Multivariate Chaotic Mappings

• Multivariate chaotic mappings are discrete-time systems on n-dimensional spaces that rigorously exhibit chaos as defined by Devaney and Li–Yorke.
• They encompass diverse models such as polynomial maps, set-valued systems, and coupled chaotic constructions that generate hyperchaos through mechanisms like cross-coupling and invariant cone-field dynamics.
• These systems are crucial for applications in cryptography, secure communications, and modeling complex physical or biological processes, with computational methods for detecting Lyapunov spectra and unstable dimension variability.

A multivariate chaotic mapping is a discrete-time dynamical system acting on an $n$-dimensional phase space, typically $\mathbb{R}^n$ or a compact metric manifold, in which the map's structure, interaction among components, or multiplicity of mappings induces chaotic dynamics in a rigorous sense (e.g., Devaney or Li–Yorke chaos). These mappings serve as canonical models for high-dimensional chaos, hyperchaos, and unstable dimension variability, and play a vital role in both theoretical dynamics and applications such as encryption, secure communications, and modeling of complex physical or biological systems.

1. Mathematical Models and Classes

The principal mathematical paradigms for multivariate chaotic mappings can be clustered into several families:

• Multivariate Polynomial Maps: These generalize logistic and Hénon-type recursions to $n$ dimensions. A canonical form is given by

$$F(x_1,\ldots,x_n) = \big(f_1(x_1,\dots,x_n),\dots,f_n(x_1,\dots,x_n)\big)$$

where

$$f_k(x_1,\ldots,x_n) = a_{kk} p_k(x_k) + \sum_{j\neq k} a_{kj} x_j$$

and each $p_k$ is a one-variable polynomial and $a_{kj}$ real parameters. Special cases include the logistic map for $n=1$ and the Hénon map for $n=2$ with quadratic nonlinearity in some coordinates and linear coupling in others (Zhang, 2015).

• Set-Valued and Multiple-Mapping Systems: Given a finite family $\mathcal{F} = \{f_1,\dots,f_k\}$ of maps on a compact metric space $X$, define an induced set-valued map $F : 2^X \to 2^X$ by $F(A) = \bigcup_{i=1}^k f_i(A)$, iterating subsets under all maps (Alvarez et al., 2024Alvarez, 2024). This framework captures both deterministic switching and concurrent application of multiple rules.
• Coupled and Cross-Coupled Chaotic Systems: These combine several low-dimensional maps (e.g., logistic, sine, or ICMIC maps) using algebraic cross-coupling. The 3D Cascaded Cross-Coupling (3D-CCC) method constructs a 3D hyperchaotic system of the form

$$\begin{aligned} x_{n+1} &= f(u_n), \quad u_n = x_n g(y_n) + (1-x_n) h(z_n) \ y_{n+1} &= f(v_n), \quad v_n = y_n g(z_n) + (1-y_n) h(x_{n+1}) \ z_{n+1} &= f(w_n), \quad w_n = z_n g(x_{n+1}) + (1-z_n) h(y_{n+1}) \end{aligned}$$

where each $f,g,h$ is a seed chaotic map (Sun, 31 Mar 2025).

• Higher-Dimensional Digital Chaotic Systems (HDDCS): Designed for fixed-precision hardware, these use random bit-masks and bitwise logic to update a vector state, ensuring that true Devaney chaos persists in the finite-precision environment (Wang et al., 2015).
• Multi-Chaos and Unstable Dimension Variability: Certain toral automorphisms and skew-product maps (e.g., $F(x,y) = (m x, a x+y + g(x,y))$ mod $1$ on $\mathbb{T}^2$) exhibit "multi-chaos," where periodic points with different unstable dimensions are simultaneously dense, often facilitated by quasiperiodic invariant curves (Das et al., 2015).

2. Rigorous Notions of Chaos in Multivariate Settings

Chaoticity for multivariate maps is most commonly formalized using one of the following definitions:

• Devaney Chaos: A mapping $F$ (single or set-valued) on a metric space $V$ is chaotic if:
  1. The set of periodic points is dense in $V$.
  2. $F$ is topologically transitive: for any pair of open sets $U,V$, some iterate $F^n(U)$ intersects $V$.
  3. There is sensitive dependence on initial conditions: some $\delta>0$ such that for any $x$ and any neighborhood, some $y$ and $n$ exist with $d(F^n(x),F^n(y))>\delta$ (Zhang, 2015Alvarez et al., 2024Alvarez, 2024).

