Morse Decompositions for DelayDifference Equations
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Abstract
Scalar difference equations \(x_{k+1}=f(x_k,x_{kd})\) with delay \(d\in {\mathbb {N}}\) are wellmotivated from applications e.g. in the life sciences or discretizations of delaydifferential equations. We investigate their global dynamics by providing a (nontrivial) Morse decomposition of the global attractor. Under an appropriate feedback condition on the second variable of f, our basic tool is an integervalued Lyapunov functional.
Keywords
Delaydifference equation Global attractor Morse decomposition Discrete Lyapunov functionalMathematics Subject Classification
37B25 37C70 34K28 39A11 39A30 92A151 Introduction

It allows to disassemble an attractor into finitely many invariant, compact subsets (the Morse sets) and their connecting orbits,

the recurrent dynamics in \({\mathcal A}\) occurs entirely in the Morse sets,

outside the Morse sets the dynamics of (1.1) on \({\mathcal A}\) is gradientlike.
All the above applications have in common to address problems in continuous time, that is differential equations. A first approach to tackle difference equations via discrete Lyapunov functionals is due to MalletParet and Sell [17]. In showing that such an integervalued functional V decreases along forward solutions, they lay the foundations of our present work. Yet, [17] is primarily motivated by timediscretizations of delaydifferential equations, while we are furthermore interested in applications being timediscrete right from the beginning by means of models originating e.g. in life sciences. Note that [17] prove the decay of V for a larger class of difference equations than (1.1), which combines (1.1) with (cyclic) tridiagonal systems (see also our Remark 3.4). Nonetheless, up to our knowledge, this paper is the first contribution using a discrete Lyapunov functional to actually understand the dynamics of discretetime models.
 \((m_1)\)

\(\alpha (y)\subseteq {\mathcal M}_i\) and \(\omega (\xi )\subseteq {\mathcal M}_j\),
 \((m_2)\)

\(i=j\)\(\Rightarrow \)\(\xi \in {\mathcal M}_i\) (thus, \(y_k\in {\mathcal M}_i\) for every \(k\in {\mathbb {Z}}\)).
Example 1.1
The paper is organized as follows: In many cases the discrete Lyapunov functional is applied to a variational equation. This allows to argue that it decreases along the difference of two solutions. Hence, the next two sections include both preparatory work and crucial related results on linear difference equations; for instance Lemma 2.1 provides a description of the eigenvalue distribution for the linearized equation. In Sect. 3 we formulate our main assumptions, the feedback conditions and introduce an integervalued Lyapunov functional, whose properties are given by Proposition 3.2 and Theorem 3.3. Our main result is Theorem 4.1, which yields a Morse decomposition of the global attractor of (1.2) (and in turn (1.1)) under appropriate assumptions on the righthand side f. Some of the arguments are given only in the negative feedback case, which seems to have more applications and requires slightly more involved proofs. For the convenience of the reader, the proofs for positive feedback are given in the Appendix. Section 5 contains applications of the theory, mainly from the life sciences, which underline that both the positive and negative feedback case are relevant. Finally some open questions and perspectives are addressed in Sect. 6.
We conclude with our notation: A discrete interval\({\mathbb {I}}\) is the intersection of a real interval with the integers and Open image in new window . Special cases are the positive halfaxis Open image in new window and the negative halfaxis Open image in new window .
2 Spectrum of the Linearization
Lemma 2.1
 (a)
If \(b>0\), then for all \(j\in {\mathbb {N}}\), \(j< \frac{d+1}{2}\), there exists exactly one pair of characteristic roots \(\lambda _j, \overline{\lambda _j}\) of (2.2) with \(\lambda _j\in S^+_j\), and all other roots are real. For even d there exists a unique real root, namely \(\lambda _+>0\) and for odd d there exist exactly two real roots, namely \(\lambda _<0<\lambda _+\).
 (b)If \(b<0\), then for all \(j\in {\mathbb {N}}\), \(j< \frac{d}{2}\), there exists exactly one pair of characteristic roots \(\lambda _j, \overline{\lambda _j}\) of (2.2), such that \(\lambda _j\in S^_j\).
 (1)
If \(m_0\le b<0\), then all other roots are real. Exactly two of those are positive and they are denoted by \(0<\lambda _{+,2}\le \lambda _{+,1}\).
 (2)
If \(b<m_0\), then there is also a pair of complex roots \(\lambda _0,\overline{\lambda _0}\), such that \(\lambda _0\in S^_0\), and then there exists no positive characteristic root.
 (3)
If d is odd, then there is no negative real root, otherwise there is exactly one, which is denoted by \(\lambda _\).
 (1)
 (c)
If \(b\ne 0\), and \(z_1=r_1 e^{i\varphi _1}\) and \(z_2=r_2 e^{i\varphi _2}\) are two distinct roots of (2.2) such that \(0\le \varphi _1 < \varphi _2 \le \pi \) holds, then \(r_1>r_2\).
Proof
3 A Lyapunov Functional
 (\(\mathbf H _\mathbf 1 \))

f is continuously differentiable and 0 is an inner point of J;
 (\(\mathbf H _\mathbf 2 \))

\(D_1 f(x_d,x_0)>0\) for all \(x_d, x_0 \in J\);
 (\(\mathbf H _\mathbf 3 \))

\(\delta ^*D_2f(0,0)>0\), \(\delta ^*f(0,x_0)x_0>0\) for all \(x_0\in J\setminus \{0\}\).
Remark 3.1
The subsequent Proposition 3.2 and Theorem 3.3 give some nice properties of the functionals \(V^\pm \) that will be essential in the proof of our main result. We state them for both positive and negative feedback, however, we only present the proofs for the negative feedback case \(\delta ^*=1\). For positive feedback, the proofs are rather similar to the ones presented below, and require only straightforward modifications in the arguments (cf. Appendix).
Proposition 3.2
 (a)
\(V^\pm (y)\le \liminf _{n\rightarrow \infty }V^\pm (y^n)\),
 (b)
\(V^\pm (y)=\lim _{n\rightarrow \infty }V^\pm (y^n)\), if \(y\in \mathcal {R^\pm }\).
Proof
 (a)
follows from the lower semicontinuity of the function \({{\mathrm{sc}}}\) on \(\mathbb {R}_*^{d+1}\).
 (b)
is shown only in the negative feedback case, since the positive feedback case can be handled by straightforward modifications in the argument.
