Terminal response and worst-case bounded-run switching near a parabolic critical branch
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Terminal response and worst-case bounded-run switching
near a parabolic critical branch
7 November 2026
Abstract
We study planar monotone switching maps in which one mode omits a quadratic contraction,
the modes share an increasing second-coordinate update, and both preserve the first coordinate
after threshold crossing. From an exact baseline critical state with threshold deficit ρ, we
prove the schedule-uniform terminal estimate |R − S| ≤ C0 ρ7 + C0 ρ2 S, where S is the sum
of omitted contractions before crossing. It yields a delayed-intervention margin asymptotic
Mn ∼ Cn−5/2 . An elementary interchange inequality identifies the exact minimum-margin
schedule under a bound of K − 1 consecutive baseline updates. A complete-tail estimate then
gives GK = PK ∼ Cζ(5/2)K −5/2 . The result allows arbitrary higher coefficients in two finite-
regularity functions while retaining quadratic flatness and terminal freezing. The prototype’s
critical geometry has established antecedents; the contribution concerns terminal transfer and
constrained switching. A leakage example shows why leading weighted local terms alone cannot
determine the terminal observable.
1 Introduction and relation to prior work
A small intervention near a slowly attracting critical orbit may leave a terminal effect that is much
smaller than the state itself. Local orbit asymptotics suggest the size of an intervention, but do
not by themselves identify its eventual effect after a long excursion. Repeated interventions pose
a second difficulty: convergence on each fixed collection of blocks does not justify an asymptotic
formula for the whole infinite schedule. We address these issues for an explicit class in which an
intervention omits contraction in one coordinate and leaves the other update unchanged.
The local critical geometry in the simplest member of this class has a precise antecedent. For
the pre-crossing map r′ = r(1 − ad2 ), d′ = d − br, the change x = d, y = −br gives (x, y) 7→
(x + y, y − ax2 y). This is a member of the analytic normal-form family in Jamieson and Merino [1].
Their Theorem A, recalling a result of Casasayas, Fontich and Nunes, gives the cubic invariant-
curve coefficient; their Theorem 3 studies the surrounding sectors. This comparison concerns the
analytic pre-crossing map, not its frozen continuation. Cufı́-Cabré and Fontich [2] develop invariant-
curve approximation and finite-regularity theory for nilpotent parabolic points. In particular, high
endpoint regularity of an invariant curve should not be presumed. Our proof uses elementary order
arguments and a polynomial approximate invariant, without differentiating a critical graph.
Comparison of ordered dynamics belongs to the general theory of monotone systems; see Angeli
and Sontag [3] for a continuous-time control framework. The finite-word interchange used here
is proved directly. The accumulation of local perturbative effects also has a broad precedent:
canonical discrete Melnikov theory [4] uses orbit sums to describe manifold displacement in a
normally hyperbolic setting. It does not directly give the parabolic terminal estimate proved below.
We do not claim new general methods for invariant manifolds, monotone control, or summing
perturbations. Nor is the appearance of a zeta value itself the contribution.
Our main estimate controls terminal transfer uniformly in the schedule and its possibly long
crossing time. It implies a one-intervention law. In combination with a legal-word interchange and
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a summable bound for the entire remaining intervention tail, it also gives an exact bounded-run
optimizer and its asymptotic margin. These statements concern the declared structured class. They
are not a universality theorem for arbitrary monotone maps or perturbations that change the order
of flatness.
2 Model, observables, and results
Fix 0 < θ < 1. Initially let f ∈ C 6 ([0, δ0 ]) and g ∈ C 4 ([0, δ0 ]), with one-sided derivatives at zero,
satisfy
f (d) = ad2 + O(d3 ), g(d) = b + O(d), a, b > 0. (1)
Extend g to a positive Lipschitz function ḡ on [θ −1, δ0 ], agreeing with g for d ≥ 0. Put F (d) = f (d)
for d ≥ 0 and F (d) = 0 for d < 0. Choose r∗ > 0 and then δ > 0 sufficiently small, as specified in
Lemma 3. On
R = (0, r∗ ] × [θ − δ, 1]
define
W (r, s) = min{1, s + ḡ(θ − s)r}, (2)
B(r, s) = (r[1 − F (θ − s)], W (r, s)), A(r, s) = (r, W (r, s)).
