Periodic Non-Autonomous Dynamical Systems

• Periodic non-autonomous discrete dynamical systems are defined by sequences of maps with a fixed period, allowing reduction to an autonomous system via composite mappings.
• They exhibit a spectrum of phenomena including minimality, sensitivity, mixing, and bifurcations, showcasing transitions from order to chaotic behavior.
• These systems are vital in modeling real-world scenarios, such as ecological and epidemiological dynamics, where periodic forcing can lead to counterintuitive stability reversals.

A periodic non-autonomous discrete dynamical system is defined by a sequence of maps $\{f_n\}_{n\in\mathbb{N}}$ on a compact metric or topological space $X$, where the time-dependent rule $x_{n+1} = f_n(x_n)$ governs the evolution and $f_{n+p} = f_n$ for some minimal period $p$. Such systems interpolate between classical autonomous dynamics (where $f_n$ is identical for all $n$) and the general class of time-varying, possibly aperiodic, non-autonomous systems. The periodic assumption yields a rich interplay between the time-dependent structure and classical dynamical phenomena including periodicity, minimality, sensitivity, mixing, and bifurcation theory.

1. Definitions, Basic Structure, and Periodicity

A periodic non-autonomous discrete system is specified by a sequence of continuous maps $f_n: X \to X$ satisfying $f_{n+p} = f_n$ for all $n \geq 1$, with period $p\geq 1$. The orbit of a point $x\in X$ is given by $$\mathcal{O}_{f_{1,\infty}}(x) = \{ f_1^n(x) : n \geq 0 \}$$, where $f_1^n = f_n \circ f_{n-1} \circ \cdots \circ f_1$ defines the forward composition. The associated autonomous system is given by the composite map $g = f_p \circ f_{p-1} \circ \cdots \circ f_1$, and for multiples of the period, $f_1^{mp} = g^m$. This reduction often allows classical results for autonomous dynamics to be transferred to the periodic non-autonomous scenario when additional structure, such as commutativity of the family $\{f_n\}$, is imposed (Salman et al., 2018, Malik et al., 27 Dec 2025).

Periodic points are defined in several non-equivalent ways in non-autonomous systems. One canonical definition states that $x$ is $T$-periodic if for all $n$, $$f_{n+T}\circ f_{n+T-1}\circ \cdots \circ f_{n+1}(x) = x$$, or, equivalently, $\omega_{n+T}(x) = \omega_n(x)$ for all $n\in\mathbb{Z}$. Alternative notions, including pseudo-periodic, strictly periodic, and asymptotically periodic points, have subtle differences and varying robustness properties, especially under perturbations of the initial segment of the sequence $\{f_n\}$ (Pravec, 2018).

2. Minimality, Equicontinuity, and Sensitivity

Minimality in periodic non-autonomous systems departs from the autonomous setting. Given the orbital hull $\mathcal{O}_H(x)$, minimality is characterized by the orbit closure properties of this hull rather than a single forward orbit: the system is minimal if, for every nonempty open set $U$, there exists $k$ such that $\mathcal{O}_H^k(x) \cap U \neq \varnothing$ for every $x$, where $\mathcal{O}_H^k(x)$ collects compositions of bounded length from the commutative family (Yadav et al., 2023).

A crucial dichotomy, generalizing the classical Auslander–Yorke result, holds in the periodic non-autonomous setting: every minimal system is either equicontinuous or sensitive to initial conditions. This is sharpened to syndetic equicontinuity versus thick sensitivity, and, on compact spaces, to equicontinuity versus eventual sensitivity. These dichotomies extend to topological and uniform frameworks, with various refined notions such as syndetically topological equicontinuity and thick Hausdorff sensitivity defined in terms of return time sets and open cover configurations (Malik et al., 27 Dec 2025).

The following table summarizes key dichotomies in various settings:

Setting	Dichotomy	Reference
Uniform space, minimal system	Sensitivity or equicontinuity	(Malik et al., 27 Dec 2025)
Compact space	Eventual sensitivity or equicontinuity	(Malik et al., 27 Dec 2025)
$T_3$ space, minimal system	Hausdorff sensitivity or topological equicontinuity	(Malik et al., 27 Dec 2025)

This dichotomy demonstrates that minimality precludes any intermediate behavior between uniform regularity and global chaos.

3. Periodic Orbits, Propagation, and Almost Periodicity

In periodic, particularly commutative, non-autonomous systems, periodicity of a point propagates to the entire orbit closure: if $x$ is $T$-periodic, then every point in $\overline{\mathcal{O}_H(x)}$ is also $T$-periodic (Yadav et al., 2023). This contrasts with autonomous dynamics, where minimal sets need not be periodic.

Almost periodic points, precisely those whose return times to any neighborhood are syndetic, inherit strong structure in the equicontinuous regime: equicontinuity together with almost periodicity implies uniform almost periodicity on the entire orbit closure. That is, for a commutative and equicontinuous family, if $x$ is almost periodic, every $y \in \overline{\mathcal{O}_H(x)}$ is almost periodic, and the hull itself is uniformly almost periodic (Yadav et al., 2023).