Theorem 3.3
 (a)
\(V^\pm (y_{k+1})\le V^\pm (y_k)\) holds for any \(k\in {\mathbb {I}}'\),
 (b)
if \(k\in {\mathbb {I}}\) is such that \(k+4d+2\in {\mathbb {I}}\) and moreover \(y_k\ne 0\) and \(V^\pm (y_k)=V^\pm (y_{k+4d+2})\) hold, then \(y_{k+4d+2}\in \mathcal {R^\pm }\).
According to Remark 3.1 this applies to solutions of (1.2) as well. We additionally remark that Theorem 3.3 remains valid in the nonautonomous situation when f depends on k and our assumptions hold for every \(k\in {\mathbb {I}}\).
The next proof is valid for any \(d\in {\mathbb {N}}\) (using notations Open image in new window where necessary in the \(d=1\) case). However it is worth mentioning that Theorem 3.3 almost trivially holds for \(d=1\), and some parts of the following proof can be omitted even in the case \(d=2\).
Proof of Theorem 3.3
For better readability, we give the proof for negative feedback \(\delta ^*=1\), the positive feedback case can be found in the Appendix.
(b) Before we prove the statement, we need to introduce some auxiliary notions. Let us say that a component y(j) \((0\le j \le d)\) of a vector \(y\in {\mathbb {R}}^{d+1}\) is irregular if \(y(j)=0\), and Open image in new window . Using this terminology, \(y\in {\mathcal {R^}}\) (i.e. y is regular) holds if and only if y has no irregular component. Furthermore, we call a vector \((y(i),\ldots ,y(j))\) with \(0\le i\le j\le d\) an irregular block (of zeros) in \(y\in {\mathbb {R}}^{d+1}\) if \(y(i)=\cdots =y(j)=0\), and moreover it has maximal length in the sense that either \(i=0\) or \(i\ge 1\) and \(y(i1)\ne 0\) hold, and similarly either \(j=d\) or \(j<d\) and \(y(j+1)\ne 0\) hold. The dimension of the block will be regarded as the length of it. Note also that consecutive zero components are irregular by definition. The proof of (b) consists of several, yet elementary, steps. From now on we always assume that \(V^(y_k)=V^(y_{k+4d+2})\), which implies, in the light of statement (a), that \(V^(y_k)=V^(y_{k+\ell })\) holds for all \(0\le \ell \le 4d+2\).
(I) If for some index k, \(y_{k+1}(j)=0\) is irregular (\(0\le j<d\)), then so is \(y_{k}(j+1)=0\) irregular.
This is trivial in case \(1\le j\le d2\).
If \(y_{k+1}(d1)=0\) is an irregular component of \(y_{k+1}\) (i.e. \(y_{k+1}(d1)=0\) and \(y_{k+1}(d2)y_{k+1}(d)\ge 0\) hold), then on the one hand \(y_k(d)=0\) holds and on the other hand, \(b_k<0\) implies \({\text {sgn}}y_{k+1}(d)={}{\text {sgn}}y_k(0)\). Combining this with \(y_k(d1)=y_{k+1}(d2)\) yields \({{\text {sgn}}(y_k(d1)y_k(d+1))=}\)\({\text {sgn}}(y_k(d1)y_k(0))={\text {sgn}}(y_{k+1}(d2)y_{k+1}(d))\), meaning \(y_k(d)\) is also irregular in \(y_k\).
For the case when \(y_{k+1}(0)=0\) is irregular, assume to the contrary that \(y_{k}(1)=0\) is regular, i.e. \(y_k(0)y_k(2)<0\). W.l.o.g. we may assume \(y_k(0)<0<y_k(2)\). Irregularity of \(y_{k+1}(0)\) combined with \(y_{k+1}(1)=y_k(2)>0\) yields \(y_{k+1}(d)\le 0\). Now, from \(a_k>0\) and \(b_k<0\) it follows that \(y_k(d)<0\). These all together imply that \({{\mathrm{sc}}}y_k\) is an even number, moreover \({{\mathrm{sc}}}y_k={{\mathrm{sc}}}y_{k+1}+1\), which gives \(V^(y_k)=V^(y_{k+1})+2\), a contradiction.
(III) As a consequence of the first two steps, if \(y_k\in {\mathcal {R^}}\) and \(V^(y_k)=V^(y_{k+1})\), then \(y_{k+1}\in {\mathcal {R^}}\). Therefore, in order to prove the statement, it is sufficient to show that there exists \(0\le \ell \le 4d+2\) such that \(y_{k+\ell }\in {\mathcal {R^}}\).
(IV) Note that if \(y_k(d)=0\) and \(y_k\ne 0\in {\mathbb {R}}^{d+1}\), then since \(a_j>0\) for all \(j\ge k\), there exists \(k+1\le k_1 \le k+d\) such that \(y_{k_1}(d) \ne 0\). Thus we may assume that \(k_1\), \(k\le k_1\le k+d\) is such that \(y_{k_1}(d)\ne 0\).
(V) This in particular implies that \(y_{k_1}(d)\) is regular. The rest of the vector \(y_{k_1}\) may contain several irregular blocks, which are separated by at least one regular coordinate from each other. The coordinate \(y_{k_1}(0)\) may also be zero, or even irregular.
Next we will study how the irregular blocks of \(y_{k_1+d+1}\) can be described by the irregular blocks of \(y_{k_1}\). Let us use notation \(k_2=k_1+d+1\).
(V.1) If there exists \(1\le j \le d\) so that \(y_{k_1}(0)=\cdots = y_{k_1}(j1)=0\) and \(y_{k_1}(j)\ne 0\), then, as \(a_j>0\) for all \(j\ge k\), \({\text {sgn}}y_{k_1+j}(d)=\cdots ={\text {sgn}}y_{k_1+j}(dj)={\text {sgn}}y_{k_1}(d)\) holds, and consequently one has \({\text {sgn}}y_{k_2}(0)=\cdots ={\text {sgn}}y_{k_2}(j1)={\text {sgn}}y_{k_1}(d)\ne 0\).
(V.2) If \(y_{k_1}(i)=\cdots =y_{k_1}(j)=0\) is an irregular block with \(1\le i< j<d\) such that \(y_{k_1}(i1)y_{k_1}(j+1)<0\), then for \(k'=k_1+i1\) one has that \(y_{k'}(1)=\cdots =y_{k'}(ji+1)= 0\) and \(y_{k'}(0)y_{k'}(ji+2)<0\) hold. Due to the argument seen in (II), and that neither \(y_{k'}(d)=y_{k'}(0)=0\), nor (3.7) holds (with \(k=k'\)), we obtain that \(y_{k'+1}(d)\) is regular, which implies in this case that it is also nonzero. Going a bit further, to \(k''=k'+ji+2=k_1+j+1\), one infers that \({\text {sgn}}y_{k''}(d)=\cdots ={\text {sgn}}y_{k''}(dj+i1)\ne 0\). Then it is straightforward that \({\text {sgn}}y_{k_2}(i)=\cdots ={\text {sgn}}y_{k_2}(j)\ne 0\) holds. Observe that in case \(i=j\), \(y_{k_1}(i)=0\) would not be irregular due to assumption \(y_{k_1}(i1)y_{k_1}(i+1)<0\).