Here B is the baseline mode and A is an intervention. We choose r∗ < θ − δ, Lg r∗ < 1, and
0 < f (d) < 1, f ′ (d) > 0 for 0 < d ≤ δ, where Lg is a Lipschitz constant of ḡ. Both maps preserve R
and are isotone in the coordinatewise order on (r, s). Indeed, F is nondecreasing, and the uncapped
second update has increments in s bounded below by (1 − Lg r∗ ) times the increment in s; it is
also increasing in r. Taking a minimum with 1 preserves isotonicity. The first coordinate never
increases and the second never decreases. For s ≥ θ, both modes freeze r.
We write d = θ − s. At prestates with d > 0, the uncapped updates are
B : (r, d) 7→ (r[1 − f (d)], d − g(d)r), A : (r, d) 7→ (r, d − g(d)r). (3)
The order used throughout is that on (r, s), not on (r, d). A critical state is a state whose all-B
orbit has rt → 0 and dt ↓ 0 without reaching d = 0 at any finite time. Lemma 3 proves existence
and uniqueness on each small positive-d slice.
A schedule is an infinite word ω ∈ {A, B}N0 . Its symbol ωt updates state zt to zt+1 . Fix a
critical initial state z0 . Define its full-trajectory margin by
m(ω; z0 ) = 21 inf min{rt , st }.
t≥0
This is an observation functional of an exact trajectory, with normalization 1/2; no disturbance is
fed back into the recurrence. Since r0 < s0 , write R = limt→∞ rt and obtain m(ω; z0 ) = R/2. If
τ is the first threshold hit, terminal freezing gives R = rτ ; otherwise R still denotes the monotone
limit. Define
Mn = inf m(B n Aη; z0 ), pK = (B K−1 A)∞ , PK = m(pK ; z0 ),
η∈{A,B}N0
and, for integers K ≥ 2,
GK = inf{m(ω; z0 ) : ω has no K consecutive B’s, including initially}.
All prefixes, including time zero, are part of these observables.
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Theorem 1 (Uniform terminal transfer). For the local choices above, let a trajectory start at any
critical state (r, θ − ρ), 0 < ρ ≤ δ, and use a schedule containing at least one A. It crosses s ≥ θ
at a finite time τ and has a positive terminal first coordinate R = rτ . Set
X
S= rt f (dt ).
0≤t<τ
ωt =A
There is a constant C0 , depending only on the fixed model and local rectangle, such that, uniformly
in ρ, the schedule, and τ ,
|R − S| ≤ C0 ρ7 + C0 ρ2 S. (4)
In particular R = S + O(ρ7 ) for any family with S = O(ρ5 ). The latter condition is not asserted
for arbitrary schedules.
Theorem 2 (Delayed intervention and bounded runs). Fix any exact critical initial state in R.
Put r
3 a2 D5
D= , C= .
2a 6b
Then B ∞ attains the tail infimum defining Mn , and
Mn ∼ Cn−5/2 , GK = PK ∼ Cζ(5/2)K −5/2 . (5)
The equality GK = PK holds for every integer K ≥ 2. The asymptotics fix the model, the continu-
ation, and the exact initial state; they do not assert uniformity under varying parameters or initial
states.
3 Critical states and their rates
Lemma 3 (Local critical branch). The choices of r∗ , δ can be made so that, for every 0 < d ≤ δ,
there is a unique critical first coordinate q(d) > 0. States on that slice with r > q(d) cross under
B, while those with r < q(d) are noncrossing and have a positive limiting deficit. For some fixed
L0 , q(d) ≤ L0 d3 .
