Dynamics of Differential Equations with Piecewise Uniformly Continuous Functions

I. Introduction to BPUC Functions and Dynamical Systems

In the study of dynamical systems, understanding the behavior of differential equations is paramount. This article delves into the dynamics of a specific class of differential equations, focusing on the role of Piecewise Function Uniformly Continuous functions, often abbreviated as BPUC functions (Bounded, Piecewise Uniformly Continuous). These functions, denoted as (Gamma ,p:mathbb Rrightarrow mathbb R), are characterized by having finite asymptotic limits, (gamma _pm :=lim _{trightarrow pm infty }Gamma (t)). This condition is fundamental to our initial analysis, setting the stage for exploring the dynamical possibilities of equations in the form:

$$begin{aligned} y’=-big (y-Gamma (t)big )^2+p(t),, end{aligned}$$

where (y(t,s,y_0)) represents the solution with initial value (y_0) at time (t=s). We interpret (Gamma (t)) as a transition function, smoothly bridging the system’s behavior from a past state (gamma _-) to a future state (gamma _+).

To understand the long-term behavior of equation (3.1), we introduce two “limit” equations:

Future Equation:
$$begin{aligned} y’ =-(y-gamma _+)^2+p(t),, end{aligned}$$

Past Equation:
$$begin{aligned} y’ =-(y-gamma _-)^2+p(t) end{aligned}$$

These equations represent the asymptotic states of (3.1) as (trightarrow infty ) and (trightarrow -infty ) respectively. Notably, their global dynamics are intrinsically linked to the simpler equation:

$$begin{aligned} x’=-x^2+p(t) end{aligned}$$

through straightforward variable substitutions. This connection allows us to leverage the analysis of (3.4) to understand the more complex dynamics of (3.1), (3.2), and (3.3).

Definition 3.1: Dynamical Cases A, B, and C

Building upon Theorem 2.11, we categorize the dynamical behavior of equation (3.1) into three exhaustive cases: A, B, and C. These cases are crucial for understanding the qualitative behavior of solutions. Theorem 2.9 further establishes that Case A is characterized by the existence of an attractor–repeller pair, which fundamentally determines the global dynamics of (3.1). We will explore these cases in greater depth, particularly under specific conditions outlined in Hypothesis 3.2.

Hypothesis 3.2: Existence of an Attractor–Repeller Pair for the Reduced Equation

A crucial assumption for further analysis is given by Hypothesis 3.2:

Hypothesis 3.2: The equation (3.4), (x’=-x^2+p(t)), possesses an attractor–repeller pair ((widetilde{a},widetilde{r})).

This hypothesis has significant implications for the dynamics of the original and limit equations.

Remark 3.3: Implications of Hypothesis 3.2

Hypothesis 3.2 is equivalent to the following assertions:

((widetilde{a}+gamma _+,widetilde{r}+gamma _+)) constitutes an attractor–repeller pair for the future equation (3.2).
((widetilde{a}+gamma _-,widetilde{r}+gamma _-)) constitutes an attractor–repeller pair for the past equation (3.3).
((widetilde{a}+Gamma (0),widetilde{r}+Gamma (0))) constitutes an attractor–repeller pair for (y’=-(y-Gamma (0))^2+p(t)).

These equivalences highlight the interconnectedness of the dynamics across the different equations under consideration.

Theorem 3.4: Attractor-Repeller Pairs and Asymptotic Behavior

Theorem 3.4, along with Remark 3.5, provides foundational insights into the dynamics of equation (3.1) within cases A, B, and C, assuming Hypothesis 3.2.

Theorem 3.4: Assume Hypothesis 3.2, and let ((widetilde{mathfrak {a}}_pm ,widetilde{mathfrak {r}}_pm ):=(widetilde{a} +gamma _pm ,widetilde{r}+gamma _pm )) be the attractor–repeller pairs for the future and past equations (3.2) and (3.3). Then:

(i) Existence of Functions (mathfrak {a}) and (mathfrak {r}): Functions (mathfrak {a}) and (mathfrak {r}) associated with equation (3.1) exist, as described by Theorem 2.5.
(ii) Asymptotic Convergence:
- (lim _{trightarrow -infty }|mathfrak {a}(t)-widetilde{mathfrak {a}}_-(t)|=0)
- (lim _{trightarrow -infty }|y(t,s,y_0)-widetilde{mathfrak {r}}_-(t)|=0) whenever (mathfrak {a}(s)) exists and (y_0
- (lim _{trightarrow +infty }|mathfrak {r}(t)-widetilde{mathfrak {r}}_+(t)|=0)
- (lim _{trightarrow +infty }|y(t,s,y_0)-widetilde{mathfrak {a}}_+(t)|=0) whenever (mathfrak {r}(s)) exists and (y_0>mathfrak {r}(s)).
(iii) Pullback Attractivity and Repulsivity: The solutions (mathfrak {a}) and (mathfrak {r}) exhibit local pullback attractivity and local pullback repulsivity, respectively.
(iv) Attractor–Repeller Pair Formation: If (mathfrak {a}) and (mathfrak {r}) are globally defined and distinct, they are uniformly separated, forming an attractor–repeller pair ((widetilde{mathfrak {a}},widetilde{mathfrak {r}}):=(mathfrak {a},mathfrak {r})) for equation (3.1).
(v) Bounded Solutions without Hyperbolicity: If equation (3.1) lacks hyperbolic solutions, it possesses at most one bounded solution, where (mathfrak {a}=mathfrak {r}).

