ||ϕ(n_1,n_0,y) - x_n_1^r|| = 0.


We postpone the proof of <ref> until the end of <ref>, since some notation and results given in <ref> will be used in the proof.

Based on <ref>, it is appropriate to refer to {x_n^r} as the local pullback attractor to X_- or simply the pullback attractor to X_-. If it's clear which backward limit point is being referred to from context, we may not mention X_-.



§ ENDPOINT TRACKING FOR SMALL RATES

As established in <ref>, if <ref> has a stable path (s,X(s)) with X_± = lim_s →±∞ X(s), the pullback attractor {x_n^r} is the unique solution of <ref> that approaches X_- as n → -∞. When we eventually define rate-induced tipping, we will be looking at the behavior of {x_n^r} as n →∞. In particular, we want to know whether lim_n →∞ x_n^r = X_+ or not:


Let {x_n^r} be the pullback attractor to X_-. If lim_n →∞ x_n^r = X_+, then the pullback attractor endpoint tracks the path (s,X(s)).


Our goal in this section is to establish that the pullback attractor will endpoint track the path as long as r > 0 is sufficiently small; in particular, we can force the pullback attractor to stay as close to the path as desired by choosing r small. This result is similar to Fenichel's Theorem for the perturbation of invariant manifolds (see Theorem 1 of <cit.>). Unfortunately, we cannot rely on the classic result for what we want to show here for two reasons: we are working with a map (which is not necessarily a diffeomorphism) and the stable path is not compact. Nevertheless, we can prove our result using similar methods to the proof of Fenichel's Theorem (for example, see Chapter 2 of <cit.>). 

To be precise, we want to prove


Let ϵ > 0. When r > 0 is sufficiently small, there is a solution {y_n} of <ref> such that ||y_n - X(rn)|| < ϵ for all n ∈. 


As long as ϵ is chosen small enough, this orbit {y_n} must be the pullback attractor to X_- since it is the only orbit that stays within an ϵ-neighborhood of X_- as n → -∞. To prove <ref>, it will be easier to work with equation <ref>. We will find a solution to <ref> satisfying the conditions of <ref> and show that it corresponds to a solution of <ref>.

Our plan for the proof of <ref> is as follows: In <ref> we prove some necessary bounds on the functions of interest. Then we introduce the graph transform map, which acts on curves close to the stable path (s,X(s)). We prove that the graph transform is a contraction mapping and hence that there is a unique curve close to (s,X(s)) which is invariant under <ref>. From this curve we will be able to construct the solution {y_n} mentioned in <ref>.

We begin with some setup and notation. Let M = {(s,X(s)) : s ∈} denote the entire stable path. Define F_r: ^ℓ + 1→^ℓ + 1 by

    F_r(s,x) = (s + r,f(x,Λ(s)))

so that <ref> can be written as (s_n+1,x_n+1) = F_r(s_n,x_n). Notice that M is invariant under the map F_0 (when r = 0); see <ref>. We will think of F_r (for r > 0) as a perturbation of F_0 and look for an invariant manifold under F_r that is close to M. This manifold will give us a solution to <ref> close to the stable path.




For notational convenience, define

    B_n(s) = [D_x f(X(s),Λ(s))]^n = D_x f^n(X(s),Λ(s)).


Let || · || denote the standard Euclidean norm as well as the induced matrix norm on square matrices. Then we have the following lemma, similar to the Uniformity Lemma of <cit.>:


There are constants κ > 0 and c ∈ (0,1) such that

    ||B_n(s)|| < κ c^n

for every s ∈ and n ∈.


Since (s,X(s)) is a stable path, there is some c ∈ (0,1) satisfying ρ(D_x f(X(s),Λ(s))) < c for all s ∈∪{±∞} (where ρ denotes the spectral radius). By Gelfand's formula (Corollary 5.6.14 of <cit.>),

    lim_n →∞ ||B_n(s)||^1/n = ρ(D_x f(X(s),Λ(s)) < c.

We do not know a priori that this limit is uniform in s; this is what we want to show.

For each s, there exists an N(s) ∈ such that ||B_N(s)(s)|| < c^N(s). By the continuity of B_N(s), there is a neighborhood U(s) of s such that for all s' ∈ U(s),

    ||B_N(s)(s')|| < c^N(s).