(V.3) Thus it remains to consider the case when \(y_{k_1}(i)=\cdots =y_{k_1}(j)=0\) is an irregular block with \(1\le i\le j<d\) such that \(y_{k_1}(i1)y_{k_1}(j+1)>0\). Choosing \(k'=k_1 +i1\) just as in the previous step one obtains \(y_{k'}(1)=\cdots =y_{k'}(ji+1)= 0\) and \(y_{k'}(0)y_{k'}(ji+2)>0\).
Now there are two possibilities. Either \(y_{k'+1}(d)\ne 0\) and then we have the same situation as in (V.2), i.e. \({\text {sgn}}y_{k_2}(i)=\cdots ={\text {sgn}}y_{k_2}(j)\ne 0\), or \(y_{k'+1}(d)= 0\). In the latter case one can easily see that \({\text {sgn}}y_{k_2}(i1)=\cdots ={\text {sgn}}y_{k_2}(j)= 0\), moreover, if \(i\ge 2\), then \(y_{k_2}(i2)y_{k_2}(j+1)<0\) holds. Note that the index of the last components of the corresponding irregular blocks in \(y_{k_1}\) and \(y_{k_2}\) coincide (it is j).
(VI) Conversely, Steps (I), (II) and (V) combined show that if \(y_{k_2}\) has an irregular block, whose last component is at index j (with \(0\le j<d\)), i.e. \(y_{k_2}(j)=0\) is irregular and \(y_{k_2}(j+1)\ne 0\), then \(y_{k_1}(j)=0\) is irregular and \(y_{k_1}(j+1)\ne 0\).
Our aim is to apply the arguments presented in (V) now for \(y_{k_3}\), where \(k_3\) is to be defined soon. For this reason we need to distinguish two cases with respect to the regularity of \(y_{k_2}(d)\).
If it is regular, then it may be zero or nonzero. If it is nonzero, then (V) shows that all irregular blocks of \(y_{k_2}\) are of the type considered in (V.1) and (V.2) (now with \(k_2\) instead of \(k_1\)). If \(y_{k_2}(d)=0\), then from its regularity it follows that \(y_{k_2}(0)\ne 0\), so all irregular blocks of zeros are of the type studied in (V.2). Note that in that step we did not use that \(y_{k_1}(d)\ne 0\). For regular \(y_{k_2}(d)\) let us define \(k_3=k_2\).
If \(y_{k_2}(d)=0\) is irregular, then \(y_{k_21}(0)=y_{k_1}(d)\ne 0\) implies that (3.7) holds with \(k={k_21}\) and some \(1\le j\le {d1}\). Then for \(k_3=k_2+j+1\le {k_2+d}\) one infers that \(y_{k_3}(d)\ne 0\), and all irregular blocks are of the types handled in (V.1) and (V.2).
Applying now the arguments of (V.1) and (V.2) for \(y_{k_3}\) instead of \(y_{k_1}\) we obtain that, for \(k_4=k_3+d+1\), \(y_{k_4}(i)\) is regular for all \(0\le i<d\). We claim that the last coordinate \(y_{k_4}(d)\) is also regular. Arguing by way of contradiction, if \(y_{k_4}(d)\) is irregular, then according to (II) there are two possibilities: either \(y_{k_41}(0)=y_{k_41}(d)=0\) or (3.7) holds for \(k=k_41\). In the former case \(y_{k_4}(d1)=y_{k_4}(d)=0\) are both irregular, which is a contradiction. If the latter holds, then \(y_{k_41}(1)=0\) is irregular and \(y_{k_41}(0)\ne 0\), thus from (I) we obtain that \(y_{k_3+1}(d)=y_{k_41(d1)}(d)=0\) is also irregular and \(y_{k_3+1}(d1)=y_{k_41}(0)\ne 0\). Thus, (3.7) must hold for \(k=k_3\), as well, which contradicts the fact that every irregular block of \(y_{k_3}\) is of the types seen in (V.1) and (V.2). This proves that \(y_{k_4}(d)\) is regular.
Thus, we proved that \(y_{k_4}\in {\mathcal {R^}}\) for some \(0\le k_4\le k+4d+{2}\). From (III) we obtain that \(y_{k+4d+{2}}\in {\mathcal {R^}}\). \(\square \)
Assertion (a) of Theorem 3.3 is also true, if we extend \(V^\pm \) into the origin as \(V^+(0)=0\) resp. \(V^(0)=1\). In this case we may also allow \(a_k\ge 0\) and \(\delta ^*b_k\ge 0\) instead of strict inequalities. The proof of this slightly modified statement is essentially identical to the one presented above.
Remark 3.4
With the modified assumptions, Theorem 3.3 (a) is a special case of a general result by MalletParet and Sell [17, Theorem 2.1]. Yet, there are two reasons for presenting an independent proof: First, our simpler situation allows a much shorter argument. Second, in the more general setting of [17] extra (smallness) conditions on the sequences \(a_k\) and \(b_k\) are necessary. As their motivation was to give the Lyapunov property for timediscretization of delaydifferential equations and tridiagonal ODEs, those extra conditions were always met for sufficiently small stepsizes. However, our motivation comes not only from timediscretization of continuoustime equations, therefore it is important that we can omit such extra conditions.
4 The Morse Decomposition
From now on, we suppose that the conditions (\(H_1\))–(\(H_3\)) are fulfilled. Furthermore, (1.2) is assumed to possess a global attractor \({\mathcal A}\).
If Open image in new window and Open image in new window , then (\(H_1\))–(\(H_3\)) imply \(a,\delta ^*b>0\).
Throughout this section we will treat the positive and negative feedback cases in a parallel way. Both statements and their proofs are formulated in this manner. The only exception in this regard is the proof of Proposition 4.2, whose positive feedback case is postponed to the Appendix.
Theorem 4.1
Note that due to definitions of \(V^\pm \) and \(N^*_\pm \), \({\mathcal M}_{N^*_\pm }\ne \{0\}\) can only occur if the origin is nonhyperbolic or \(N^*_\pm =d+1\), and d is odd (resp. even) in the positive (resp. negative) feedback case.
