This paper is devoted to the study of the global dynamics of the scalar difference equations \[ x_{k+1} = f (x_{k} , x_{k-d} ) \tag{1}\] involving an arbitrary delay \(d \in \mathbb{N}\), where \(f\) fulfills a suitable positive or negative feedback condition in the second variable. Such problems are of interest not only as time-discretizations of delay-differential equations, but they also intrinsically arise in a multitude of models in the life sciences. Particularly in the latter applications, much work so far concentrated on the problem to provide (sufficient) conditions for the global asymptotic stability of a positive equilibrium of (1). Nevertheless, being a discrete-time model it is no surprising that much more complicated dynamics can be expected. Under quite natural and frequently met assumptions the delay-difference equations (1) are dissipative. This means that their forward dynamics eventually enters a bounded subset of the state space. Whence, a global attractor \(\mathcal{A}\) exists, which contains all bounded entire solutions and therefore all dynamically relevant objects, like for instance equilibria, periodic solutions, as well as homo- or heteroclinics. The dynamics on the attractor itself can be highly nontrivial due to, e.g., a cascade of period doubling bifurcations and even chaotic dynamics might arise. Beyond the pure existence of a global attractor, one is rather interested in an as detailed as possible picture of its interior structure. An adequate tool for this endeavor is a Morse decomposition of \(\mathcal{A}\) due to the following reasons:

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

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

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outside the Morse sets the dynamics of (1) on \(\mathcal{A}\) is gradient-like.

On the one hand, identifying a nontrivial Morse decomposition of the attractor (along with the connecting orbits) provides a more detailed picture of the long-term behavior of (1), since every solution is attracted by exactly one Morse set. On the other hand, obtaining a Morse decomposition is a difficult task and requires further tools. In our case, this is an integer-valued (or discrete) Lyapunov functional, which roughly speaking counts the number of sign changes and decreases along solutions. This allows to quantize solutions to (1) in terms of their oscillation rates. Such a concept is not new and actually turned out to be very fruitful to understand the global behavior of other finite and infinite dimensional dynamical systems. For instance, discrete Lyapunov functionals are used to obtain convergence to equilibria in tridiagonal ODEs and scalar parabolic equations, but also a Poincaré-Bendixson theory for a class of ODEs in \(\mathbb{R}^{n} , n > 2\), reaction-diffusion equations and delay-differential equations. In conclusion, the existence of such a discrete Lyapunov functional imposes a serious constraint on the possible long-term behavior of various systems. 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 J. Mallet-Paret and G. R. Sell [J. Differ. Equations 125, No. 2, 441–489 (1996; Zbl 0849.34056)].
This paper is (probably) the first contribution using a discrete Lyapunov functional to actually understand the dynamics of discrete-time models.