This is also true for s = ±∞, so we can get a finite subcovering {U(s_i)}_i = 1^K for the extended real line. Set

    κ = sup_s ∈,  r ∈ [0,max N(s_i)]||B_r(s)||/c^r.

Note that the supremum must be finite because B_r(s) approaches constant matrices as s →±∞. 

Pick s ∈ and find s_i such that s ∈ U(s_i). Then pick any n ∈ and write n = d N(s_i) + r, where d,r ∈ and 0 ≤ r < N(s_i). Then

    B_n(s) = B_dN(s_i) + r(s) = [B_N(s_i)(s)]^d B_r(s).


If we take the norm, we get

    ||B_n(s)||    ≤ ||B_N(s_i)(s)||^d ||B_r(s)|| 
        < c^dN(s_i) ||B_r(s)|| 
        = c^n ||B_r(s)||/c^r
       ≤κ c^n.



We can immediately conclude


There exists an N ∈ such that ||B_n(s)|| < 1/4 for all s ∈ and n ≥ N.


Define a small closed neighborhood around M by

    𝒩_ϵ = {(s,x): s ∈, ||x - X(s)|| ≤ϵ}.

See <ref> for an illustration.

Our next lemma shows that F_0^n (F_0 composed with itself n times) contracts 𝒩_ϵ toward M by at least 1/3 for certain n.


For the N in <ref> and for ϵ > 0 small enough, F_0^n: 𝒩_ϵ→𝒩_ϵ/3 for all n ∈{N, N+1, …, N(N+1)}.


<ref> implies that the linear part of F_0^n contracts 𝒩_ϵ toward M by at least 1/4. The nonlinear part of F_0^n is 𝒪(ϵ^2) and approaches a constant function in the limits as s →±∞, so if we take ϵ small enough, we will get that F_0^n: 𝒩_ϵ→𝒩_ϵ/3 for all n ∈{N, N+1, …, N(N+1)}.


Define π: ^ℓ + 1→^ℓ to be the projection onto the last ℓ coordinates, so π(s,x) = x. Then π∘ F_r^n: ^ℓ + 1→^ℓ, and if D_x denotes the derivative with respect to the last ℓ coordinates, then D_x [π∘ F_r^n] is an ℓ×ℓ matrix.

The next lemma gives an estimate of how close together F_0^n and F_r^n (as well as their derivatives) must be in terms of r and n.


Let 𝒦 be a neighborhood of M that is uniformly bounded in the x-directions. Then there exist constants κ_n depending only on F_0 and n such that

    ||F_0^n(s,x) - F_r^n(s,x)||, ||D_x[π∘ F_0^n](s,x) - D_x[π∘ F_r^n](s,x)|| ≤ r κ_n,

for all (s,x) ∈𝒦 as long as F_0^i(s,x) and F_r^i(s,x) remain in 𝒦 for all i ≤ n-1.


We will prove the result for F_0,  F_r; the result for the derivatives can be proven similarly.

Let η be a Lipschitz constant for F_0 on 𝒦 (which exists because F_0 is C^1, 𝒦 is uniformly bounded in the x-directions, and F_0 approaches (s,x) ↦ (s, f(x,λ_±)) as s →±∞). 

We will use induction to show that ||F_0^n(s,x) - F_r^n(s,x)|| ≤ r ∑_i = 0^n-1η^i for n ∈. The base case when n = 1 is true because

    ||F_0(s,x) - F_r(s,x)|| = ||(s,f(x,Λ(s))) - (s+r,f(x,Λ(s)))|| = ||(-r,0)|| = r.


Then, if we assume the statement is true for a given n ≥ 1,

    ||F_0^n+1(s,x) - F_r^n+1(s,x)||    ≤ ||F_0^n+1(s,x) - F_0(F_r^n(s,x))|| + ||F_0(F_r^n(s,x)) - F_r^n+1(s,x)|| 
       ≤η||F_0^n(s,x) - F_r^n(s,x)|| + r 
       ≤ r η(∑_i = 0^n-1η^i) +r 
        = r ∑_i = 0^n η^i



Now we can apply <ref> to show that F_r^n maps 𝒩_ϵ into itself for certain n.


For r and ϵ sufficiently small, we have

    F_r^n( 𝒩_ϵ) ⊂𝒩_ϵ

for any n ∈{N,N+1,…,N(N+1)}.