Proof of Theorem 3.4

(i) We apply Proposition 2.1 to the attractor–repeller pair ((widetilde{mathfrak {a}}_-,widetilde{mathfrak {r}}_-)) of the past equation (3.3). For any (varepsilon >0), there exists (delta _-=delta _-(varepsilon )>0) such that if (left| Sigma -gamma ^-right| le delta _-) then the equation (y’=-(y-Sigma (t))^2+p(t)) also has an attractor–repeller pair ((widetilde{mathfrak {a}}_Sigma ,widetilde{mathfrak {r}}_Sigma )) with (left| widetilde{mathfrak {a}}_Sigma -widetilde{mathfrak {a}}_-right| le varepsilon ) and (left| widetilde{mathfrak {r}}_Sigma -widetilde{mathfrak {r}}_-right| le varepsilon ). This proposition also guarantees a common dichotomy pair ((k_varepsilon ,beta _varepsilon )) for all such functions (Sigma ), valid for both hyperbolic solutions.

Choose (t^-=t^-(varepsilon )) such that (|Gamma (t)-gamma ^-|le delta _-) for (tle t^-), and define (Sigma ^-(t)) as (Gamma (t)) on ((-infty ,t^-)) and (Gamma (t^-)) on ([t^-,infty )). Then, (big Vert Sigma ^–gamma ^-big Vert le delta ), and the equation (y’=-(y-Sigma ^-(t))^2+p(t)) has an attractor–repeller pair ((widetilde{mathfrak {a}}_{Sigma ^-},widetilde{mathfrak {r}}_{Sigma ^-})) with (big Vert widetilde{mathfrak {a}}_{Sigma ^-}-widetilde{mathfrak {a}}_-big Vert le varepsilon ) and (big Vert widetilde{mathfrak {r}}_{Sigma ^-}-widetilde{mathfrak {r}}_-big Vert le varepsilon ). Specifically:

$$begin{aligned} exp int _s^t(-2,widetilde{mathfrak {a}}_{Sigma ^-}(l)+2,Sigma ^-(l)),dlle k_varepsilon ,e^{-beta _varepsilon (t-s)} quad text {for}quad tge s,. end{aligned}$$

Let (widehat{mathfrak {a}}_{Sigma ^-}) be the solution of (3.1) with (widehat{mathfrak {a}}_{Sigma ^-}(t^-)=widetilde{mathfrak {a}}_{Sigma ^-}(t^-)). Since (widehat{mathfrak {a}}_{Sigma ^-}(t)=widetilde{mathfrak {a}}_{Sigma ^-}(t)) for (tle t^-), it remains bounded as t decreases, indicating the existence of (mathfrak {a}) and (widehat{mathfrak {a}}_{Sigma ^-}le mathfrak {a}). For the converse, consider (y_0>widehat{mathfrak {a}}_{Sigma ^-}(t^-)). Then (y(t,t^-,y_0)) becomes unbounded as t decreases, due to:

$$begin{aligned} frac{1}{k_varepsilon },e^{beta _varepsilon (t^–t)}le exp int _{t^-}^t(-2,widehat{mathfrak {a}}_{Sigma ^-}(l)+2,Sigma ^-(l)),dlle frac{y(t,t^-,y_0)-widehat{mathfrak {a}}_{Sigma ^-}(t)}{y_0-widehat{mathfrak {a}}_{Sigma ^-}(t^-)} end{aligned}$$

for (tle t^-), leveraging (3.5) and a result from [29]. A similar approach using ((widetilde{mathfrak {a}}_+,widetilde{mathfrak {r}}_+)) defines (mathfrak {r}) at least on ([t^+,infty )).

(ii) From the proof of (i), given (varepsilon >0), we find (t^-) such that (|mathfrak {a}(t)-widetilde{mathfrak {a}}_-(t)|=|widetilde{mathfrak {a}}_{Sigma ^-}(t)-widetilde{mathfrak {a}}_-(t)| le varepsilon ) for (tle t^-), proving the first asymptotic assertion for (mathfrak {a}). If (y_0 then there exists (t_0 such that (y(t_0,s,y_0). Since (y(t,s,y_0)=y(t,t_0,y(t_0,s,y_0))) solves (y’=-(y-Sigma ^-(t))^2+p(t)) for (tle t_0), Theorem 2.9(i) implies (lim _{trightarrow -infty }|y(t,s,y_0)-widetilde{mathfrak {r}}_-(t)|=0). The remaining asymptotic assertions follow analogously.