The proof of Theorem 4.1 is based on several tools, whose logical structure is borrowed from [19]. However, there are certain differences: For instance, the finite dimensional state space of (1.2) allows to simplify various arguments based on the Arzelà–Ascoli theorem. Another simplification in our finite dimensional setting is the nonexistence of superexponentially decaying solutions established in Lemma 4.5. On the flip side, the nonconnectedness of orbits gives rise to technical difficulties in some arguments.
The next two results play a significant role in the sequel and show how bounded solutions (on either of the halfaxes) can be characterized with the aid of \(N^*_\pm \) and functional \(V^\pm \). Although they correspond to continuous time results in [15, 19], we give a detailed proof.
Proposition 4.2
Proof
We present only the proof (a) in the negative feedback case, i.e. when (\(H_1\))–(\(H_3\)) with \(\delta ^*=1\) hold. The proof of (b) is analogous. The proof for the positive feedback case is also rather similar, but simpler (see Appendix), as there are less cases for the distribution of eigenvalues of (4.1) (cf. Lemma 2.1).
The eigenvalues of (4.1) are precisely the solutions of the characteristic equation (2.2). If \(M^*\) is equal to 0, 1 or \(d+1\), then the statement is trivial. Otherwise, according to Lemma 2.1, \(M^*\) is an even number, say \(M^*=2n'+2\), where \(0\le n'\le \frac{d2}{2}\).
As for the eigensolutions corresponding to real eigenvalues, let us first recall that determined by the value of b, either there are exactly two positive eigenvalues of (4.1) (\(\lambda _{+,2}\le \lambda _{+,1}\)), and in that case there is no eigenvalue in the sector \(S^_0\), or there are no positive eigenvalues and there exists a pair of complex eigenvalues \(\lambda _0,\overline{\lambda _0}\), where \(\lambda _0\in S^_0\). In the former case let \(z_{0}^1\), \(z_0^2\) and \(z_{0}^3\) be defined by \(z_{0,k}^1(j)=\lambda _{+,1}^{k+j}\), \(z_{0,k}^2(j)=\lambda _{+,2}^{k+j}\) and \(z_{0,k}^3(j)=(k+j)\lambda _{+,2}^{k+j}\) for all \(k\in {\mathbb {Z}}_\) and \(j\in {\mathbb {N}}_0,\ j\le d\), respectively. Clearly, for any \(i=1,2,3\) and for all \(k\in {\mathbb {Z}}_,\ k< d\), every component of \(z_{0,k}^i\) has the same sign, and in particular \(V^(z_{0,k}^i)=1\) for all \(k< d\). Concerning negative eigenvalues, if d is even, say \(d=2\ell \) for some \(\ell \in {\mathbb {N}}\), then any eigensolution corresponding to the unique negative eigenvalue Open image in new window can be written in the form \(c_\ell z_\ell \), where \(c_\ell \in {\mathbb {R}}\) and \(z_{\ell ,k}(j)=(r_\ell )^{k+j}\) for all \(k\in {\mathbb {Z}}_\) and \(j\in {\mathbb {N}}_0,\ j\le d\). Obviously \({{\mathrm{sc}}}z_{\ell ,k}=d\) and therefore also \(V^(z_{\ell ,k})=d+1=2\ell +1\) hold for all \(k\in {\mathbb {Z}}_\). According to Lemma 2.1 there are no other real eigenvalues.
We claim that \(V^(y_{k})\le V^(z_{n,k})=2n+1< 2n'+2=N^*_\) holds for all \(k\le k_1\), where \(k_1\) is to be defined soon. Then by monotonicity of \(V^\) (proved in Theorem 3.3) we get statement (a). If \(c_0,\dots , c_n\) are all zeros, then the claim trivially holds. Otherwise we may assume w.l.o.g. that \(c_n\ne 0\).
If \(c_0 \ne 0\), then there are no positive eigenvalues of (4.1), so the first sum in (4.4) is zero, and we readily obtain that for \(n=0\), \(V^(y_{k})= V^(z_{n,k})=1< 2n'+2=N^*_\) holds.
For brevity, let us say that the \(\ell \)th component of \(z_{n,k}\), i.e. \(z_{n,k}(\ell )=r_n^{k}\sin ((k+\ell )\varphi _n+ \omega _n)\) is small if \(\sin ((k+\ell )\varphi _n+ \omega _n)\le \sin \vartheta _n\), otherwise the component will be said to be big. Inequality (4.8) yields that if \(z_{n,k}(\ell )\) is big for some \(\ell =0,1,\ldots ,d\) and \(k\le k_1\), then \({\text {sgn}}y_k(\ell )={\text {sgn}}z_{n,k}(\ell )\ne 0\) holds. Note that inequality (4.5) guarantees that there is no \(k\le k_1\) and component \(0\le \ell <d\) such that \(z_{n,k}(\ell )\) and \(z_{n,k}(\ell +1)\) are both small. Similarly, (4.6)–(4.7) imply that \(z_{n,k}(0)\) and \(z_{n,k}(d)\) cannot be small at the same time. Moreover, by (4.5) one obtains that if \(z_{n,k}(\ell )\) is small for some \(k\le k_1\) and \(\ell \in \{1,\ldots , d1\}\), then \({\text {sgn}}z_{n,k}(\ell 1)={\text {sgn}}z_{n,k}(\ell +1)\).
Combining them means that, if \(k\le k_1\) and \(z_{n,k}(0)\) and \(z_{n,k}(d)\) are both big, \({{\mathrm{sc}}}y_k={{\mathrm{sc}}}z_{n,k}\) holds, and \(V^(y_k)=V^(z_{n,k})=2n+1<N^*_\) follows.
If the origin is nonhyperbolic, and \(M^*\ge 2\), then \({N^*_}=M^*+1\). Keeping our notations from above, there is a pair of eigenvalues \(\lambda _{n'+1},\overline{\lambda _{n'+1}}\) on the unit circle, or \(1\) is a simple eigenvalue (if d is even and \(M^*=d\)). In the latter case, statement (a) of the lemma is trivial. In the former case all backward bounded solutions of (4.2) can be written as (4.4), this time with \(n\le n'+1\), and an argument similar to the one applied in the hyperbolic case shows that \(V^(y_k)=2n+1\le 2n'+3=N^*_\) for all \(k\in {\mathbb {Z}}_\). \(\square \)
Proposition 4.3
 (a)
If \(y_k\in \overline{U}\cap {\mathcal A}\) for all \(k\le 0\), then \(V^\pm (y_k)\le N^*_\pm \) for all \(k\in {\mathbb {Z}}\).
