ChaosComp: Chaos-Based Classification & Complexity

• ChaosComp is a framework that models each class as a one-dimensional chaotic system, enabling compression-based classification via symbolic dynamics.
• It extends to using chaotic systems as computational resources for quantitative complexity analysis, secure communications, and analog signal processing.
• Key challenges include sensitivity to threshold and block-length parameters, risk of floating-point underflow, and the need for long data series for stable statistics.

ChaosComp is used in two closely related senses in the cited literature. In the narrow sense, it denotes a supervised classification framework grounded in symbolic dynamics and data compression using chaotic maps: each class is modeled by a one-dimensional chaotic dynamical system, test data are thresholded and symbolized, and the predicted label is the one yielding the shortest compressed representation under class-wise symbolic statistics (Naik et al., 4 Aug 2025). The available literature also suggests a broader sense in which “ChaosComp” denotes the use of chaotic dynamics, multiscale complexity measures, or chaos-enabled physical substrates as computational resources for classification, diagnosis, sensing, signal processing, and decision-making (Brechtl et al., 2017, Li, 2017, Bhattacharyya et al., 2020).

1. Terminology and conceptual range

In its explicit, named form, ChaosComp is a compression-based classifier that “models each class as its own one-dimensional chaotic dynamical system.” It sits at the intersection of dynamical systems, symbolic representations, and Minimum Description Length: real-valued features are thresholded into binary sequences, class-specific symbolic statistics define return maps, and classification is performed by backward iteration and code-length comparison (Naik et al., 4 Aug 2025).

A broader usage appears across several neighboring lines of work. In one direction, “ChaosComp” functions as a label for quantitative complexity analysis of fluctuating phenomena, where refined composite multiscale entropy is used as a unifying measure across deterministic fractals, stochastic power-law noise, and low-dimensional chaos (Brechtl et al., 2017). In another direction, it denotes hardware or physical-computation paradigms that exploit chaotic carriers, random media, chaotic optical cavities, or comb-based photonic chaos as the operative computational substrate rather than as a disturbance to be suppressed (Li, 2017, Hougne et al., 2018, Shi et al., 4 Jan 2026).

The available literature therefore suggests that ChaosComp is not a single standardized doctrine. A plausible implication is that the term names a family of approaches in which chaos is made computable: sometimes as a generative model for inference, sometimes as a measurable complexity index, and sometimes as a physical resource for analog or photonic information processing.

2. Complexity as a computable observable

A key antecedent to ChaosComp is the treatment of “complexity” as the unpredictability or information-richness of a time series across multiple temporal scales. In the refined composite multiscale entropy framework, sample entropy is the negative logarithm of the conditional probability that two length-$m$ sequences remain close at the next point, multiscale entropy extends this over coarse-graining scales $\tau$, and refined composite averaging improves stability and reduces undefined values:

$$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$

In the cited study, the implementation uses $m=2$, $r=0.15\,\sigma_X$, and $\tau$ up to $20$ for Weierstrass and Logistic data or $30$ for colored noise (Brechtl et al., 2017).

For the real part of the Weierstrass function,

$$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$

with $y=2$, $\tau$0, and $\tau$1, the sample-entropy curves increase approximately exponentially with the fractional dimension $\tau$2. For colored noise with $\tau$3, the complexity is highest for pink noise, $\tau$4, and the fitted exponent in

$\tau$5

obeys the nearly perfect linear law $\tau$6 with $\tau$7. For the logistic map,

$\tau$8

the RCMSE map is negligible for $\tau$9, small nonzero values appear for $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$0, and sample entropy rises rapidly for $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$1, with local dips at periodic islands such as period-3 near $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$2 (Brechtl et al., 2017).

Within this perspective, ChaosComp is a quantitative “complexity compass.” The cited work shows that refined composite multiscale entropy distinguishes higher fractal roughness in Weierstrass functions, peaks near $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$3 in $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$4 noise, and mirrors the logistic map’s bifurcation structure. It also makes explicit that the choice of $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$5, $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$6, and $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$7 affects absolute values, that long data series are needed for stable statistics, and that computational cost grows with $$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$8 and ensemble size (Brechtl et al., 2017).

3. ChaosComp as compression-based classification

The named ChaosComp framework recasts supervised classification as class-wise chaotic modeling and symbolic compression. Its core map is the skewed Baker’s map

$$\mathrm{RCMSE}(\tau)= -\ln\!\left[\frac{\sum_{k=1}^{\tau}A_k^m(\tau)}{\sum_{k=1}^{\tau}B_k^m(\tau)}\right].$$9

with symbolic dynamics generated by thresholding at $m=2$0. Rather than encoding one bit at a time, the method groups symbols into non-overlapping length-$m=2$1 blocks $m=2$2, estimates empirical probabilities

$m=2$3

and forms a class-specific return map whose subinterval widths are exactly those block frequencies. With Laplace smoothing,

$m=2$4

The resulting interpretation is arithmetic-coding-like: the class whose symbolic model best fits the test symbols also yields the shortest description length (Naik et al., 4 Aug 2025).