By <ref>, F_0^n: 𝒩_ϵ→𝒩_ϵ/3 for all n ∈{N, N+1, …, N(N+1)} if ϵ > 0 is small enough. Pick some set 𝒦 that is uniformly bounded in the x-directions such that 𝒦 contains an open neighborhood of F_0^n(𝒩_ϵ) for all n ∈{0,1,…,N(N+1)}. Let κ_n be as in <ref>. Pick δ > 0 so that if (s,x) ∈𝒩_ϵ/3, then the δ-ball around (s,x) is contained in 𝒩_ϵ. Make r > 0 as small as needed so that r κ_n < δ and F_r^n(𝒩_ϵ) ⊂𝒦 for all n ∈{0,1,…,N(N+1)}.

Pick some (s,x) ∈𝒩_ϵ. Then F_0^n(s,x) ∈𝒩_ϵ/3 for all n ∈{0,1,…,N(N+1)}. By <ref>,

    ||F_0^n(s, x) - F_r^n(s, x)|| < r κ_n < δ.

This implies that F_r^n(s, x) ∈𝒩_ϵ. Hence, F_r^n(𝒩_ϵ) ⊂𝒩_ϵ.


Finally, we combine previous results to put a bound on the derivative of π∘ F_r^n.


Assume that r and ϵ are sufficiently small. Then for all n ∈{N, N+1, …, N(N+1)} and (s,x) ∈𝒩_ϵ,

    ||D_x [π∘ F_r^n](s,x)|| < 1/2.



By the triangle inequality, we have

    ||D_x [π∘ F_r^n](s,x)|| ≤ ||D_x [π∘ F_r^n](s,x) - D_x [π∘ F_0^n](s,x)|| + ||D_x [π∘ F_0^n](s,x) - B_n(s)|| + ||B_n(s)||.

By <ref>, we can make ||D_x[π∘ F_0^n](s,x) - D_x[π∘ F_r^n](s,x)|| < 1/8 on 𝒩_ϵ for all n ≤ N(N+1) if r is sufficiently small, and ||D_x [π∘ F_0^n](s,x) - B_n(s)|| < 1/8 if ϵ is small for all n ∈{N, N+1,…, N(N+1)}. Finally, we know from <ref> that ||B_n(s)|| < 1/4 for all s ∈ and n ≥ N. Therefore, we get the desired inequality.


Let S denote the space of functions u: →𝒩_ϵ. It is a complete metric space under the sup-norm: ||u||_∞ = sup_s ∈ ||u(s)||. The graph of u is (u) = {(s,u(s)): s ∈}. The map F_r^n defines a map on S in the following way:

    (s,u(s)) ↦ (s + nr, π∘ F_r^n(s,u(s))) =: (ξ, (G_n u)(ξ)),

where G_n: S → S is called the graph transform (see <ref>). The graph transform is well-defined because s ↦ s +nr = ξ is a bijection on , and <ref> implies that the image of S under G_n truly is a subset of S. A fixed point under G_n satisfies

    F_r^n((u)) = (u).




We will use the contraction mapping theorem to show that there is a unique fixed point of G_n in S for each n ∈{N,N+1,…,N(N+1)}.


For ϵ and r sufficiently small and u,u' ∈ S,

    ||G_nu - G_nu'||_∞≤1/2 ||u - u'||_∞

for all n ∈{N, N+1,…, N(N+1)}.


Choose s ∈. Then

    ||(G_nu)(s) - (G_nu')(s)||     = ||π∘ F_r^n(s-nr,u(s-nr)) - π∘ F_r^n(s-nr,u'(s-nr))|| 
        = ||D_x [π∘ F_r^n](s-nr,u^*)(u(s-nr) - u'(s-nr))|| , for some  u^*
        < 1/2 ||u(s-nr) - u'(s-nr)|| , by <ref>
       ≤1/2 ||u-u'||_∞

Therefore,

    ||G_nu - G_nu'||_∞≤1/2 ||u - u'||_∞.



And now we are ready to prove <ref>:


By <ref>, each G_n for n ∈{N, N+1,…, N(N+1)} is a contraction mapping of the complete metric space S. Therefore, by the contraction mapping theorem, there is a unique fixed point u_n of G_n. This implies

    F_r^n((u_n)) = (u_n).