(iii) Choose (varepsilon in (,0,,inf _{tin mathbb R}(widetilde{a}(t)-widetilde{r}(t)),)). From (i), we have (t^-) and functions (widetilde{mathfrak {a}}_{Sigma ^-}) and (widetilde{mathfrak {r}}_{Sigma ^-}) satisfying (inf _{sin (-infty ,t^-]}(mathfrak {a}(s)-widetilde{mathfrak {r}}_{Sigma ^-}(s))= inf _{sin (-infty ,t^-]}(widetilde{mathfrak {a}}_{Sigma ^-}(s)-widetilde{mathfrak {r}}_{Sigma ^-}(s))>varepsilon ). Theorem 2.5(ii) for (y’=-(y-Sigma ^-(t))^2+p(t)) indicates that solutions (y^-(t,s,mathfrak {a}(s)pm varepsilon )) are defined for (tge s) if (sle t^-). For fixed (tle t^-) and (sle t), (mathfrak {a}(l)=widetilde{mathfrak {a}}_{Sigma ^-}(l)) and (y(l,s,mathfrak {a}(s)pm varepsilon )) match solutions (y^-(l,s,widetilde{mathfrak {a}}_{Sigma ^-}(s)pm varepsilon )) of (y’=-(y-Sigma ^-(t))^2+p(t)). Theorem 2.9(i) and (varepsilon ) yield, for (beta _0in (0,beta _varepsilon )) and (k_0=k_0(beta _0,varepsilon )ge 1) (independent of s):

$$begin{aligned} |mathfrak {a}(t)-y(t,s,mathfrak {a}(s)pm varepsilon )| =|widetilde{mathfrak {a}}_{Sigma ^-}(t)-y^-(t,s,widetilde{mathfrak {a}}_{Sigma ^-}(s)pm varepsilon )| le k_0,e^{-beta _0(t-s)}varepsilon ,, end{aligned}$$

proving local pullback attractivity of (mathfrak {a}). The proof for (mathfrak {r}) is analogous.

(iv) If globally defined (mathfrak {a}) and (mathfrak {r}) exist with (mathfrak {r}, their asymptotic convergence to ((widetilde{mathfrak {a}}_-,widetilde{mathfrak {r}}_-)) and ((widetilde{mathfrak {a}}_+,widetilde{mathfrak {r}}_+)) as (trightarrow pm infty ) (from (ii)) implies their distance is bounded below on ((-infty ,0]) and uniformly separated. Theorem 2.9 then confirms they form an attractor–repeller pair.

(v) From (iv), the sole possibility for bounded solutions without hyperbolic ones is (mathfrak {a}=mathfrak {r}), proving (v). (square )

Remark 3.5: Dynamical Cases A, B, and C Revisited under Hypothesis 3.2

Given the conditions on (Gamma ) and p and Hypothesis 3.2, Theorem 3.4, combined with Theorems 2.11 and 2.9, elucidates the dynamical cases:

Case A (Attractor-Repeller Pair): Equation (3.1) falls into Case A if and only if it possesses an attractor–repeller pair ((widetilde{mathfrak {a}},widetilde{mathfrak {r}})) (Definition 2.10), or equivalently, two distinct bounded solutions. This pair connects ((widetilde{mathfrak {a}}_-,widetilde{mathfrak {r}}_-)) to ((widetilde{mathfrak {a}}_+,widetilde{mathfrak {r}}_+)) via end-point tracking: (lim _{trightarrow pm infty }|widetilde{mathfrak {a}}(t)-widetilde{mathfrak {a}}_pm (t)|=0) and (lim _{trightarrow pm infty }|widetilde{mathfrak {r}}(t)-widetilde{mathfrak {r}}_pm (t)|=0). (widetilde{mathfrak {a}}(t)) is the unique solution approaching (widetilde{mathfrak {a}}_-) as time decreases, and (widetilde{mathfrak {r}}(t)) is the unique solution approaching (widetilde{mathfrak {r}}_+) as time increases.
Case B (Unique Bounded Solution): Equation (3.1) is in Case B if and only if it has a unique bounded solution (mathfrak {b}). This solution is locally pullback attractive and repulsive (Sect. 2.3), connecting (widetilde{mathfrak {a}}_-) to (widetilde{mathfrak {r}}_+): (lim _{trightarrow -infty }|mathfrak {b}(t)-widetilde{mathfrak {a}}_-(t)|=0) and (lim _{trightarrow +infty }|mathfrak {b}(t)-widetilde{mathfrak {r}}_+(t)|=0). No other solution exhibits these properties.
Case C (No Bounded Solutions – Tipping): Equation (3.1) is in Case C if and only if it lacks bounded solutions. There exists a locally pullback attractive solution (mathfrak {a}), unique and bounded as (trightarrow -infty ), approaching (widetilde{mathfrak {a}}_-) (i.e., (lim _{trightarrow -infty }|mathfrak {a}(t)-widetilde{mathfrak {a}}_-(t)|=0)). Similarly, a locally pullback repulsive solution (mathfrak {r}), unique and bounded as (trightarrow +infty ), approaches (widetilde{mathfrak {r}}_+) (i.e., (lim _{trightarrow +infty }|mathfrak {r}(t)-widetilde{mathfrak {r}}_+(t)|=0)). This scenario represents a loss of connection, often termed tipping.