Proof. First fix positive bounds g− ≤ ḡ ≤ g+ on a reduced compact interval, and f (d) ≤ q+ d2 . Set
L0 = q+ /g− , choose M > L0 and 0 < k < a/(3b). Choose r∗ < θ/2 with Lg r∗ < 1. Then reduce
δ ≤ 1 so that θ − δ > r∗ , M δ 3 < r∗ , the stated conditions on f hold, and
g+ kδ 2 < 1, g+ L0 δ 2 ≤ 1, θ + g+ L0 δ 3 < 1.
All bounds were fixed before this last reduction. On r = kd3 the baseline successor has d′ > 0, and
k(d′ )3 − r′ = k(a − 3bk)d5 + O(d6 ) > 0
after a further reduction of δ. For smaller r at the same s, isotonicity lowers both successor
coordinates in (r, s); the allowed bound k(θ − s′ )3 increases as s′ decreases. Thus 0 < r ≤ kd3 is a
forward invariant noncrossing region.P
On any noncrossing all-B orbit, t rt ≤ d0 /g− , so rt → 0. Consequently
X X f (dt )
r0 = rt f (dt ) = (dt − dt+1 ) ≤ L0 d30 . (6)
t t
g(dt )
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The initial state with r0 = M d30 < r∗ must therefore cross. The set of noncrossing slice points is
downward closed by comparison. It is relatively closed: a finite strict crossing is open in initial
conditions, and a hit of d = 0 is followed by a strict crossing since r > 0 and g(0) > 0. Its maximal
point q(d0 ) is attained and positive. If its limiting deficit were d∞ > 0, then rt → 0 would force
eventual entry into the strict interior r < kd3 of the invariant region. Continuity of finitely many
updates would allow a slightly larger noncrossing initial first coordinate, contradicting maximality.
Hence its limiting deficit is zero.
For two ordered noncrossing orbits starting on the same slice with distinct r, their s gap becomes
positive after the first update. Subsequently
shi lo hi hi lo
t+1 − st+1 ≥ (1 − Lg rt )(st − st ).
The product of these factors is positive, because Lg r∗ < 1 and t rthi < ∞. Both orbits cannot
P
tend to s = θ. This proves uniqueness and also that every smaller slice point has a positive limiting
deficit. Estimate (6) gives the bound on q. In particular, a baseline successor of a critical point is
the critical point on its new slice. No graph regularity is used.
Lemma 4 (Polynomial drift). There is a polynomial E(r, d) = rH(d)+P (d), with H(d) = 1+O(d2 )
and P (d) = −ad3 /(3b) + O(d4 ), such that on 0 ≤ d ≤ ρ, 0 < r ≤ L0 ρ3 ,
|E(B(r, d)) − E(r, d)| ≤ C1 rρ6 . (7)
The expression B(r, d) here uses the uncapped local formula, including a possible negative successor
deficit. Along a switched segment from a critical state of deficit ρ up to its first crossing,
X
|∆B E(rt , dt )| ≤ C2 ρ7 . (8)
t<τ
The analogous bound holds for every finite noncrossing segment and every infinite noncrossing tail.
Proof. Write the Taylor polynomials 2 3 4 5 2 3
2 3
P6 asi f5 = ad + cd + ud + vd , g3 = b + ed + jd + kd . Take
H = 1 + k2 d + k3 d and P = i=3 vi d , with
k2 = −a/2, k3 = (ae − 3bc)/(6b),
v3 = −a/(3b), v4 = (ae − bc)/(4b2 ),
v5 = (a2 b2 + 2abj − 2ae2 − 2b2 u + 2bce)/(10b3 ),
v6 = (−2a2 b2 e + 3ab3 c + 3ab2 k − 6abej + 3ae3
− 3b3 v + 3b2 cj + 3b2 eu − 3bce2 )/(18b4 ).
These symbols denote Taylor coefficients, not derivatives without their factorials. Expanding in r,
the coefficients of r and r2 in the polynomial baseline drift are respectively
−f5 H − g3 P ′ , −(1 − f5 )g3 H ′ + 12 g32 P ′′ .
Substitution makes these O(d6 ) and O(d3 ). All remaining terms are polynomials divisible by r3 .