In particular,

    F_r^N((u_N))     = (u_N) 
    
    F_r^N+1((u_N+1))     = (u_N+1)

so

    F_r^N(N+1)((u_N)) = (u_N) 
    
    F_r^N(N+1)((u_N+1)) = (u_N+1)

By the uniqueness of the fixed point under F_r^N(N+1), (u_N) = (u_N+1), so u_N = u_N+1. Let this common function be called u. Then

    (u) = F_r^N+1((u)) = F_r(F_r^N((u))) = F_r((u)).

This means that

    F_r(s,u(s)) = (s+r, u(s+r))

for all s ∈. For k ∈, set y_k = u(rk). Then,

    F_r(rk,y_k) = F_r(rk,u(rk)) = (r(k+1),u(r(k+1))) = (r(k+1),y_k+1).

This implies that {(rk,y_k)}_k ∈ is a solution of <ref> and hence that {y_k}_k ∈ is a solution of <ref>. Finally, we know

    ||y_k - X(rk)|| = ||u(rk) - X(rk)|| < ϵ

since u ∈𝒩_ϵ.


Based on the discussion immediately following the statement of <ref>, we can phrase this result in terms of the pullback attractor to X_-.


For any ϵ > 0, there exists an r_0 > 0 such that if r ∈ (0,r_0), then the pullback attractor {x_n^r} to X_- satisfies ||x_n^r - X(rn)|| < ϵ for all n ∈.






As an example of <ref>, consider the 1-dimensional logistic map

    x_n+1 = f(x_n,λ) = λ x_n (1-x_n).

When 1 < λ < 3, there is a unique stable fixed point at x = 1 - 1/λ. By setting Λ(rn) = 2 + .9 tanh(rn) and plugging this in for λ in the autonomous logistic map, we get a nonautonomous map of the form <ref>. As s varies over , Λ(s) varies between 1.1 and 2.9, and X(s) = 1 - 1/Λ(s) is a stable path. <ref> demonstrates that the smaller r > 0 is, the closer the pullback attractor {x_n^r} must be to the path.




We still have yet to show that for small r > 0 the pullback attractor will endpoint track the path (s,X(s)). Intuitively, we might expect that choosing ϵ > 0 small enough and forcing the pullback attractor to stay within ϵ of the path would force endpoint tracking. That is indeed the case, and we prove it <ref>. First, we need some notation and a lemma.

Let (X(s),Λ(s)) denote the basin of attraction for X(s) in the autonomous map <ref> when λ = Λ(s).


Let K ⊂(X_+,λ_+) be compact. Then there exists an S > 0 such that if x_N ∈ K for rN ≥ S, then lim_n →∞ x_n = X_+.

A nearly identical statement is made in Lemma 5 of <cit.> for flows instead of maps, but the proof is the same. We omit the details here. Finally, we can conclude


There exists an r_0 > 0 such that if r ∈ (0,r_0), then the pullback attractor {x_n^r} to X_- will endpoint track the path (s,X(s)).


There is some ϵ > 0 such that B_ϵ(X_+)⊂(X_+,λ_+). By <ref>, there is an r_0 > 0 such that for all r ∈ (0,r_0), x_n^r ∈B_ϵ(X_+) for sufficiently large n. By <ref>, if x_n^r ∈B_ϵ(X_+) for large n, then lim_n →∞ x_n^r = X_+. Therefore, if r ∈ (0,r_0), lim_n →∞ x_n^r = X_+.


Finally, we finish this section with a proof of <ref>, which states that what we have been calling the pullback attractor is indeed a local pullback attractor:


Take U = B_ϵ(X_-), where ϵ > 0 is the same as in <ref>. Let y ∈ U. Pick an element u ∈ S satisfying u(s) = y for all sufficiently small s. Let v ∈ S denote the unique fixed point under the graph transform G guaranteed by the proof of <ref> so v(n) = x_n for all n ∈. We know G acts as a contraction mapping on S, so

    ||G_n u - v||_∞→ 0

as n →∞. Fix any n_1 ∈, and let δ > 0. Then there exists an N ∈ such that for all n ≥ N, ||G_n u - v||_∞ < δ. Making N larger if necessary, we can also guarantee that u(n_1 - n) = y for all n ≥ N. Then in particular, 

    ||G_nu(n_1) - v(n_1)|| = ||ϕ(n_1,n_1 - n, y) - x_n_1|| < δ

for all n ≥ N. Therefore, ϕ(n_1,n_0,y) → x_n_1 as n_0 → -∞.