• Li–Yorke Chaos: There exists an uncountable "scrambled set" where pairs $(x,y)$ realize both $\liminf_{n} d(F^n(x),F^n(y))=0$ and $\limsup_{n} d(F^n(x),F^n(y))>0$ (Zhang, 2015).

• Symbolic Dynamics: When a map on an invariant set is topologically conjugate to a full shift on $m$ symbols, it inherits strong chaos properties in both the Devaney and Li–Yorke senses (Zhang, 2015).
• Set-Valued Generalizations: For a family $\mathcal{F}$ of maps, Devaney chaos is defined for the induced set-valued map on the hyperspace of compact sets $2^X$ or $K(X)$, using the Hausdorff metric (Alvarez et al., 2024Alvarez, 2024). Periodicity and transitivity are correspondingly defined for compact sets, not just points.
• Multi-Chaos and UDV: In multi-chaotic sets, periodic points of distinct unstable dimensions are both dense, yielding "unstable dimension variability". This sharpens the topological and ergodic complexity of the invariant set (Das et al., 2015).

3. Conditions and Mechanisms for Multivariate Chaos

The generation of multivariate chaos is governed by structural and parameter criteria:

• Uniform Hyperbolicity: For multivariate polynomial maps, a cone-field and nondegenerate expansion/contraction in subspaces establish the presence of a Smale horseshoe, which is conjugate to a shift on $m$ symbols. For instance, for

$$\begin{cases} f_1(x,y,z)=a_1\,p(x)+a_2\,y+a_3\,z,\ f_2(x,y,z)=b_1\,x,\ f_3(x,y,z)=c_2\,y \end{cases}$$

with large $|a_1|$, $|c_2|<1$, and certain inequalities, one obtains a horseshoe in $$U=[\xi_1,\xi_2]\times[b_1\xi_1,b_1\xi_2]\times[c_2b_1\xi_1,c_2b_1\xi_2]$$ (Zhang, 2015).

• Expansion Regimes:
  • Unstable dimension $=1$: Smale horseshoe conjugate to shift on $2$ symbols.
  • Unstable dimension $=2$: Horseshoe yields shift on $4$ symbols.
  • All expanding: Forward-invariant set semi-conjugate to one-sided shift on $8$ symbols (Zhang, 2015).
• Nonlinearity-Induced Chaos: If at least one map $f_k$ in a family $\mathcal{F}$ is nonlinear and strongly sensitive, while others may merely be transitive or even nonsensitive, the entire set-valued system can be Devaney-chaotic. Density of periodic points for the family ensures the rigorous lifting of chaos to the multivariate setting (Alvarez, 2024).
• Cone-Field and Skew-Product Constructions: For toral and affine maps, invariant cone-fields and quasiperiodic invariant tori allow strong transitivity, ensuring that periodic points of differing indices (e.g., saddle, repeller) are each dense (Das et al., 2015).
• Cross-Coupling: Algebraic interweaving of one-dimensional chaotic maps with cross-coupling ensures ergodicity, positive multiple Lyapunov exponents, and mixing, producing hyperchaos in discrete time (Sun, 31 Mar 2025).

4. Computational Detection and Quantification

Empirical identification and analysis of multivariate chaos employ several algorithmic schemes, often relying on samples from discretizations of the phase space:

• Transitivity Detection: For sample pairs within an $\varepsilon$-grid, iterate under all map compositions for a fixed horizon $N$, marking a "hit" if orbits intersect small neighborhoods of each other (Alvarez et al., 2024Alvarez, 2024).
• Periodicity Detection: For a sample $x$, iterate the family and track returns within $\delta$ of $x$ to estimate the density of periodic points.
• Sensitivity/Separation Testing: For near pairs $(x,y)$, iterate compositions and detect separation above a divergence threshold.
• Lyapunov Spectrum Estimation: Compute the growth rates of tangent vectors via the Jacobian or direct numerical differentiation along orbits; hyperchaos is confirmed by at least two positive Lyapunov exponents (Sun, 31 Mar 2025).