For visual representations of these dynamical behaviors, refer to Figures 1-6 in [29]. Note a typo in the figure captions: cases A and C are interchanged.

In all three cases, constants (beta _0) and (k_0) in (3.6) can be chosen such that:

$$begin{aligned} |mathfrak {a}(t)-y(t,s,y_0)| le k_0,e^{-beta _0(t-s)}|mathfrak {a}(s)-y_0| quad text {for}quad y_0ge widetilde{mathfrak {r}}_-(s)+varepsilon quad hbox {and if} quad sle tle t^-,. end{aligned}$$

This indicates that (mathfrak {a}(t)) forwardly attracts, exponentially fast, solutions (y(t,s,y_0)) starting above (widetilde{mathfrak {r}}_-(s)+varepsilon ) for (s while (tle t^-). Similar bounds apply to (mathfrak {r}).

3.1 Fundamental Inequalities for (lambda ^*(2,Gamma ,,p-Gamma ^2))

Recall from Theorem 2.11 that (lambda ^*(2,Gamma ,,p-Gamma ^2)) is associated with equation (3.1) as the bifurcation point in (lambda ) of (x’=-(x-Gamma (t))^2+p(t)+lambda ). We now explore properties of this value under various assumptions on (Gamma ) and p. Note that Hypothesis 3.2 is not assumed in this subsection.

Our first comparison relates (lambda ^*(0,q)) to (lambda ^*(2,Gamma ,,p-Gamma ^2)) for specific functions q. Recall that the hull (Omega _p) of a BPUC function p, as detailed in Appendix A and Sect. 2.3, is constructed through flow orbits. A function p is recurrent if every orbit within its hull is dense. Almost periodic functions are always recurrent, and the hull of any BPUC function contains recurrent functions. A function (qin Omega _p) belongs to the alpha limit (omega limit) of p if a sequence ((t_n)_{nge 1}) with limit (-infty ) (resp. (+infty )) exists such that (q=lim _{nrightarrow infty }p_{t_n}) on (Omega _p), where (p_t(s):=p(s+t)).

Proposition 3.6: Comparison of Bifurcation Values

Proposition 3.6: Let (Gamma , p:mathbb Rrightarrow mathbb R) be BPUC functions with (Gamma ) having finite asymptotic limits. If (q:mathbb Rrightarrow mathbb R) is in the alpha or omega limit of p, then (lambda ^*(0,q)le lambda ^*(2,Gamma ,,p-Gamma ^2)). In particular, if p is recurrent, then (lambda ^*(0,p)le lambda ^*(2,Gamma ,p-Gamma ^2)).

Proof of Proposition 3.6

Let (lambda ^*:=lambda ^*(2,Gamma ,p-Gamma ^2)). Theorem 2.11 ensures a globally bounded solution (mathfrak {b}) for (y’=-(y-Gamma (t))^2+p(t)+lambda ^*). We aim to show a bounded solution for (x’=-x^2+q(t)+lambda ^*), which by Theorem 2.11 implies (lambda ^*(0,q)le lambda ^*).

Consider the case where ((t_n)uparrow infty ) and (q=lim _{nrightarrow infty }p_{t_n}) in (Omega _p). Define (mathfrak {b}_{t_n}(t):=mathfrak {b}(t+t_n)), which solves (y’=-(y-Gamma _{t_n}(t))^2+p_{t_n}(t)+lambda ^*), with (Gamma _{t_n}(t):=Gamma (t+t_n)). Assume, without loss of generality, (lim _{nrightarrow infty }mathfrak {b}_{t_n}(0)=:mathfrak {b}_0). Then, (lim _{nrightarrow infty }(-2,Gamma _{t_n},p_{t_n}-Gamma _{t_n}^2+lambda ^*)= (-2gamma _+,q-gamma _+^2+lambda ^*)) in (Omega _{-2Gamma ,,p-Gamma ^2+lambda ^*}). Theorem A.2 guarantees uniform convergence of ((mathfrak {b}_{t_n})_{nge 1}) on compact sets to the solution (mathfrak {b}_{gamma _+}) of ( y’ =-(y-gamma _+)^2+q(t)+lambda ^*) with (mathfrak {b}_{gamma _+}(0)=mathfrak {b}_0). (mathfrak {b}_{gamma _+}) is bounded on (mathbb R). Thus, (b:=mathfrak {b}_{gamma _+}-gamma _+) is a bounded solution of (x’=-x^2+q(t)+lambda ^*), proving the assertion.

The proof is analogous for (q=lim _{nrightarrow infty }p_{t_n}) in (Omega _p) with ((t_n)downarrow -infty ), using (gamma _-) instead of (gamma _+). The final assertion follows directly. (square )

Theorem 3.7: Comparison with Different Transition Functions

Theorem 3.7: Let (Gamma _1,Gamma _2,p:mathbb Rrightarrow mathbb R) be BPUC functions with (Gamma _2-Gamma _1) nondecreasing. Let (lambda _i:=lambda ^*(2,Gamma _i,,p-(Gamma _i)^2)).