§ DEFINITION AND CONDITIONS FOR RATE-INDUCED TIPPING IN MAPS


Now that we have proven some preliminary results in <ref>, we are ready to define rate-induced tipping. As before, suppose <ref> has a stable path (s,X(s)) with X_± = lim_s →±∞ X(s). <ref> guarantees that as long as r > 0 is sufficiently small, the pullback attractor {x_n^r} to X_- endpoint tracks the path; that is, that lim_n →∞ x_n^r = X_+. However, for some maps and paths there may be a rate r_0 > 0 for which the pullback attractor does not endpoint track the path, or lim_n →∞ x_n^r_0 X_+ or does not exist. This will be our definition for rate-induced tipping in a map.


If there is some r_0 > 0 for which the pullback attractor to X_- does not endpoint track the path (s,X(s)), then rate-induced tipping has occurred.




To demonstrate that rate-induced tipping can happen in maps, consider

    f(x,λ)     = 2(x-λ)/(1+(x-λ)^2)^2 + λ
    Λ(s)     = tanh(s).

Then there are two stable paths, X(s) = tanh(s) + √(-1 + √(2)) and Z(s) = tanh(s) - √(-1 + √(2)), as well as an unstable path, Y(s) = tanh(s). The limiting values are X_± = ± 1 + √(-1 + √(2)), Y_± = ± 1, and Z_± = ± 1 - √(-1 + √(2)), corresponding to λ_± = ± 1. In <ref> we plot these paths along with the pullback attractor {x_n^r} to X_- for two different values of r>0. When r is small, the pullback attractor tracks (s,X(s)), but when r is large, it does not, and there is rate-induced tipping.




The rest of this section will be devoted to exploring which kinds of parameter shifts lead to rate-induced tipping and which don't. The same thing is done for flows in <cit.> and <cit.>, and we will compare and contrast these results with R-tipping results for maps. As we will see, some of the R-tipping results are the same between continuous and discrete dynamical systems, but some are quite different.

We begin with the notion of forward basin stability, introduced in <cit.>. We say a stable path (s,X(s)) is forward basin stable if

    {X(s_1): s_1 < s_2}⊂(X(s_2),Λ(s_2))

for all s_2 ∈.

It is shown in <cit.> that forward basin stability prevents R-tipping in 1-dimensional flows, but from <cit.> we know it does not necessarily prevent R-tipping in flows of higher dimensions. As the following example shows, forward basin stability does not prevent R-tipping even in maps of one dimension.

Intuitively, the reason for this is that in maps of any dimension (as in flows of dimension higher than 1), a point x may be in the basin of attraction of some other point p, but that does not mean that ϕ(n,n_0,x) moves in the direction of p or even gets strictly closer to p as n increases. We use this fact to our advantage to construct the following example.



In the nonautonomous map

    f(x,λ) = .25((x-λ)^2 - (x-λ)) + λ


    Λ(rn) = 3 ((rn)^10)

there is a unique stable path (s,X(s)) with X(s) = Λ(s) and a unique unstable path (s,Y(s)) with Y(s) = 5 + Λ(s). For any s ∈, (X(s),Λ(s)) = (-4 + Λ(s), Y(s)). With the given parameter shift Λ(s), (s,X(s)) is forward basin stable. However, the pullback attractor {x_n^r} to X_- = 0 does not endpoint track the path for r = 2 (see <ref>).




In flows, if X_- ⊂(Y_+,λ_+) for an attracting equilibrium Y_+, then the pullback attractor to X_- will converge to Y_+ for all sufficiently large r > 0 (see Theorem 2 of <cit.>). However, <ref> shows that this is not the case in maps. We have X_- ⊂(X_+,λ_+) (in fact, X_- = X_+ = 0), but for all large values of r, the pullback attractor to X_- does not endpoint track. If we let {x_n^∞} denote the pointwise limit of the {x_n^r} as r →∞, then x_-1^∞ = 0, x_0^∞ = 0, x_1^∞ = 6, x_2^∞ = 7.5, etc. which diverges to infinity. The next Theorem gives a general result about what happens to the pullback attractor for large r in maps.