Thus the drift is O(rρ6 ) on the stated region. This also follows from weight 3 for r, weight 1 for d,
and cancellation of all terms of weight at most eight.
For the actual maps, f − f5 = O(d6 ) and g − g3 = O(d4 ). Their successor changes relative to the
polynomial maps are O(rρ6 ) in r and O(rρ4 ) in d. The interpolating successors have r′ = O(ρ3 )
and |d′ | = O(ρ), including at exit; reducing δ makes this immediate from g+ L0 δ 2 ≤ 1 and a fixed
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bound on g3 . Here Er = O(1) and Ed = O(ρ2 ). The mean-value estimate gives (7). The polynomial
E is defined also for negative d; no negative-d Taylor assumption on f or g is used.
On a switched segment from the indicated critical state, rt ≤ L0 ρ3 and 0 < dt ≤ ρ at all
prestates. The crossing overshoot satisfies −g+ L0 ρ3 ≤ dτ ≤ 0, and the cap is inactive up to and
including crossing by the radius choice above. Hence
X ρ + g+ L0 ρ3
rt ≤ = O(ρ).
t<τ
g−
This estimate, which does not involve τ , proves (8). For noncrossing segments the same telescoping
estimate has no overshoot.
Lemma 5 (Critical rates). On any fixed critical baseline orbit,
a 3 √ aD3
q(d) = d + O(d4 ), n dn −→ D, n3/2 rn −→ .
3b 3b
Proof. On an infinite critical tail E tends to zero. Lemma 4 therefore gives |E(q(d), d)| ≤ C2 d7 .
Using q(d) = O(d3 ) in the polynomial formula gives the stated expansion. Thus dn+1 = dn −
(a/3)d3n +O(d4n ), so d−2 −2 −2
n+1 −dn = 2a/3+O(dn ). Cesàro averaging gives dn /n → 2a/3. Substitution
yields the limit for rn . These statements concern values of q, not its derivatives.
4 From interventions to terminal values
Proof of Theorem 1. At the first A, the successor has the same s as the baseline successor, but its
first coordinate is larger by ϵ = rf (d) > 0. It lies strictly above the critical slice endpoint. Lemma 3
therefore implies finite crossing for its baseline tail. Since B ≤ A and both maps are isotone, every
tail crosses no later than that baseline tail. Positivity of r at finite times and terminal freezing give
R > 0. This proves crossing independently of the energy estimate.
At each prestate, the two local successor deficits coincide. Affinity of E in r gives the exact
identity
E(A(r, d)) − E(B(r, d)) = rf (d)H(d − g(d)r). (9)
The initial energy is O(ρ7 ) by the preceding critical-tail argument. At exit, R = O(ρ3 ) and
|dτ | = O(ρ3 ), so E(R, dτ ) = R + O(ρ9 ). Summing (9), and treating the baseline drift hypothetically
even at A prestates, gives
R = S + O(ρ7 ) + O(ρ2 S).
Lemma 4 makes the constants independent of schedule and crossing time. This proves (4).
For a first intervention after n baseline updates, order comparison shows that the all-B tail
minimizes every later state and hence the margin. On that tail S = rn f (dn ) = O(d5n ), and therefore
2Mn = rn f (dn ) + O(d7n ). (10)
Lemma 5 implies n5/2 rn f (dn ) → a2 D5 /(3b), proving the first asymptotic in Theorem 2.
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5 Exact ordering under a run constraint
Lemma 6 (Finite-word interchange). For every N , the length-N prefix of pK produces a state no
greater coordinatewise than any length-N word with no K consecutive B’s.
Proof. We have B ≤ A and
A ◦ B ≤ B ◦ A. (11)
Indeed W (r, s) ≥ s, so the first coordinate of A ◦ B is r[1 − F (θ − s)], no greater than r[1 − F (θ −
W (r, s))], the first coordinate of B ◦ A. Their second coordinates are ordered by isotonicity of W
in r. The argument includes crossing and capping because F is nondecreasing and vanishes at
nonpositive arguments.