Empirical results—e.g., in Baker's-type maps or coupled logistic/rotation systems—show transitivity and sensitivity indicators close to 1, density of periodic points $\gtrsim$ 0.95 for moderate samples, and Lyapunov exponents $\sim0.2$–$0.7$ in hyperchaotic regimes (Alvarez et al., 2024Alvarez, 2024Sun, 31 Mar 2025).

5. Examples, Regimes, and Structural Features

Representative systems and phenomena are:

Map Family/Structure	Key Example or Formula	Chaotic Regime
Multivariate Polynomial Map	$F(x,y,z)=\big(7(x-1)(2-x)+0.2y+z,\,x,\,0.2y\big)$	Horseshoe, shift on $2$ syms (Zhang, 2015)
Cross-Coupled (3D-CCC)	See 3D-ICCCLS equations in Section 1	Hyperchaos, Lyap > 0 (Sun, 31 Mar 2025)
Set-Valued (Multiple Mappings)	$F=\{f_1, f_2\}$, $f_1$ nonlinear, $f_2$ transitive	Devaney chaotic (Alvarez, 2024)
Toral Skew-Product (Multi-Chaos)	$F(x,y)=(m x, a x + y + g(x,y))$ mod 1	Multi-chaos, UDV (Das et al., 2015)

Notable features:

• Baker’s-type maps: Multidimensional, invertible, and mixing; well suited for both theory and applications.
• Digital implementations: HDDCS uses bitwise operations and random masking to ensure reproducible chaos at fixed precision for cryptographic applications (Wang et al., 2015).
• Hyperchaotic cross-coupling: 3D-CCC yields positive Lyapunov spectrum and strong ergodicity, not achievable by simple tensor-product maps without genuine interaction (Sun, 31 Mar 2025).
• Multi-chaos: Achieved via coexistence and density of periodic orbits with distinct unstable dimensions, often facilitated by nontrivial topology or invariant tori (Das et al., 2015).

6. Applications and Implementation Contexts

• Cryptography and Secure Communications: High-dimensional and digital chaos improves key-space size, increases sensitivity, and impedes cryptanalysis—applied for real-time media encryption and hardware RNG (Wang et al., 2015Sun, 31 Mar 2025).
• Physical and Biological Modeling: Multivariate chaos is inherent in laser cavities (mode switching), particle accelerators (alternating lattices), seasonally forced population models, and neural networks with multiple phases (Alvarez, 2024).
• Symbolic Coding and Information Theory: Full-shift conjugacy provides a foundation for symbolic coding, data hiding, and compressed representations.
• Hardware Realization: FPGA implementations leverage bitwise parallelism for high-throughput chaotic stream generation, requiring low logic resources and well-controlled cycle length (Wang et al., 2015).

7. Extensions, Limitations, and Open Directions

• Scalability: Algorithms for detection scale well up to $d\leq4$ with modern sampling and search strategies; higher dimensions pose computational challenges for set separation and Lyapunov estimation (Alvarez et al., 2024).
• Parameter Exploration: Many cross-coupled and polynomial map regimes exhibit chaos only for specific parameter domains, often requiring numerical analysis for validation (Sun, 31 Mar 2025).
• Genericity and Persistence: Multi-chaos and UDV depend on structural conditions—existence and robustness of quasiperiodic tori, non-degeneracy of coupling—which remain active areas of investigation (Das et al., 2015).
• Symbolic and Non-Euclidean Contexts: Generalization to networks, semigroup actions, or non-Euclidean geometries is the subject of ongoing research.

Multivariate chaotic mappings, in their diverse formulations, thus constitute a core paradigm in the analysis and engineering of high-dimensional chaotic behavior, extending classical one-dimensional and two-dimensional theories to settings where dimensionality, multiplicity of governing rules, and algebraic or digital structure fundamentally shape the system’s dynamics (Zhang, 2015Wang et al., 2015Das et al., 2015Alvarez et al., 2024Alvarez, 2024Sun, 31 Mar 2025).