Let Z_1 = f(X_-,Λ(0)) and Z_2 = f(Z_1,λ_+), and let {x_n^r} be the pullback attractor to X_-. If Z_2 ∈𝔹(Y_+,λ_+) for some attracting fixed point Y_+, then x_n^r → Y_+ as n →∞ for all sufficiently large r>0.


By Lemma <ref>, there exists an r_1 > 0 such that |x_-1^r - X_-| < ϵ_1 for all r ≥ r_1. If ϵ_1 is small enough and r is made larger, if necessary, then

    |x_0^r - X_-| ≤ |f(x_-1^r, Λ(-r)) - f(X_-,Λ(-r))| + |f(X_-,Λ(-r))-f(X_-,λ_-)| < ϵ_2

can be made as small as desired. By making ϵ_2 small and r larger if necessary,

    |x_1^r - Z_1| = |f(x_0^r,Λ(0)) - f(X_-,Λ(0))| < ϵ_3

can be made as small as desired. By making ϵ_3 small and r larger if necessary, we can ensure that

    |x_2^r - Z_2| ≤ |f(x_1^r,Λ(r))- f(x_1^r,λ_+)| + |f(x_1^r,λ_+) - f(Z_1,λ_+)| < ϵ_4

can be made small. For small ϵ_4 and large r, <ref> implies that x_n^r → Y_+ as n →∞.


In the context of <ref>, X_- = 0, Z_1 = 6, and Z_2 = 7.5. Loosely, we can think of 7.5 as being in the basin of attraction of infinity when λ= λ_+, and indeed we find that x_n^r diverges to infinity as n →∞ for all large r.

The above examples and discussion demonstrate some ways in which conditions for R-tipping in maps are different from those for flows (specifically, forward basin stability does not prevent R-tipping, and the behavior of the pullback attractor as r →∞ is different). However, some results between discrete and continuous dynamical systems are the same. We conclude this section with two results: one (<ref>) giving conditions for when R-tipping is guaranteed to happen in maps, and one (<ref>) in which R-tipping cannot happen. The corresponding results for flows are given in Theorem 2 and Proposition 1 of <cit.>, respectively.


Suppose (s,X(s)) and (s,Y(s)) are two distinct stable paths, where X(s) exists for all s ∈ and Y(s) for all sufficiently large s. If X(u) ∈𝔹(Y(v),Λ(v)) for some u < v, then (s,X(s)) is not forward basin stable and there is a parameter shift Λ̃ such that there is R-tipping away from X_- to Y_+ for some r>0.


We will construct a parameter shift Λ̃ that gives R-tipping away from X_-. Since solutions to maps are only evaluated at integer multiples of r and since r is not yet determined, we introduce the functions

    σ(s)     = s+u 
    τ(s)     = s+v

which will enable us to make sure the pullback attractor we introduce will be evaluated at both u and v. Notice lim_s →±∞ X(σ(s)), X(τ(s)) = X_±, lim_s →±∞ Y(σ(s)), Y(τ(s)) = Y_±.

Now, let x_n^r,σ denote the pullback attractor to X_- under the parameter shift Λ∘σ. Then

    x_n+1^r,σ = f(x_n^r,σ,Λ (σ(rn))).

Since X(u) ∈(Y(v),Λ(v)), we can pick ϵ > 0 such that K = B_ϵ(X(u)) satisfies f(K,Λ(u)) = K_1 ⊂𝔹(Y(v),Λ(v)). Then by <ref> there is an r_0 > 0 such that for all r ∈ (0,r_0), |x_n^r,σ - X(σ(rn))| < ϵ/2 for all n ∈. By similar reasoning, we can find an r_1 > 0 such that for all r ∈ (0,r_1), if {y_n} is an orbit under the map y_n+1 = f(y_n,Λ(τ(rn))) with y_0 ∈ K_1, then y_n → Y_+ as n →∞. Now fix r ∈ (0,min{r_0,r_1,v-u}). 