Transform a legal word to the periodic prefix from left to right. At a first mismatch, the target
cannot be A: that position follows K − 1 already matched B’s, which force A in any legal word.
The target is thus B and the current symbol is A. If a later B exists, move its first occurrence left
across the intervening A’s. Each replacement of a temporal AB by BA lowers the resulting state
by (11); isotonicity preserves the inequality through the rest of the word. The moved B has an
A before it until its last move. At that last move the already matched preceding run has length
less than K − 1. The run left at its old location shortens. Thus legality is preserved. If no later
B exists, replace the current A by B; the following symbols are all A and the same preceding-run
bound ensures legality. This replacement also lowers the state. Iteration terminates at the desired
prefix.
Apply this lemma to the prefix of each admissible infinite schedule at every N . Taking limits of
the first coordinates proves PK ≤ m(ω; z0 ), while pK is itself admissible. Thus GK = PK for every
K ≥ 2.
6 The complete periodic tail
Let tj = jK − 1 be the prestate timeP of the jth A under pK . Set ϵj = rtj f (dtj ) before crossing and
ϵj = 0 after crossing. Then S = j≥1 ϵj . Prior to crossing, rt and dt decrease, so their product
rt f (dt ) decreases. Each ϵj with j > 1 is bounded above by each of the preceding K − 1 baseline
losses. These disjoint blocks give, for every J ≥ 1,
X rtJ
ϵj ≤ . (12)
K −1
j>J
The estimate remains valid if crossing truncates the nonzero kicks. At t1 = K − 1 the trajectory is
exactly critical. Writing ρ = dK−1 ∼ DK −1/2 , we obtain
rt1
S ≤ ϵ1 + = O(K −5/2 ) = O(ρ5 ).
K −1
Theorem 1, applied from that prestate, gives
2PK = S + O(ρ7 ). (13)
The omitted prefix does not change the margin, because r decreases and s increases throughout.
For completeness we prove the needed fixed-block limit for the switched orbit. Use the scaled
time x = 1 + (t − t1 )/K and variables u = K 3/2 r, z = K 1/2 d. At t1 , the initial values tend to
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(aD3 /(3b), D) by Lemma 5. For each fixed integer J ≥ 1, the baseline recurrence on a bounded
stopped region is
auz 2 bu
u+ = u − + O(K −3/2 ), z+ = z − + O(K −3/2 ).
K K
At an A update the first equation omits its O(K −1 ) loss; there are at most J such updates on
1 ≤ x ≤ J. The limiting ODE and solution are
aD3 −3/2
u̇ = −auz 2 , ż = −bu, u(x) = x , z(x) = Dx−1/2 .
3b
On a compact neighborhood of this solution the vector field is Lipschitz. Euler comparison gives
em+1 ≤ (1 + LJ /K)em + CJ K −3/2 + (CJ /K)1{A update} ,
with the ODE local truncation error O(K −2 ) absorbed. Discrete Gronwall over at most JK steps
yields max em = O(e0 + K −1/2 ) = o(1). Stop provisionally at the first crossing or exit from
that neighborhood.
√ Its boundary can be chosen a positive distance from the limiting path, and
z(x) ≥ D/ J > 0; the estimate therefore excludes either stop for all sufficiently large K. This
argument only fixes J; it assumes no uniformity as J grows. Consequently, for every fixed j,
a2 D5 −5/2 aD3 −3/2
K 5/2 ϵj −→ j , K 3/2 rtj −→ j . (14)
3b 3b
By (12) and (14),
X aD3 −3/2
lim sup K 5/2 ϵj ≤ J −→ 0.
K→∞ 3b
j>J
Truncate at J, pass K → ∞, then let J → ∞. Equations (13)–(14) yield
X
K 5/2 PK → C j −5/2 = Cζ(5/2).
j≥1
Together with the exact optimizer, this completes Theorem 2.