There are illustrative constructions where the periodicity is global yet the dynamical complexity remains high, such as a non-autonomous system on $[0,1]$ with every point periodic of period $2$, but the underlying behavior is Li–Yorke chaotic (Yadav et al., 2023). Minimality does not guarantee density of single orbits; instead, distributions are supported on orbit closures.

4. Mixing, Specification, and Devaney Chaos

Periodic non-autonomous systems admit a rich tapestry of mixing phenomena. Under commutativity, weak mixing—defined via simultaneous return to pairs of open sets—transfers between the base system and induced systems on spaces of measures or compact subsets. For commutative periodic systems on intervals, weak mixing is equivalent to Devaney chaos: topological transitivity, dense periodic points, and sensitive dependence coincide, mirroring the classical interval map setting (Salman et al., 2018).

Specification properties—strong, weak, and quasi-weak—characterize orbit-tracing capabilities. The strong specification property (SSP) ensures that arbitrary pieces of orbits can be shadowed by a single periodic orbit with prescribed minimal period. The weak specification and quasi-weak specification properties give progressively weaker variants. SSP implies Devaney chaos in all compact metric spaces, while WSP or QSP suffices for Devaney chaos on intervals. Specification properties are preserved under finite products and semi-conjugacy, and extend to induced systems on hyperspaces and measure spaces (Salman et al., 2018).

Sensitivity in periodic non-autonomous systems admits several variants (syndetic, thick, ergodic, collective, multi-) that, under periodicity and commutativity, become equivalent. However, without these constraints, sensitivity can exist in weaker forms that fail to imply stronger versions, reflecting fundamental departures from the autonomous case (Salman et al., 2018).

5. Entropy, Recurrence, and Pseudo-Orbits

Topological entropy for periodic non-autonomous systems can be characterized through pseudo-orbits. The pseudo-entropy and periodic-pseudo-entropy, defined via the exponential growth rates of separated (periodic) pseudo-orbits, coincide with the topological entropy of the system. Under chain transitivity, the periodic-pseudo-entropy equals the actual topological entropy. For expansive, chain-transitive systems with shadowing, entropy can be directly computed from counts of genuine periodic points (Nia, 2016).

Recurrence phenomena in non-autonomous systems are multifaceted and include non-wandering, recurrent, and chain-recurrent points, as well as their chain-transitive and shadowing properties. Equicontinuity localizes entropy to the chain recurrent set. Notably, if the family $\{f_n\}$ converges uniformly, the topological entropy of the periodic non-autonomous system does not exceed that of the limiting autonomous map.

6. Stability, Bifurcation, and Parrondo-type Paradoxes

Periodic non-autonomous systems display stability behaviors absent in autonomous systems, particularly at non-hyperbolic fixed points. A central phenomenon is the "Parrondo-type" dynamic paradox: a common fixed point can exhibit reversed stability under periodic alternation of maps. In dimension one, there exist triples of polynomial maps, each making a fixed point locally attracting (LAS), whose composition, however, makes it repelling, and vice versa. In higher dimensions, the paradox can occur for fewer maps. The mechanism for this reversal is rooted in higher-order terms (stability and Birkhoff constants) and cannot be predicted by linearization alone (Cima et al., 2017).

This stability reversal has significant implications for time-dependent real systems, including ecological and epidemiological models subject to seasonal forcing, where the stroboscopic map may be locally attracting even though the individual rules are repelling.

7. Bifurcations and Complex Dynamics in Periodically Forced Systems

Periodic forcing can act as a route to intricate bifurcation sequences in non-autonomous discrete systems. A canonical example is the first return map derived for a periodically forced heteroclinic network on the two-sphere, leading to a two-dimensional skew product on a cylinder. For small amplitudes, bifurcation analysis reveals the discrete-time Bogdanov–Takens phenomenon, Neimark–Sacker bifurcation (leading to invariant circles), frequency locking (Arnold tongues), coexistence of attracting fixed points and invariant curves (bistability), and horseshoe dynamics (Labouriau et al., 2018).

Maps derived in this context take the form

$$T\colon \begin{cases} x' = x - K \ln y \mod (\pi/\omega),\ y' = y^\delta + A(1 + k_1 \sin(2 \omega x)), \end{cases}$$

with parameter-dependent regions delineating saddle-node creation, invariant circle emergence, and chaotic regimes. Frequency locking and bistability are robust features of periodically forced non-autonomous systems, further complicating their qualitative dynamics.

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Periodic non-autonomous discrete dynamical systems unify rich mathematical structure with significant modeling relevance. Core results establish precise analogues and sharp departures from autonomous theory, rigorous dichotomies, and a spectrum of behaviors from uniform regularity to robust chaos, governed by the interplay of periodicity, commutativity, and equicontinuity (Yadav et al., 2023, Salman et al., 2018, Malik et al., 27 Dec 2025, Cima et al., 2017, Nia, 2016, Salman et al., 2018, Labouriau et al., 2018, Pravec, 2018).