We will construct a reparametrization

    Λ̃(s) := Λ(ρ(s))

using a monotonic increasing function ρ∈ C^1(,) that increases rapidly from ρ(s) = u to ρ(s) = v but increases slowly otherwise. Define

    ρ(s) = {[       σ(s) for  s ≤ 0;     τ(s)-r for  s ≥ r ].

and then continue ρ on (0,r) in a C^1 way. 

Let x̃_n^r be the pullback attractor to X_- with parameter shift Λ̃. By construction, we know that x̃_0^r = x_0^r, σ, so |x̃_0^r - X(u)| < ϵ/2. Hence, x̃_0^r ∈ K. By our choice of K, we know

    x̃_1^r = f(x̃_0^r,Λ̃(0)) = f(x̃_0^r, Λ(u)) ∈ K_1.

If we set y_n = x̃_n+1^r, then y_0 ∈ K_1 and {y_n}_n = 0^∞ is a forward orbit under the map y_n+1 = f(y_n,Λ(τ(rn))). Therefore, because r < r_1, x̃_n^r → Y_+ as n →∞.


The Ikeda map example in <ref> demonstrates an application of <ref>.

The idea of a forward inflowing stable path was introduced in <cit.>, and it was proved that forward inflowing stability prevents R-tipping in flows of any dimension. Likewise, here we will show that there can be no R-tipping away from a forward inflowing stable path in a map. A stable path (s,X(s)) is forward inflowing stable if for each s ∈ there exist compact sets K(s) satisfying



  * if s_1 < s_2, then K(s_1) ⊂ K(s_2);

  * f(K(s),Λ(s)) ⊂ K(s) for all s ∈;

  * X_±∈ K_± where K_- = ⋂_s ∈ K(s) and K_+ = ⋃_s∈ K(s); and

  * K_+ ⊂(X_+,λ_+) is compact.



Suppose (s,X(s)) is a forward inflowing stable path. Then there will not be R-tipping away from X_-.


Fix r > 0 and let {x_n^r} denote the pullback attractor to X_-. Let {K(s)} be the sets guaranteed by forward inflowing stability. There exists an ϵ > 0 such that B_ϵ(X_-) ⊂ K_-, and by <ref> there is some N > 0 such that x_n^r ∈ B_ϵ(X_-) for all n ≤ -N. Therefore, x_n^r ∈ K_- ⊂ K(rn) for all n ≤ -N. It follows by induction that x_n^r ∈ K(rn) for all n ∈. In particular, x_n^r ∈ K_+, which is a compact subset of (X_+,λ_+). Therefore, by <ref>, x_n^r → X_+ as n →∞.




§ FLOWS AND MAPS


In <ref> we gave some conditions for R-tipping in maps, some of which agree with related conditions for R-tipping in flows, some of which do not. In this section we will highlight the differences by exploring what happens when a map is obtained from a flow and there are two pullback attractors (one for the map and one for the flow).

Consider a continuous nonautonomous dynamical system of the form

    dx/dt = f(x,Λ(rt))

where x ∈^ℓ, Λ: →^m is C^1 and satisfies <ref>, and f: ^ℓ×^m →^ℓ is C^1. The corresponding autonomous system is

    dx/dt = f(x,λ)

for a fixed λ∈^m. Let Φ_λ denote the flow under <ref> for a given value of λ, so Φ_λ(t,x_0) = x(t) is the solution to <ref> with initial condition x(0) = x_0. Then we can define a map by evaluating the flow at integer time values, but allowing the parameter value to change with each time step:

    x_n+1 = Φ_Λ(rn)(1,x_n).

Equation <ref> is of the same form as equation <ref> because it is a nonautonomous map that depends on a time-varying parameter which changes according to <ref>.

Suppose <ref> has a stable path (s,X(s)) (all eigenvalues of D_xf(X(s),Λ(s)) have negative real part; see <cit.>). Then (s,X(s)) is also a stable path for <ref>. For any r > 0, there is a unique pullback attractor x^r(t) to X_- under <ref> (Theorem 2.2 of <cit.>), and there is a unique pullback attractor {x_n^r} to X_- under <ref> (from <ref>).

In general it is not the case that x_n^r = x^r(n) for all n ∈. To go from x_n^r to x_n+1^r in the map, we fix λ = Λ(rn) and apply <ref> to x_n^r for one time unit; over that whole time interval, λ is fixed. On the other hand, to go from x^r(n) to x^r(n+1) in the flow, we apply <ref> for one time unit, where Λ(rn) can change over time. Since the parameter change affects the flow and the map differently, the pullback attractors can have different behavior. The following example illustrates this.