 (b)
If \(y_k\in \overline{U}\cap {\mathcal A}\) for all \(k\ge 0\), then \(V^\pm (y_k)\ge N^*_\pm \) for all \(k\in {\mathbb {Z}}\).
Proof
The proof relies mainly on Proposition 4.2. We only prove statement (a), since the proof of part (b) is analogous.
We argue by way of contradiction. If the statement is not true, then there exists a sequence of nontrivial, bounded entire solutions \(y^n:{\mathbb {Z}}\rightarrow J^{d+1}\) of equation (1.2) such that \(\sup _{k\le 0}\Vert y_k^n\Vert \rightarrow 0\), as \(n\rightarrow \infty \) and \(V^\pm (y_{k_n}^n)>N^*_\pm \) holds for some \(k_n\in {\mathbb {Z}}\).
Since \(y^n\) is bounded, there exists an integer \(j_n\le k_n\), such that \(\Vert y^n_k\Vert <2\Vert y^n_{j_n}\Vert \) holds for all integers \(n\in {\mathbb {Z}}_+\) and \(k\le k_n\).
On the other hand, by our assumptions, for any fixed integer \(k\le 0\), \(y^{n_\ell }_k\rightarrow 0\), as \(\ell \rightarrow \infty \). Thus \(a^{n_\ell }_k\rightarrow a=D_1f(0,0)\) and \(b^{n_\ell }_k\rightarrow b=D_2f(0,0)\) hold as \(\ell \rightarrow \infty \).
From the Proposition 4.3 we immediately infer the next result.
Corollary 4.4
 (a)
If \(N^*_\pm >1\), then there exists no heteroclinic solution towards the trivial solution.
 (b)
If there exists a homoclinic solution \((\tilde{y}_k)_{k\in {\mathbb {Z}}}\) to the trivial equilibrium, then \(V^\pm (\tilde{y}_k)\equiv N^*_\pm \) on \({\mathbb {Z}}\).
 (c)
In particular, if the trivial solution is hyperbolic, then there exists no homoclinic solution to it in the positive feedback case, while in case of negative feedback it may only exist if \(M^*=N^*_=1\).
Proof
 (a)
Fixedpoints \(y^*\ne 0\) of (1.2) have equal, nonzero components, so \(V^\pm (y^*)\le 1\) and \(y^*\in \mathcal {R^\pm }\). Assume now to the contrary that \((\tilde{y}_k)_{k\in {\mathbb {Z}}}\) is an entire solution with \(\lim _{k\rightarrow \infty }\tilde{y}_k=0\) and \(\lim _{k\rightarrow \infty }\tilde{y}_k=y^*\). Then from Proposition 3.2 (b) we get \(\lim _{k\rightarrow \infty }V^\pm (\tilde{y}_k)\le 1\), while Proposition 4.3 (b) yields \(\lim _{k\rightarrow \infty }V^\pm (\tilde{y}_k)\ge N^*_\pm >1\), a contradiction to Proposition 3.3 (a).
 (b)
 (c)
Let the trivial solution be hyperbolic. If \(M^*\) equals 0 or \(d+1\), then the statement holds trivially since either the local unstable or the local stable manifold is trivial. Otherwise, except the case of negative feedback with \(M^*=N^*_=1\), \(N^*_\pm =M^*\) is an odd (resp. even) number in the positive (resp. negative) feedback case. Now, recall that \(V^\pm \) takes on only even (resp. odd) values, then apply (b) to conclude the proof.\(\square \)
The next technical lemma shows that solutions of (1.2) can neither grow nor decrease faster than exponentially in some neighborhood of the origin.
Lemma 4.5
Proof
Remark 4.6
From the above proof and the \(C^1\)smoothness of f it is clear that for any solution \((y_k)_{k\in {\mathbb {I}}}\) of (3.2), (3.4) on some discrete interval \({\mathbb {I}}\) and any compact set \(K\subset {\mathbb {R}}^{d+1}\), there exists \(C>0\), such that \(y_k\in K\) for some \(k\in {\mathbb {I}}\) implies \(\Vert y_{k+1}\Vert \le C \Vert y_k\Vert \). Moreover, if K is a sufficiently small neighborhood of 0, then there exists also \(c>0\) such that \(c\Vert y_k\Vert \le \Vert y_{k+1}\Vert \).
We combine the next three lemmas with Proposition 4.3 in order to prove that the family of sets \({\mathcal M}_n\) fulfills the “Morse properties” (\(m_1\)) and (\(m_2\)).
Lemma 4.7
 (a)
If \(\lim _{k\rightarrow \infty }V^\pm (y_k)=N\), then \(V^\pm (\eta )=N\) for any \(\eta \in \omega (\xi )\setminus \{0\}\).
 (b)
If \(\lim _{k\rightarrow \infty }V^\pm (y_k)=K\), then \(V^\pm (\eta )=K\) for any \(\eta \in \alpha (y)\setminus \{0\}\).
Proof
We only prove statement (a), since (b) can be shown analogously.
For an arbitrary \(\eta \in \omega (\xi )\setminus \{0\}\), there exist a monotone sequence \(k_n\rightarrow \infty \), as \(n \rightarrow \infty \), such that \(y_{k_n}\rightarrow \eta \), and – by the invariance of \(\omega (\xi )\setminus \{0\}\) – there exists a bounded entire solution \((z_k)\) of (1.2) such that \(z_0=\eta \) and \(z_k \in \omega (\xi )\setminus \{0\}\) holds for all \(k\in {\mathbb {Z}}\). By Theorem 3.3 (b), there exist integers \(\ell _1<0 < \ell _2\) such that \(z_{\ell _1}\) and \(z_{\ell _2}\) both belong to \(\mathcal {R^\pm }\). Since \(z_{\ell _1}\in \omega (\xi )\setminus \{0\}\), there exists a monotone sequence \(k'_n\rightarrow \infty \), as \(n \rightarrow \infty \), such that \(y_{k'_n}\rightarrow z_{\ell _1}\), and from the continuity of the righthand side of equation (1.2) one infers that \(y_{k_n+\ell _2}\rightarrow z_{\ell _2}\), as \(n \rightarrow \infty \).
Lemma 4.8
 (a)
If \(\lim _{k\rightarrow \infty }V^\pm (y_k)\ne N^*_\pm \), then either \(\omega (\xi )=\{0\}\) or \(0 \notin \omega (\xi )\).