7 Examples and limitations
The prototype is f (d) = ad2 , g(d) = b. Its polynomial energy simplifies to
a 3 a 2 a2 5
E =r− d − rd + d ,
3b 2 10b
and gives q(d) = ad3 /(3b) + O(d5 ). A nonconstant example is f (d) = d2 (1 + d), g(d) = 1 + d for
d ≥ 0, with ḡ = 1 for d < 0 and any fixed 0 < θ < 1. For a sufficiently small rectangle it satisfies
all hypotheses and has a = b = 1, hence the same leading constant as the corresponding prototype.
More generally, higher coefficients in f and g may vary arbitrarily, subject to the stated regularity
and a sufficiently small local domain.
The continuation is part of the theorem. Freezing preserves a positive crossing value; it does
not itself force crossing or determine the leading value. To see the limitation of local information,
for λ > 0 set
r[1 − F (θ − s)] r
e s) =
B(r, , W (r, s) , e s) =
A(r, , W (r, s) .
1 + λr 1 + λr
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These maps remain isotone: r 7→ r/(1 + λr) is increasing, 1 − F (θ − s) is nonnegative and non-
decreasing in s, and W is unchanged. With weight 3 for r and weight 1 for d, their pre-crossing
expansions retain the leading weighted contraction ard2 and accumulation br. The added O(r2 )
loss has weight six, above the weight-five contraction, and is smaller on r = O(d3 ). The ordinary
Taylor jet does change; the comparison here concerns these weighted terms. Nevertheless every
schedule satisfies
rt 1
rt+1 ≤ , rt ≤ −1 −→ 0.
1 + λrt r0 + λt
All margins are then zero. This is a counterexample to an extension based only on leading weighted
local terms and monotonicity, not to the theorem: it violates terminal freezing and the exact
omitted-contraction form.
Quadratic flatness is also structural. A perturbation adding a linear term to f is outside the
hypotheses. The exponent 5/2 therefore does not revise the exponent of any linear-loss model.
The exact critical initialization is essential to the stated asymptotics; we have not proved uniform
results for approximate critical states, parameter limits, or noisy recurrences. Adaptive choices
between the two specified modes are covered pathwise: each resulting trajectory has a schedule to
which the theorems apply, with their stated intervention and run conditions. Interventions that
change the update maps or inject disturbances are outside the scope. The arbitrary-tail transfer
bound permits S larger than O(ρ5 ), as occurs for repeated A updates. Only the single-intervention
and sparse periodic schedules use the sharper remainder.
Reproducibility and review status
The package includes validation/verify final.py, a coefficient snapshot, and the validation receipts. The
script matches the six printed energy coefficients to the snapshot, checks nine exact drift cancellations and
the scaled ODE identities, and exhaustively checks the finite-word transformation for lengths at most twelve
and 2 ≤ K ≤ 5. Its citation-key, label, delimiter, and environment checks are syntactic; source attribution is
assessed separately in the literature-review record. A workspace-only option checks preservation hashes; the
mathematical and source checks can also run on the standalone package. These computations supplement
the analytic proof and are not formal verification. No numerical approximation to a critical state is used as
proof.
This manuscript was prepared with AI assistance. The accompanying records identify two independent AI
reviews of the complete supplied source, including a hostile review, and a separate scoped editorial/literature
review. They are not human peer review or external endorsement. Compilation and page-by-page visual
inspection are documented in the accompanying production receipt.
References
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exhibiting 1–1 resonance, Advances in Difference Equations (2018), Article 142.
doi:10.1186/s13662-018-1595-x.
[2] C. Cufı́-Cabré and E. Fontich, Differentiable invariant manifolds of nilpotent parabolic points, Discrete
and Continuous Dynamical Systems 41(10) (2021), 4667–4704. doi:10.3934/dcds.2021053.
[3] D. Angeli and E. D. Sontag, Monotone control systems, IEEE Transactions on Automatic Control
48(10) (2003), 1684–1698. doi:10.1109/TAC.2003.817920.
[4] H. E. Lomelı́, J. D. Meiss and R. Ramı́rez-Ros, Canonical Melnikov theory for diffeomorphisms,
Nonlinearity 21 (2008), 485–508. doi:10.1088/0951-7715/21/3/007.
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