Let

    dx/dt = -(x-Λ(rt))(x-Λ(rt) - 1)(x-Λ(rt)-2)


    Λ(s) = (s) + 0.4 tanh(s)

and define the map

    x_n+1 = Φ_Λ(rn)(1,x_n)

where Φ_λ is the flow generated by the continuous autonomous system. Then there are stable paths at (s,X(s)) = (s,Λ(s)+2) and (s,Z(s)) = (s,Λ(s)) and an unstable path at (s,Y(s)) = (s,Λ(s) + 1). <ref> shows a plot of the two pullback attractors x^r(t) and {x_n^r} to X_- = 1.6 when r = 3. They have different behavior as time goes to infinity; x^r(t) endpoint tracks the path (s,X(s)) while {x_n^r} does not. (In fact, this is true for all sufficiently large r.)






§ A 2-DIMENSIONAL EXAMPLE: THE IKEDA MAP


Up until now, all of our examples of rate-induced tipping in maps have involved 1-dimensional maps, even though our results have not specified anything about dimension. In part this is because it is easier to understand the complete dynamics in a 1-dimensional map; also it is easier to plot. In this section we will explore possibilities for rate-induced tipping in the Ikeda map, which is a 2-dimensional map that is used to model light in a ring cavity containing a dispersive nonlinear medium. The version of the map we will use here is the same as that used in <cit.>:

    z_n = a + R exp[ i( ϕ - p/1 + |z_n-1|^2) ]z_n-1,

where z_j ∈ and a, ϕ, p , R ∈ are parameters. If we let x_j = (z_j) and y_j = (z_j), then we can rewrite (<ref>) as

    x_n     = a + Rx_n-1cosθ - R y_n-1sinθ
    
    y_n     = R x_n-1sinθ + R y_n-1cosθ.

where θ = ϕ - p/1 + x_n-1^2 + y_n-1^2. As shown in <cit.>, there are generically an odd number of fixed points of <ref>. When there are three, let X, Y, and Z denote the fixed point with the largest, middle, and smallest norm, respectively. Then X is always stable, Y is always a saddle point, and Z is sometimes stable, sometimes a saddle. When Z is stable, the 1-dimensional stable manifold for Y splits the plane into basins of attraction for X and Z. See <ref> for an illustration.




Of the four parameters in the map, a is the one that is the most realistic to change, as it represents the amplitude of the inputted light. The other three parameters give information about the ring cavity itself and make sense to leave fixed. So, to get an example of R-tipping, we will fix R = 0.9, ϕ =1, and p = 6, letting a=a(s) be the time-varying parameter. From <ref>, we know we can get R-tipping away from some X_- if X(u) ∈(Z(v),a(v)) for some u < v, where (s,X(s)) and (s,Z(s)) are distinct stable paths. In particular, this will be the case if X_- ∈(Z_+,a_+). If a_+ = 0.5 and Z_+ corresponds to the stable fixed point of <ref> with smallest norm, then we already know what (Z_+,a_+) looks like from <ref>. So, we just need to find a value for a_- such that the fixed point with largest norm, X_-, is in (Z_+,a_+). One such value is a_- = 4.5 (X_- ≈ (2.9698, 5.0274); see <ref>). Thus, we will decrease a(s) from a_- = 4.5 to a_+ = 0.5 according to the function

    a(s) = 5/2 - 2 tanh(s).

The bifurcation diagram for this parameter change is shown in <ref>. There is one stable path of fixed points for all time, (s,X(s)), which corresponds to the fixed point with largest norm. When a < 1.6, there are two other paths, (s,Y(s)) and (s,Z(s)), which are the unstable middle-norm and stable smallest-norm fixed points, respectively. Since we have carefully set up our parameter shift in a way that satisfies the conditions of <ref>, we hope to see R-tipping away from X_- to Z_+ for some rate r>0, which indeed we do when r = 2; see <ref>.






§ ACKNOWLEDGMENTS

The author is greatly indebted to Chris Jones for his guidance on this project, in pointing out the Ikeda map and for his suggestions on some proof techniques.










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