A mathematical preprint reports that some linear maps on the two-dimensional real plane meet the conditions for Devaney chaos when they are examined under a weak topology. In this paper, Devaney chaos is the combined classification built from transitivity, sensitivity and dense periodic points. The study asks how those properties behave under a topology generated by a family of seminorms, and how the answers compare with the strong version of the topology.
This is a theorem-driven study, not an experiment. Its unit of analysis is a matrix map that applies a two-by-two real matrix to a state in two-dimensional real space. There is no empirical participant sample or observational dataset. The comparisons are between specified matrix forms, their eigenvalue regimes and the weak or strong properties they satisfy.
Real eigenvalues split the answers
For a diagonal matrix with real eigenvalues, weak transitivity depends on whether the two eigenvalues are equal. When they are unequal, the paper reports weak transitivity for the specified linear-functional case in which the two displayed coordinate terms have a nonzero product. Equal eigenvalues do not meet that result, and strong transitivity is absent for every eigenvalue choice.
Jordan-block matrices follow another pattern. The paper states weak transitivity for its specified nonzero-functional case for every value of the scalar eigenvalue, while strong transitivity fails for every value.
With conjugate complex eigenvalues, weak and strong transitivity are both stated to hold outside an exceptional set of rotation angles, while weak transitivity fails on that set. The supplied extraction does not render the exceptional set reliably, so its exact angles need checking. For this class, the paper states that strong transitivity and weak transitivity are equivalent.
Periodic points bring a sharper dividing line
Periodic points are states that return after repeated applications of the map. For diagonal real matrices, the classification turns on whether each eigenvalue's modulus, or absolute value, equals one. When both moduli equal one, all points are periodic. In either mixed case, the coordinate family tied to the unit-modulus eigenvalue is periodic and strong density is stated. When both moduli differ from one, no nonzero periodic points remain and the periodic points are not weakly dense.
For the rotation-scaling form used for conjugate complex eigenvalues, a rational rotation angle together with unit scaling is stated to make all points periodic and strongly dense. If the angle is irrational, or the scaling factor differs from one, only the zero point is periodic and weak density is absent.
Across the cases, strong density of periodic points is stated to be equivalent to weak density.
The supplied analysis flags a separate uncertainty in the mixed diagonal cases: their density claims are not fully reconciled with the functional conditions used in the definition.
The Jordan-block result comes with a serious qualification. Theorem 4.2 states that unit scalar modulus gives no nonzero periodic points and no weakly dense periodic points, while nonunit modulus gives a displayed periodic-point family and strong density. But the supplied analysis reports that the proof reverses those two case labels. The periodic-point result is therefore internally inconsistent, and the Jordan conclusions that depend on it require correction or rechecking.
Sensitivity does not simply track transitivity
Sensitivity asks whether nearby starting states can eventually be separated by repeated applications of the map. For diagonal matrices, strong sensitivity is reported only when both eigenvalue moduli exceed one. Weak sensitivity is absent when the eigenvalues are equal and their common modulus is at most one, and is stated to hold otherwise.
For Jordan blocks, strong sensitivity is reported only when the scalar eigenvalue modulus exceeds one. Weak sensitivity has a broader condition: it holds whenever the scalar is nonzero, and fails when the scalar equals zero.
Complex eigenvalues again make the strong and weak tests line up. Strong sensitivity is stated to hold outside the exceptional angle set and also on that set when the scaling factor exceeds one. When the angle is exceptional and the scaling factor is at most one, weak sensitivity is denied. The paper states that strong and weak sensitivity are equivalent for this class.
The paper's version of chaos
Putting the tests together, diagonal real matrices are never strongly Devaney chaotic. Weak Devaney chaos is reported only when the eigenvalues differ and at least one eigenvalue modulus equals one.
Jordan blocks are also stated never to be strongly Devaney chaotic. Their theorem gives weak Devaney chaos when the scalar modulus is neither one nor zero. Because the periodic-point theorem is internally inconsistent, this part of the classification should be treated as a stated result awaiting verification.
For the complex-eigenvalue form, strong Devaney chaos is stated for a rational rotation angle outside the exceptional set when the scaling factor equals one. In all other cases, the paper states that weak Devaney chaos does not occur.
The distinction matters for a broader theorem. In this two-dimensional weak-topology setting, the paper says the Banks et al. theorem holds for strong Devaney chaos but not for weak Devaney chaos in general.
The paper also reports a linear system that is weakly transitive and has a fixed point but is not weakly Li-Yorke chaotic.
Read as a mathematical classification
These findings concern mathematical maps, not evidence that a physical, biological or human system is chaotic. The work classifies specified linear maps in two-dimensional real space rather than reporting results from an empirical sample.
The document is an arXiv version 1 preprint dated 27 August 2026. No funding statement is reported in the supplied document. The central question left by the paper is whether the Jordan periodic-point cases can be corrected without changing the classifications that rely on them.
Paper data and sources
Original title: Devaney chaos in a Two-Dimensional Space with Weak Topology
Authors: Hongbo Zeng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-27
DOI: Not available
Original paper · Full text