 (b)
If \(\lim _{k\rightarrow \infty }V^\pm (y_k)\ne N^*_\pm \), then either \(\alpha (y)=\{0\}\) or \(0 \notin \alpha (y)\).
Proof
Lemma 4.9
 (a)
If Open image in new window , then either \(\omega (\xi )=\{0\}\) or \(\omega (\xi )\subseteq {\mathcal M}_N\).
 (b)
If Open image in new window , then either \(\alpha (y)=\{0\}\) or \(\alpha (y)\subseteq {\mathcal M}_K\).
Proof
In order to prove assertion (a), let us assume that \(\omega (\xi )\ne \{0\}\) and that Open image in new window \(V^\pm (y_k)\ne N^*_\pm \). According to Lemma 4.8, \(0\notin \omega (\xi )\). Let \(\eta \in \omega (\xi )\) be arbitrarily chosen, then using the invariance of \(\omega (\xi )\), there exists a bounded entire solution \((z_k)\), such that \(z_0=\eta \) and \(z_k\in \omega (\xi )\) holds for all \(k\in {\mathbb {Z}}\). From compactness of \(\omega (\xi )\), \(\alpha (z)\cup \omega (\eta )\subseteq \omega (\xi )\) follows. This implies that \(0\notin \alpha (z)\cup \omega (\eta )\). On the other hand, \(z_k\in \omega (\xi )\) and Lemma 4.7 together ensure that \(V^\pm (z_k)=N\) for all \(k\in {\mathbb {Z}}\), meaning that \(\eta \in {\mathcal M}_N\) holds.
The proof of statement (b) is analogous. \(\square \)
The next lemma is a step towards the compactness proof of Morse sets.
Lemma 4.10
For every \(N\in 2{\mathbb {N}}_0\setminus N^*_+\) (resp. \(N\in (2{\mathbb {N}}_0+1)\setminus N^*_\)) there exists an open neighborhood \(\tilde{U}\) in \({\mathcal A}\) of the origin, such that \({\mathcal M}_N\cap \tilde{U}=\emptyset \).
Proof
Assume to the contrary that there is a nonnegative even (resp. odd) \(N\ne N^*_\pm \) and a sequence \((\xi ^n)_{n\ge 0}\) in \({\mathcal M}_N\) such that \(\xi ^n\rightarrow 0\) as \(n\rightarrow \infty \). For each \(n \ge 0\) let \(y^n\) be a bounded entire solution such that \(y^n_0=\xi ^n\), \(0\notin \alpha (y^n)\cup \omega (\xi ^n)\) and \(V^\pm (y^n_k)=N\) hold for all \(k\in {\mathbb {Z}}\).
We claim that \(\omega (\xi ^n)\) cannot be a subset of \(\overline{U}\), where U is from Proposition 4.3. To see this, suppose that \(\omega (\xi ^n)\subseteq \overline{U}\). Let \(\eta \in \omega (\xi ^n)\). By invariance of \(\omega (\xi ^n)\) there exist a bounded entire solution \((z_k)\) through \(\xi ^n\), such that \(z_k\in \omega (\xi ^n)\subseteq \overline{U}\) holds for all \(k\in {\mathbb {Z}}\). Thus Proposition 4.3 implies that \(V^\pm (z_k)=N^*_\pm \) holds for all \(k\in {\mathbb {Z}}\). However, \(V^\pm (\eta )=N\) should also hold by Lemma 4.8, which is a contradiction, so \(\omega (\xi ^n)\not \subseteq \overline{U}\) for all \(n\ge 0\).
The proof for the case \(N<N^*_\pm \) is analogous. \(\square \)
Lemma 4.11
The set \({\mathcal M}_N\) is closed for all \(N\in 2{\mathbb {N}}_0\cup \{N^*_+\}\) (resp. all \(N\in (2{\mathbb {N}}_0+1)\cup \{N^*_\}\)).
Proof
Let \((\xi ^n)_{n\ge 0}\) be a sequence in \({\mathcal M}_N\) for some \(N\in 2{\mathbb {N}}_0 \cup \{N^*_+\}\) (resp. \(N\in (2{\mathbb {N}}_0+1)\cup \{N^*_\}\)), such that \(\lim _{n \rightarrow \infty }\xi ^n=\xi \) for some \(\xi \in {\mathcal A}\). We claim that \(\xi \in {\mathcal M}_N\).
If \(\xi =0\), then Lemma 4.10 yields that \(N=N^*_\pm \). By definition of \({\mathcal M}_{N^*_\pm }\), \(0\in {\mathcal M}_{N^*_\pm }\) also holds, so the statement is proved in this particular case.
Assume now that \(\xi \ne 0\). By definition of \({\mathcal M}_N\) and using \(\xi ^n\in {\mathcal M}_N\), there exist bounded entire solutions \((y^n_k)\) of (1.2), such that \(y^n_0=\xi ^n\), \(y^n_k\in {\mathcal M}_N\) and \(V^\pm (y^n_k)=N\) hold for all \(n\ge 0\) and \(k\in {\mathbb {Z}}\). The compactness of \({\mathcal A}\) and a Cantor diagonalization argument leads to a subsequence \((y^{n_\ell })_{\ell \in {\mathbb {N}}}\), and a solution \((y_k)_{k\in {\mathbb {Z}}}\) of (1.2), so that \(y^{n_\ell }_k \rightarrow y_k\) holds for every \(k\in {\mathbb {Z}}\), as \(\ell \rightarrow \infty \).
Now, we are finally in the position to prove our main theorem.
Proof of Theorem 4.1
By definition, the sets \({\mathcal M}_n\) are pairwise disjoint and invariant. Since \({\mathcal A}\) is compact, Lemma 4.11 yields that \({\mathcal M}_n\) is compact for all possible n.
It remains to prove that these sets fulfill the “Morse properties” (\(m_1\)) and (\(m_2\)), i.e. for all \(\xi \in {\mathcal A}\) and any bounded entire solution \((y_k)_{k\in {\mathbb {Z}}}\) for which \(y_0=\xi \) holds, there exist \(i\ge j\) with \(\alpha (y)\subseteq {\mathcal M}_i\) and \(\omega (\xi )\subseteq {\mathcal M}_j\), and in case \(i=j\), then \(\xi \in {\mathcal M}_i\) (thus, \(y_k\in {\mathcal M}_i\) for every \(k\in {\mathbb {Z}}\)).