(i) Continuous Difference: If (Gamma _2-Gamma _1) is continuous, then (lambda _1le lambda _2). If additionally, (Gamma _2-Gamma _1) is absolutely continuous and nonconstant on a nondegenerate interval, and (lambda _1=lambda _2), then (y’=-(y-Gamma _1(t))^2+p(t)+lambda _1) has infinitely many bounded solutions (non-hyperbolic), similarly for (y’=-(y-Gamma _mu (t))^2+p(t)+lambda _mu ) for (mu in (0,1)), where (Gamma _mu :=mu ,Gamma _1+(1-mu ),Gamma _2) and (lambda _mu :=lambda ^*(2,Gamma _mu ,,p-(Gamma _mu )^2)).
(ii) Finite Asymptotic Limits: If (Gamma _1) and (Gamma _2) have finite asymptotic limits, then (lambda _1le lambda _2).

Proof of Theorem 3.7

(i) If (Gamma _2-Gamma _1) is continuous and nondecreasing, it is of bounded variation, so ((Gamma _2-Gamma _1)'(t)ge 0) for Lebesgue-a.a. (tin mathbb R) (Remark 2.4). Let (mathfrak {b}_2) be a bounded solution of (y’=-(y-Gamma _2(t))^2+p(t)+lambda _2). Then (b_2:=mathfrak {b}_2-(Gamma _2-Gamma _1)) is bounded and continuous with nonincreasing singular part, satisfying (b_2′(t)=-(b_2(t)-Gamma _1(t))^2+p(t)+lambda _2 -(Gamma _2-Gamma _1)'(t)le -(b_2(t)-Gamma _1(t))^2+p(t)+lambda _2) for almost all (tin mathbb R). Theorem 2.5(v) ensures a bounded solution for (x’=-(x-Gamma _1(t))^2+p(t)+lambda _2), thus (lambda _1le lambda _2) by Theorem 2.11.

If (Gamma _2-Gamma _1) is also absolutely continuous and nonconstant on an interval [s, t], then ((Gamma _2-Gamma _1)'(t_0)>0) for some (t_0in mathbb R). Theorem 2.5(v) implies (x’=-(x-Gamma _1(t))^2+p(t)+lambda _2) has multiple bounded solutions. If (lambda _1=lambda _2), then (x’=-(x-Gamma _1(t))^2+p(t)+lambda _1) has infinitely many bounded nonhyperbolic solutions (Theorem 2.11, Remark 2.13). For (Gamma _mu ) and (lambda _mu ), (lambda _1le lambda _mu le lambda _2). If (mu in (0,1]) and (Gamma _2-Gamma _1) is nonconstant, (Gamma _2-Gamma _mu ) is also nonconstant and absolutely continuous, thus the last assertion holds for (Gamma _mu ).

(ii) Fix (varepsilon >0). We want to show (lambda _1le lambda _2+varepsilon ). Let (kappa ) bound (left| Gamma _1right| ) and (left| Gamma _2right| ). Theorem 2.12 provides (delta _varepsilon =delta _varepsilon (varepsilon ,kappa )>0) such that if (widetilde{Gamma }_1, widetilde{Gamma }_2) are BPUC with norm bounded by (kappa ) and (big Vert widetilde{Gamma }_1-widetilde{Gamma }_2big Vert le delta _varepsilon ), then (|lambda ^*big (2,widetilde{Gamma }_1,,p-(widetilde{Gamma }_1)^2big )- lambda ^*big (2,widetilde{Gamma }_2,,p-(widetilde{Gamma }_2)^2big )|. Let (gamma _i^pm :=lim _{nrightarrow pm infty }Gamma _i(t)). Find common (t_varepsilon >0) such that (|Gamma _i(t)-gamma _i^pm |le delta _varepsilon /2) for (pm tge t_varepsilon ) and (Gamma _i) continuous at (pm t_varepsilon ). Define BUPC functions (Gamma _{i,varepsilon }^infty ):

$$begin{aligned} Gamma _{i,varepsilon }^infty (t):=left{ !begin{array}{ll} Gamma _i(-t_varepsilon )&{}quad text {if}quad ;t

Then (big Vert Gamma _i-Gamma ^infty _{i,varepsilon }big Vert le delta _varepsilon /2), so

$$begin{aligned} |lambda ^*(2,Gamma _i,,p-Gamma _i^2big )- lambda ^*(2,Gamma ^infty _{i,varepsilon },,p-(Gamma ^infty _{i,varepsilon })^2big )|

