Existence of chaos for partial difference equations via tangent and cotangent functions

Picture a cartographer drawing a coastline who keeps zooming in. Instead of smoothing out, the edge just becomes more jagged, more folded, endlessly complex. That's what chaos looks like mathematically — structure that never settles, no matter how closely you look. Haihong Guo and Wei Liang just found a way to manufacture it on demand inside a class of equations that most people thought were too structured to misbehave. The equations in question are partial difference equations — systems that evolve in both discrete time and discrete space, indexed by a time step n and a lattice index m. Think of a grid where each point gets updated at every tick according to its neighbors. These systems show up in imaging, digital filters, and spatial modeling. They're not exotic. And yet proving that a specific one is chaotic — rigorously, not just visually — is hard. Guo and Liang wanted to go further: not just identify chaos where it already exists, but deliberately inject it into systems that start out tame. To do that, they needed a precise definition of what they were aiming for. They use two definitions, and they are worth keeping straight. Li–Yorke chaos requires the existence of an uncountable scrambled set — an uncountable collection of points such that every distinct pair of trajectories starting there neither converges together nor flies apart to infinity, but stays perpetually entangled in complicated relative motion. Devaney chaos is stricter: the map must be topologically transitive, meaning orbits can wander anywhere in the region, and periodic points must be dense throughout it. Guo and Liang note that for continuous maps on infinite sets, transitivity and dense periodic points together imply sensitive dependence on initial conditions — the famous butterfly effect property. Devaney chaos is the stronger condition. Both appear in this paper, and which schemes achieve which is one of the main results. Now, the technical bridge that makes the whole approach work. A partial difference equation defined on a spatial grid with k plus one points and a specified boundary condition can be rewritten as an ordinary map on the space of R to the power of k plus one. Guo and Liang call this the induced system, and they define the original partial difference equation as chaotic precisely when its induced map is chaotic. That's the reduction step — it converts a spatially extended problem into something you can analyze with finite-dimensional tools. The key tool is what the paper calls coupled-expanding maps. Imagine a map acting on a domain that contains several compact subsets. The map is coupled-expanding if each subset gets stretched and folded so that its image covers all the other subsets. The strictly coupled-expanding version adds a geometric gap: every pair of these subsets must be separated by a positive distance. Two lemmas then do the heavy lifting. Lemma five says that a continuous, strictly coupled-expanding map on a finite family of disjoint compact sets is Li–Yorke chaotic. Lemma six goes further: if you add a quantitative expansion condition — specifically, that the control parameter epsilon minus a contraction bound L times the larger of one and lambda exceeds one — then the map has an invariant Cantor set. On that Cantor set, the dynamics is topologically conjugate to a full symbolic shift. That means you get both Devaney and Li–Yorke chaos simultaneously, living on a fractal subset of the state space. The Cantor set is the precise mathematical address of the chaos. So why use tangent and cotangent as controllers? The choice is geometrically motivated. Both functions have vertical asymptotes and oscillate with unbounded slopes near those asymptotes. That steep, divergent behavior is exactly what you need to create the stretching required for coupled expansion. You take a base system that may be perfectly well-behaved, add a tangent or cotangent term scaled by a parameter epsilon, and for epsilon large enough, the induced map becomes strictly coupled-expanding on a pair of disjoint compact sets. The chaos is provably manufactured by the controller. Guo and Liang construct four distinct chaotification schemes. They differ in whether the controller uses tangent or cotangent, and in which boundary conditions govern the spatial grid. Not all four deliver the same strength of chaos. Two of the four schemes produce Li–Yorke chaos alone. The other two produce chaos in both the Li–Yorke and Devaney senses, because they meet the stronger expansion condition required by Lemma six. The threshold for Theorem one, built on tangent, requires the spatial parameter r to exceed five pi over four. For any epsilon greater than seventeen pi over sixteen, you get a Cantor set carrying Devaney chaos. Theorems two through four carry analogous thresholds — seven pi over four for Theorem two, three pi over two for Theorem four — each specifying the epsilon value at which the controller becomes strong enough to guarantee the invariant Cantor set. The generality here matters. These results hold for a broad class of base systems, not one handcrafted special case. As long as the local update function satisfies a Lipschitz-type bound — the sum of its partial derivatives bounded by L on a specified domain — the chaotification theorems apply. Then the examples, which is where you actually see this work. Guo and Liang provide three examples. In Example one, the base system has parameters r equal to four, L equal to one quarter, and lambda equal to one half, with the boundary value f of zero comma zero equal to pi over two. Theorem one applies for any epsilon exceeding seventeen pi over sixteen. The authors set epsilon to nine pi over eight, slightly above that threshold, and plot the resulting orbits in both two and three dimensions for spatial sizes k equal to one and two. What they report are complicated dynamical behaviors — trajectories that look nothing like the orderly evolution of the uncontrolled system. Example two uses a different base system with L equal to one and lambda equal to four thirds, giving a threshold of seven pi over four. Setting epsilon to two pi, the controlled system again produces highly irregular orbits on a Cantor set contained in a union of two hypercubes in the state space. Example three uses periodic boundary conditions rather than the fixed-point conditions of the earlier schemes, and for epsilon equal to two pi the cotangent controller again induces chaos verified by both simulation and the proof of Theorem four. Three different setups, three different parameter regimes — and in each case, the controller switch turns order into verified, provable chaos. To appreciate the novelty, it helps to see where this sits in the literature. Earlier work by Chen and Liu proved that a version of this class of partial difference equations in three-dimensional space is Li–Yorke chaotic by constructing spatial periodic orbits directly. Shi reformulated the same type of equation as a discrete system and applied snap-back repeller theory. More recent schemes introduced sawtooth functions, modular operations, sine, and cosine as controllers. Into that lineage, Guo and Liang bring tangent and cotangent — functions whose asymptotic behavior generates the expansion conditions more cleanly than bounded trigonometric functions like sine and cosine, and more analytically than the piecewise sawtooth. The distinction between Devaney and Li–Yorke chaos also advances the literature in a specific way. Not every chaotification scheme is powerful enough to produce Devaney chaos, which requires the map to be topologically transitive — orbits genuinely mix throughout the chaotic region — and to have dense periodic points. The fact that two of the four schemes achieve this stronger condition, while the other two achieve only Li–Yorke chaos, tells you something precise about how the controller's geometry interacts with the structure of the invariant set. Chaotification as a field isn't purely abstract. Controlled chaos has genuine applications in cryptography, secure communications, and the modeling of complex physical systems where irregular, unpredictable behavior is a design goal rather than a defect. The extension from one-dimensional difference equations to spatially extended partial difference equations is a meaningful step up in complexity — the state space grows with the lattice size, and the induced map carries the full geometry of the grid. The tangent and cotangent trick generalizes naturally to other trigonometric functions with asymptotes, suggesting a broader family of controllers waiting to be characterized. Guo and Liang have built a constructive, provable method for injecting chaos into a wide class of structured systems — and the elegance of the approach is that the chaos lives on a Cantor set you can specify in advance. This lecture was created by ennepō. Go to https://ennepo.ai to Discover, Create and Follow the latest research in your field. Read when you can. Listen when you want to.