First, observe that if \(N=N^*_\pm \), then \(\omega (\xi )\subseteq {\mathcal M}_{N^*_\pm }\). In order to prove this, choose \(\eta \in \omega (\xi )\) arbitrarily. If \(\eta =0\), then \(\eta \in {\mathcal M}_{N^*_\pm }\) holds by definition, thus we may assume now that \(\eta \ne 0\). By Lemma 4.7 we obtain that \(V^\pm (\eta )=N^*_\pm \). Moreover, by the invariance of \(\omega (\xi )\setminus \{0\}\) there exists an entire solution \((z_k)\) in \(\omega (\xi )\setminus \{0\}\), such that \(z_0=\eta \). Thus Lemma 4.7 yields that \(V^\pm (z_k)=N^*_\pm \) for all \(k\in {\mathbb {Z}}\), meaning that \(\eta \in {\mathcal M}_{N^*_\pm }\) holds.
A similar argument can be applied to prove that \(K=N^*_\pm \) implies that \(\alpha (y)\subseteq {\mathcal M}_{N^*_\pm }\) holds.
We will distinguish four cases in terms of the values of N and K.
Case 1 If \(N=K=N^*_\pm \), then \(\alpha (y)\cup \omega (\xi ) \subseteq {\mathcal M}_{N^*_\pm }\) holds by our previous observation. Moreover, from the monotonicity of \(V^\pm \) it follows that \(V^\pm (y_k)\equiv N^*_\pm \) on \({\mathbb {Z}}\). This implies \(\xi \in {\mathcal M}_{N^*_\pm }\), thus both (\(m_1\)) and (\(m_2\)) hold.
Case 2\(N=N^*_\pm < K\). As already shown, \(\omega (\xi )\subseteq {\mathcal M}_{N^*_\pm }\) holds in this case.
Moreover, observe that \(\alpha (y)\ne \{0\}\). Otherwise Proposition 4.3 would imply \(V^\pm (y_k)\le N^*_\pm \) for \(k\in {\mathbb {Z}}\), and thus \(K\le N^*_\pm =N\), which is impossible.
Therefore, Lemma 4.9 can be applied to obtain \(\alpha (y)\subseteq {\mathcal M}_K\), so property (\(m_1\)) is fulfilled. Note that (\(m_2\)) holds automatically, since the two Morse sets in question, i.e. \({\mathcal M}_{N^*_\pm }\) and \({\mathcal M}_K\), are different.
Case 3 A similar argument applies in the case when \(N<N^*_\pm =K\).
Case 4 If \(N\ne N^*_\pm \ne K\), then Lemma 4.9 yields that either \(\omega (\xi )=\{0\}\) or \(\omega (\xi )\subseteq {\mathcal M}_N\). Similarly, either \(\alpha (y)=\{0\}\) or \(\alpha (y)\subseteq {\mathcal M}_K\) holds. Note that \(\omega (\xi )\) and \(\alpha (y)\) cannot be both \(\{0\}\) in this case, because then Proposition 4.3 would imply \(V^\pm (y_k)\equiv N^*_\pm \) on \({\mathbb {Z}}\), contradicting assumption \(N\ne N^*_\pm \ne K\).
If \(\omega (\xi )\ne \{0\} \ne \alpha (y)\), then Lemma 4.9 yields that \(\omega (\xi )\subseteq {\mathcal M}_N\) and \(\alpha (y)\subseteq {\mathcal M}_K\) hold, so (\(m_1\)) is fulfilled. If \(K=N\), then their definition imply that \(V^\pm (y_k)=N=K\) for all \(k \in {\mathbb {Z}}\). Since the limit sets are now also assumed to be nontrivial, thus Lemma 4.8 ensures that \(0\notin \alpha (y)\cup \omega (\xi )\), thus \(y_k\in {\mathcal M}_N\) also holds for all \(k\in {\mathbb {Z}}\). This establishes property (\(m_2\)).
If \(\omega (\xi )=\{0\}\ne \alpha (y)\), then \(\omega (\xi )\subseteq {\mathcal M}_{N^*_\pm }\) holds by definition. Furthermore, Proposition 4.3 implies that \(V^\pm (y_k)\ge N^*_\pm \) holds for all \(k\in {\mathbb {Z}}\), and consequently \(N^*_\pm < N\le K\). On the other hand Lemma 4.9 yields that \(\alpha (y)\subseteq {\mathcal M}_K\), so (\(m_1\)) holds. Property (\(m_2\)) is fulfilled automatically.
An analogous argument applies for the case when \(\omega (\xi )\ne \{0\}=\alpha (y)\). We have taken all possible cases into consideration, so our proof is complete. \(\square \)
5 Applications
Since the existence of a global attractor is assumed in Theorem 4.1, we begin with a condition for dissipativity:
Lemma 5.1
 (i)
There exist reals \(a\in (0,1)\), \(K\ge 1\) with \(\left\ A^k\right\ \le Ka^k\) for all \(k\in {\mathbb {Z}}_+\),
 (ii)
there exist reals \(\beta _0,\beta _1\ge 0\) so that \(\left\ H(y)\right\ \le \beta _0+\beta _1\left\ y\right\ \) for all \(y\in {\mathbb {R}}^d\).
Proof
5.1 Life Sciences
In the literature often sufficient conditions for global asymptotic stability of \(x_*\) are given (cf., for example [11, 12]). In this case, only one Morse set \({\mathcal M}_{N^*_\pm }\) is obtained from Theorem 4.1. Let us consequently list several dissipative difference equations that fulfill the assumptions of our main theorem, i.e. (\(H_1\))–(\(H_3\)). Violating the conditions for global asymptotic stability might indicate bifurcations which lead to more complex dynamics and possibly Morse sets.
Throughout, suppose that \(\alpha \in (0,1)\), \(\beta >0\).
5.1.1 ClarkType Models
For bounded functions h one obtains the following result on dissipativity.
Proposition 5.2
(Dissipativity for (5.3), cf. [20]) If there exists a \(K^+\ge 0\) such that \(h(x)\le K^+\) for all \(x\in {\mathbb {R}}_+\), then (5.3) is dissipative and \(A=[0,R]\) is absorbing for every \(R>\tfrac{K^+}{1\alpha }\).
h  \(x_*\)  \(R_0\)  GAS 

(5.4)    \(\tfrac{\beta }{1\alpha }\)  \(p\le 1\) 
(5.6) for \(p<1\)  \(\left( \tfrac{\alpha +\beta 1}{1\alpha }\right) ^{1/p}\)  \(\tfrac{\tfrac{\beta \eta }{1+\eta ^p}\beta _1\eta }{\alpha ^d(1\alpha )\beta _1}\)  \(1\alpha <\beta \) 
(5.6) for \(p=1\)  \(\tfrac{\alpha +\beta 1}{1\alpha }\)  \(\tfrac{\beta }{1\alpha }\)  \(1\alpha <\beta \) 
(5.7)  \(W(\tfrac{\beta }{1\alpha })\)  \(\tfrac{\beta }{1\alpha }\)  \(\tfrac{\beta }{1\alpha }\le e\) 
Example 5.3
Our next example requires some dissipativity criterion which applies even though h is unbounded.