Take ordered points ({a_0,ldots ,a_m}) in ((-t_varepsilon ,t_varepsilon )) where (Gamma _1) or (Gamma _2) are discontinuous, with (a_0:=-t_varepsilon , a_m:=t_varepsilon ). Let (h:=inf _{jin {0,ldots ,m-1}}(a_{j+1}-a_j)>0). For (nin mathbb {N}) and (i=1,2), define (Lambda _{i,varepsilon }^n:[-t_varepsilon ,t_varepsilon ]rightarrow mathbb R): if (tin [a_j,a_{j+1}-h/n)), (Lambda _{i,varepsilon }(t):=Gamma _i(t)); if (tin [a_{j+1}-h/n,a_{j+1})), (Lambda _{i,varepsilon }(t):=Gamma _i(a_{j+1})+(a_{j+1}-t)(n/h)big (Gamma _i(a_{j+1}-h/n)-Gamma _i(a_{j+1})big )). Extend to (Gamma _{i,varepsilon }^n(t)):

$$begin{aligned} Gamma _{i,varepsilon }^n(t):=left{ !begin{array}{ll} Gamma _i(-t_varepsilon )&{}quad text {if}quad ;t

(Gamma _{i,varepsilon }^n) are continuous BPUC, and (lim _{nrightarrow infty }Gamma _{i,varepsilon }^n(t)=Gamma ^infty _{i,varepsilon }(t)). By Lebesgue’s theorem, ((Gamma _{i,varepsilon }^n)_{nge 1}) converges to (Gamma _{i,varepsilon }^infty ) in (L^1_textrm{loc}(mathbb R,mathbb R)). (Gamma _{1,varepsilon }^n-Gamma _{2,varepsilon }^n) is nondecreasing, so by (i), (lambda ^*(2,Gamma ^n_{1,varepsilon },,p-(Gamma ^n_{1,varepsilon })^2) le lambda ^*(2,Gamma ^n_{2,varepsilon },,,p-(Gamma ^n_{2,varepsilon })^2)). We show (lim _{nrightarrow infty }lambda ^*(2,Gamma ^n_{i,varepsilon },,p-(Gamma ^n_{i,varepsilon })^2) =lambda ^*(2,Gamma ^infty _{i,varepsilon },,p-(Gamma ^infty _{i,varepsilon })^2)), thus:

$$begin{aligned} lambda ^*(2,Gamma ^infty _{1,varepsilon },,p-(Gamma ^infty _{1,varepsilon })^2) le lambda ^*(2,Gamma ^infty _{2,varepsilon },,,p-(Gamma ^infty _{2,varepsilon })^2),. end{aligned}$$

This and (3.7) yield (lambda _1le lambda _2+varepsilon ).

Let (lambda _varepsilon (n):=lambda ^*(2,Gamma ^n_varepsilon ,,p-(Gamma ^n_varepsilon )^2)). We need to prove:
1: Given (lambda , (exists n_1) s.t. (lambda le lambda _varepsilon (n)) for (nge n_1).
2: Given (lambda >lambda _varepsilon (infty )), (exists n_2) s.t. (lambda ge lambda _varepsilon (n)) for (nge n_2).

Proof of 1: Assume contradiction: (exists bar{lambda } and subsequence ((Gamma _varepsilon ^k)_{kge 1}) with (bar{lambda }>lambda _varepsilon (k)). Theorem 2.11(i) gives bounded solution (mathfrak {b}_varepsilon ^k) for (y’=-big (y-Gamma _varepsilon ^k(t)big )^2+p(t)+bar{lambda }). Bounded (left| Gamma _varepsilon ^kright| ) implies bound m for (big Vert mathfrak {b}_varepsilon ^kbig Vert ) (Theorem 2.5(iv)). Subsequence ((Gamma _varepsilon ^j)_{jge 1}) of ((Gamma _varepsilon ^k)_{kge 1}) with (y_{0}:=lim _{jrightarrow infty }mathfrak {b}_varepsilon ^j(0)). Theorem A.3 implies solution (y^infty _varepsilon (t,0,y_{0})) of (y’=-big (y-Gamma ^infty _varepsilon (t)big )^2+p(t)+bar{lambda }) is bounded by m, contradicting (bar{lambda }.

Proof of 2 (Sketch): For (bar{lambda }>lambda _varepsilon (infty )), equation (3.8) for (n=infty ) has attractor–repeller pair ((widetilde{mathfrak {a}}_varepsilon ^infty ,widetilde{mathfrak {r}}_varepsilon ^infty )). For large n, functions (mathfrak {a}_varepsilon ^n, mathfrak {r}_varepsilon ^n) exist for (3.8)(_varepsilon ^n) on ((-infty ,t_{varepsilon }]) and ([t_{varepsilon },infty )) with (mathfrak {a}_varepsilon ^n(t_{varepsilon })ge mathfrak {r}^n_varepsilon (t_{varepsilon })). This implies a bounded solution, hence (bar{lambda }ge lambda _varepsilon (n)). Use proof of Theorem 3.4(i) with ((widetilde{mathfrak {a}}_varepsilon ^infty ,widetilde{mathfrak {r}}_varepsilon ^infty )) and time (-t_varepsilon ). Find (rho :=) (min _{tin [-t_varepsilon ,t_varepsilon ]} big (widetilde{mathfrak {a}}_varepsilon ^infty (t)-widetilde{mathfrak {r}}_varepsilon ^infty (t)big )>0). Theorem A.3 gives (n_2) such that for (nge n_2), (max _{tin [-t_varepsilon ,t_varepsilon ]} big |y_varepsilon ^n(t,-t_varepsilon ,widetilde{mathfrak {a}}_varepsilon ^infty (-t_varepsilon ))- y_varepsilon ^infty (t,-t_varepsilon ,widetilde{mathfrak {a}}_varepsilon ^infty (-t_varepsilon ))big |le rho ,. ) This leads to (mathfrak {a}_varepsilon ^n(t)=y_varepsilon ^n(t,-t_varepsilon ,mathfrak {a}_varepsilon ^n(-t_varepsilon )) =y_varepsilon ^n(t,-t_varepsilon ,widetilde{mathfrak {a}}_varepsilon ^infty (-t_varepsilon ))ge widetilde{mathfrak {r}}_varepsilon ^infty (t)) for (tin [-t_varepsilon ,t_varepsilon ]) and (nge n_2), thus (mathfrak {a}_varepsilon ^n(t_{varepsilon })ge mathfrak {r}^n_varepsilon (t_{varepsilon })). (square )

Corollary 3.8: Applications to Nondecreasing Functions

Corollary 3.8: Let (p:mathbb Rrightarrow mathbb R) be a BPUC function.

(i) Let (Gamma ^+,Gamma ^-:mathbb Rrightarrow mathbb R) be bounded, uniformly continuous, and nondecreasing, and (Gamma :=Gamma ^+-Gamma ^-). Then, (lambda ^*(-2,Gamma ^-!,,p-(Gamma ^-)^2)le lambda ^*(2,Gamma ,,p-Gamma ^2)le lambda ^*(2,Gamma ^+!,,p-(Gamma ^+)^2)).
(ii) Let (Gamma :mathbb Rrightarrow mathbb R) be nondecreasing, BPUC with finite asymptotic limits or bounded and uniformly continuous. Then, (lambda ^*(-2,Gamma ,,p-Gamma ^2)le lambda ^*(0,p)le lambda ^*(2,Gamma ,,p-Gamma ^2)). Moreover:
- If p is recurrent and (Gamma ) has finite asymptotic limits, (lambda ^*(-2,Gamma ,,p-Gamma ^2)=lambda ^*(0,p)).
- If (Gamma ) is continuous, absolutely continuous and nonconstant on a nondegenerate interval, and (lambda ^*(0,p)=lambda ^*(-2,Gamma ,,p-Gamma ^2)), then (y’=-(y+Gamma (t))^2+p(t)+lambda ^*(-2,Gamma ,,p-Gamma ^2)) has infinitely many bounded solutions. If (x’=-x^2+p(t)+lambda ^*(0,p)) has a unique bounded solution, then (lambda ^*(0,p)

Proof of Corollary 3.8

(i) Follows from Theorem 3.7(i).
(ii) The inequalities follow from Theorem 3.7 with (Gamma _1:=0, Gamma _2:=Gamma ) or (Gamma _1:=-Gamma , Gamma _2:=0). Recurrent p case from Proposition 3.6. Last assertions from Theorem 3.7(i). (square )

Corollary 3.9: Absence of Bounded Solutions

Corollary 3.9: Let (p:mathbb Rrightarrow mathbb R) be BPUC and (x’=-x^2+p(t)) have no bounded solutions. Then (y’=-(y-Gamma (t))^2+p(t)) has no bounded solutions if:

(a) p is recurrent, (Gamma ) is BPUC with finite asymptotic limits.
(b) (Gamma ) is nondecreasing, BPUC with finite asymptotic limits or bounded and uniformly continuous.

If (x’=-x^2+p(t)) has an attractor–repeller pair and conditions of (b) hold, then (y’=-(y+Gamma (t))^2+p(t)) has an attractor–repeller pair.

Proof of Corollary 3.9

If (x’=-x^2+p(t)) has no bounded solutions, (lambda ^*(0,p)>0) (Theorem 2.11).
(a) From Proposition 3.6, (lambda ^*(2,Gamma ,p-Gamma ^2)>0), so no bounded solutions for (y’=-(y-Gamma (t))^2+p(t)).
(b) From Corollary 3.8(ii), (lambda ^*(2,Gamma ,p-Gamma ^2)>0), same conclusion. Last assertion similarly from Corollary 3.8(ii). (square )

3.2 Tipping Induced by Local Increment of Transition Function

We analyze tipping values of c (Definition 3.10) for the family:

$$begin{aligned} y’=-big (y-c,Gamma (t)big )^2+p(t) end{aligned}$$

for (cin mathbb R), under conditions on (Gamma ) and p. Equation for fixed c is (3.9)(_c). Future and past equations also depend on c.

Tipping analysis examines dynamics change as c varies, assuming an increasing point for (Gamma ). Dynamics are in cases A, B, or C (Definition 3.1). Case A is attractor–repeller pair existence (Theorem 2.9).

Definition 3.10: Tipping Value

Definition 3.10: (c_0in mathbb R) is a tipping value for family (3.9)(_c) if (3.9)(_c) is in Case A for c in an open interval with endpoint (c_0), but not at (c_0).