Proposition 5.4
Proof
Example 5.5

For \(p<1\) choose \(\beta _1\in (0,\alpha ^d(1\alpha ))\) and from \(h'(y)=\frac{\beta (1+(1p)y^p)}{(1+y^p)^2}\rightarrow 0\) as \(y\rightarrow \infty \) there is an \(\eta >0\) such that \( h'(\eta )=\beta _1\). Hence, (5.5) holds with \(\beta _0=h(\eta )\beta _1\eta >0\) and Proposition 5.4 yields that [0, R] is absorbing for \(R>\tfrac{h(\eta )\beta _1\eta }{\alpha ^d(1\alpha )\beta _1}\).

For \(p=1\) the bounded function \(y\mapsto \frac{\beta y}{1+y}\) is strictly increasing to \(\beta \), we derive from Proposition 5.2 that [0, R] is absorbing for \(R>\frac{\beta }{1\alpha }\).
A series of further sufficient conditions guaranteeing global asymptotic stability of \(x^*>0\) is given in [3, Theorem 4.1], which for instance address the situation \(p>1\).
Example 5.6
Additional conditions for global asymptotic stability of \(x^*>0\), depending on the delay d though, can be found in [3, Theorem 4.5].
5.1.2 Models of HutchinsonType
Example 5.7
Example 5.8
Example 5.9
(Nonempty Morse sets) Let \(d=mp\) for some \(m,p\in {\mathbb {N}}\), \(p\ge 2\). We utilize the simple observation that \(\tilde{x}^p\) is a pperiodic solution of equation (1.1) if and only if it is a pperiodic solution of the corresponding undelayed equation \(x_{k+1}=f(x_k,x_k)\).
It is well known that the undelayed Ricker equation \(x_{k+1}=x_k e^{\beta x_k}\) admits a 2periodic solution if and only if \(\beta >2\), and that this solution oscillates about \(x_*=\beta \) (see [23, Proposition 3]). Now, let us fix \(\beta >2\). Then there exists \((\tilde{x}^2_k)_{k\in {\mathbb {Z}}}\), a 2periodic solution of (5.10) for any even d. Thus the corresponding first order, \(d+1\) dimensional equation (1.2) with (5.11) has a 2periodic solution \((\tilde{y}^2_k)_{k\in {\mathbb {Z}}}\), moreover, from the oscillation of \((\tilde{x}^2_k)_{k\in {\mathbb {Z}}}\) follows immediately that \(V^(\tilde{y}^2_k)\equiv d+1\) holds on \({\mathbb {Z}}\). In particular, the corresponding Morse set \({\mathcal M}_{d+1}\) contains a 2periodic orbit. Note that in this case \(d+1=N^*_\) holds, meaning that \({\mathcal M}_{N^*_}\) is nontrivial.
Now assume that \(\beta >1+\ln 9\) and let \(d=3m\) for some \(m\in {\mathbb {N}}\). Then according to [23, Proposition 5] and the comment subsequent to it, there exists a 3periodic solution \((\tilde{x}^3_k)_{k\in {\mathbb {Z}}}\) of \(x_{k+1}=x_k e^{\beta x_k}\) (by Šarkovs\({}^\prime \)kiĭ’s Theorem [26], also nperiodic solutions exist for any \(n\in {\mathbb {N}}\)). It is easy to see that \((\tilde{x}^3_k)_{k\in {\mathbb {Z}}}\) oscillates about \(x_*=\beta \). Then one readily obtains that the corresponding \(d+1\) dimensional equation (1.2) with (5.11) has a 3periodic solution \((\tilde{y}^3_k)_{k\in {\mathbb {Z}}}\), for which \(V^(\tilde{y}^3_k)\equiv 2m+1\) holds on \({\mathbb {Z}}\), and thus the corresponding Morse set \({\mathcal M}_{2m+1}\) contains a 3periodic orbit.
As a consequence, if \(\beta >1+\ln 9\) and \(d=6n\) for some \(n\in {\mathbb {N}}\), then the Morse set \({\mathcal M}_{6n+1}={\mathcal M}_{N^*_}\) is nontrivial and \({\mathcal M}_{4n+1}\) is nonempty for the \(d+1\) dimensional Ricker map (1.2) with (5.11).
Note that although one can similarly find oscillatory nperiodic solutions for \(nm+1\) dimensional Ricker maps with \(\beta >1+\ln 9\), it is nontrivial to determine the number of sign changes they have on one period, and therefore it is not clear in which Morse set they are contained.
The above example is closely related to the topic of the first open problem presented in Sect. 6.
5.2 Discretizations
Proposition 5.10
(Dissipativity, cf. [20]) If there exist \(K^,K^+\ge 0\) such that \(K^\le h(x)\le K^+\) for \(x\in {\mathbb {R}}\), then (5.14) is dissipative and every \(A=[R^,R^+]\) with \(R^+>K^+\) and \(R^<K^\) is an absorbing set.
As a prototype for equation (5.14) we study the following example.
Example 5.11
As a closing remark we point out that the existence of the global attractor for (5.3) (and therefore for Examples 5.4, 5.7 and 5.11), as well as for Examples 5.9 and 5.10 follows also from [5, Theorem 3.1].
6 Perspectives

Which Morse sets are nonempty? First note that the Morse set \({\mathcal M}_{N^*_\pm }\) is obviously nonempty, as the trivial solution is contained in it. In Example 5.9 we showed one possible method which can provide nonempty Morse sets different from \({\mathcal M}_{N^*_\pm }\). However it seems a challenging problem whether it can be shown (maybe under further assumptions on (1.1)) that \({\mathcal M}_n\ne \emptyset \) holds for all \(n<N^*_\pm \), as it was proved by MalletParet in the continuous time case with negative feedback [15].

It would be interesting to obtain a Morse decomposition in the general setting of [17] and in particular for tridiagonal equations.

Finally, what can be said in the nonautonomous case?
Notes
Acknowledgements
Open access funding provided by University of Klagenfurt. The authors would like to sincerely thank the late Professor George R. Sell for suggesting this research. We are most grateful to Professor Eduardo Liz for various hints to the literature.
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