Theorem 3.4 and Remark 3.5 detail dynamical situations under Hypothesis 3.2. Theorem 2.11 links dynamical case to sign of

$$begin{aligned} widehat{lambda }:mathbb Rcup {pm infty }rightarrow mathbb R,,quad cmapsto widehat{lambda }(c):=lambda ^*big (2,c,Gamma ,,p-c^2Gamma ^2big ),, end{aligned}$$

where (widehat{lambda }(c)) is the bifurcation point for (3.9)(_c). Case A (B, C) iff (widehat{lambda }(c)<0) (null, >0). Tipping at (c_0) implies (3.9)(_{c_0}) is in Case B.

Proposition 3.11: Continuity and Lipschitz Property of (widehat{lambda }(c))

Proposition 3.11: Let (Gamma ,p:mathbb Rrightarrow mathbb R) be BPUC, and (widehat{lambda }) as in (3.10).

(i) Local Lipschitz Continuity: For every (kappa >0), (exists m_kappa >0) such that for (c_1,c_2in [-kappa ,kappa ]), (|widehat{lambda }(c_1)-widehat{lambda }(c_2)|le m_kappa |c_1-c_2|). (widehat{lambda }) is continuous and locally Lipschitz on (mathbb R).
(ii) Lipschitz Continuity for (C^1 Gamma): If (Gamma ) is (C^1) with (left| Gamma ‘right| := sup _{tin mathbb R}|Gamma ‘(t)|, then (|widehat{lambda }(c_1)-widehat{lambda }(c_2)|le left| Gamma ‘right| ,|c_1-c_2|). (widehat{lambda }) is Lipschitz on (mathbb R).

Proof of Proposition 3.11

(i) From Theorem 2.12.
(ii) For (cin mathbb R), (x=y-c,Gamma (t)) transforms (3.9)(_c) to

$$begin{aligned} x’=-x^2+p(t)-c,Gamma ‘(t),, end{aligned}$$

preserving dynamics. Repeat proof of [29, Theorem 4.13(ii)]. (square )

Proposition 3.12: Existence of Tipping for Large c

Proposition 3.12: Let (p:mathbb Rrightarrow mathbb R) be a BPUC function. If (Gamma :mathbb Rrightarrow mathbb R) is nondecreasing and nonconstant on some interval ([0,1]), then:

(i) There exists (c_0>0) such that for all (cge c_0), equation (3.9)(_c) has no bounded solutions. Also, (lim _{crightarrow infty }widehat{lambda }(c)=+infty ).
(ii) If (Gamma ) is nondecreasing and either BPUC with finite asymptotic limits or bounded and uniformly continuous, then (widehat{lambda }(c)le widehat{lambda }(0)) for all (cge 0).

Proof of Proposition 3.12

(i) Assume (Gamma ‘(t)ge delta >0) for (tin [0,1]). For (cin mathbb R), transform (3.9)(_c) to (3.11)(_c) with (x=y-c,Gamma (t)). Find (c_0>0) such that (c_0Gamma ‘(t)ge pi ^2+p(t)) for (tin [0,1]), same for (cge c_0). For (cge c_0), solution (x_c(t,0,x_0)) of (3.11)(_c) satisfies (x_c(t,0,x_0)le pi tan (-pi t+arctan (x_0/pi ))). Unbounded for any (x_0in mathbb R) and (cge c_0).

For (k>0), find (c_k>0) such that (lambda ^*big (2,c,Gamma ,,p+k-c^2Gamma ^2big )>0) for (cge c_k), so (lambda ^*big (2,c,Gamma ,,p-c^2Gamma ^2big )>k) for (cge c_k). Thus (lim _{crightarrow infty }widehat{lambda }(c)=+infty ).

(ii) From Corollary 3.8(ii) and Hypothesis 3.2, (widehat{lambda }(c)le lambda ^*(0,p) = widehat{lambda }(0)) for (cge 0). (square )

Proposition 3.13: Unique Tipping Value

Proposition 3.13: Let (p:mathbb Rrightarrow mathbb R) be BPUC. Assume Hypothesis 3.2, and (Gamma :mathbb Rrightarrow mathbb R) has finite asymptotic limits, is (C^1), nondecreasing, and nonconstant. Then there is exactly one tipping value (widehat{c}) for (3.9)(_c), and (widehat{c}>0).

Proof of Proposition 3.13

Hypothesis 3.2 implies (widehat{lambda }(0)<0). Proposition 3.12 gives (c>0) with (widehat{lambda }(c)>0). Continuity of (widetilde{lambda }) (Proposition 3.11(i)) yields minimum (c_1>0) with (widehat{lambda }(c_1)=0). Assume contradiction: (c_2>c_1) with (widehat{lambda }(c_2)=0). Theorem 3.7(i) for (Gamma _1=c_1,Gamma , Gamma _2=c_2,Gamma ) implies (y’=-(y-c_1,Gamma (t))^2+p(t)) has infinitely many bounded non-hyperbolic solutions, contradicting Remark 3.5 case B. (square )

This tipping analysis is a bifurcation analysis depending on c. Proposition 3.13(ii) gives conditions for a global saddle-node nonautonomous bifurcation in family (3.9) (Remark 2.13).