The workflow is fully specified. Features are normalized to $m=2$5 via Min–Max scaling, thresholded at a hyperparameter $m=2$6, padded to make the bit-vector length divisible by $m=2$7, and then partitioned into non-overlapping blocks. Training accumulates class-wise counts $m=2$8 and stores the $m=2$9 probabilities per class. At test time, backward iteration on the symbol sequence $r=0.15\,\sigma_X$0 under each class map produces a consistent subinterval $r=0.15\,\sigma_X$1 with length $r=0.15\,\sigma_X$2, and the code length is

$r=0.15\,\sigma_X$3

Prediction is $r=0.15\,\sigma_X$4, with ties resolved by cosine similarity between the empirical block-frequency vector of the test sample and the class-wise block-frequency vectors. The per-sample complexity is stated as $r=0.15\,\sigma_X$5 for backward iteration over $r=0.15\,\sigma_X$6 pieces. On synthetic datasets, ChaosComp perfectly solved XOR, NAND, and NOR with $r=0.15\,\sigma_X$7; on UCI datasets it reports, among other results, macro F1-scores of $r=0.15\,\sigma_X$8 on Breast Cancer Wisconsin and $r=0.15\,\sigma_X$9 on Seeds. The authors explicitly state that, rather than aiming for state-of-the-art performance, the goal is to reinterpret classification through the lens of dynamical systems and compression (Naik et al., 4 Aug 2025).

The framework is generative rather than discriminative. Each class corresponds to a symbolic Markov model encoded as a chaotic return map, and decision-making is MDL/MAP-like rather than margin-based. The same paper also notes a direct link between the Baker-map Lyapunov exponent,

$\tau$0

and the binary entropy of the threshold distribution, and it identifies open issues: backward iteration can suffer floating-point underflow on long symbol sequences, the present formulation is limited to one-dimensional maps, and sensitivity to the threshold $\tau$1 and block length $\tau$2 requires cross-validation (Naik et al., 4 Aug 2025).

4. Physical embodiments and analog platforms

A hardware-centric lineage of ChaosComp appears in chaos-based signal processing. One example is the analog “chaos processor” built from Chua’s circuit and a matched receiver. The generator is a continuous-time, piecewise-linear chaotic oscillator whose regime is tuned by the resistor $\tau$3, while the retriever is an identical circuit that synchronizes to the transmitter and permits subtraction-based message recovery. The same architecture is used for chaotic masking in communication and for waveform generation in sound synthesis. The reported simulations show perfect synchronization and error-free recovery for matched parameters, small glitch errors for a $\tau$4 mismatch in $\tau$5, and failure of synchronization for a $\tau$6 mismatch in $\tau$7 (Li, 2017).

A second embodiment treats chaos as an analog linear-algebra substrate. In wave-based analog computation with a chaotic microwave cavity, the random medium is not fabricated to equal a target operator; instead, the overall transformation is programmed by shaping the incident wavefront with a tunable metasurface reflect-array. The formal target is

$\tau$8

while the cavity response is characterized by an impact matrix. The experimental demonstration includes a one-shot $\tau$9 discrete Fourier transform with ensemble-averaged error $20$0, a model-based prediction error reduced to $20$1 after averaging over realizations, and a sequential $20$2 implementation. The paper concludes that off-the-shelf wireless communication infrastructure together with a simple reflect-array suffices to perform analog computation with Wi-Fi waves reverberating in a room (Hougne et al., 2018).

Photonic and magnonic systems extend this logic to high-bandwidth chaos generation and sensing. In comb-injected semiconductor lasers, permutation entropy reaches $20$3 and chaos bandwidth reaches $20$4; in a single deformed optical cavity used as a chaos-assisted spectrometer, channel decorrelation of $20$5 supports reconstruction over $20$6 within a footprint of $20$7 and power consumption of $20$8. A later “chaos-on-comb” architecture reports a single-channel effective bandwidth of $20$9, a broadband terahertz noise source with excess noise ratio $30$0, and parallel photonic decision-making in a $30$1-armed bandit problem with power-law scaling exponent $30$2. In the magnonic domain, ultra-strongly coupled magnons in a synthetic antiferromagnet generate a magnonic chaotic comb through three-wave mixing, with transitions to chaos via subcritical Hopf bifurcation, torus-doubling bifurcation, and torus breakdown (Malica et al., 2024, Zhang et al., 18 Jun 2025, Shi et al., 4 Jan 2026, Sun et al., 29 May 2025).

5. Quantum, symbolic, and computational-theoretic interpretations

In quantum-chaos research, ChaosComp becomes a complexity diagnostic. One line of work defines circuit complexity through Nielsen geometry and applies it to the reduced density matrix of a subsystem in a two-oscillator model. The complexity of the effective pure state obtained from the reduced density matrix shows a short initial plateau, a linear-growth regime, and saturation; the scrambling time and linear-growth slope are controlled almost entirely by the bath parameter $30$3, and swapping $30$4 changes the reduced-state complexity. The same study finds that complexity of purification captures an early-time plateau and a single linear slope but misses the two-slope Lyapunov spectrum and the system-versus-bath asymmetry (Bhattacharyya et al., 2020). A related single-mode Gaussian analysis uses an echo-type target state and finds that complexity reproduces the Lyapunov exponent and scrambling time familiar from OTOCs, with fitted relations $30$5 and $30$6, and it introduces an additional “critical time” $30$7 marking when the system “learns” it has become chaotic (Ali et al., 2019).

A more formal computational reading treats computation itself as a chaotic dynamical process. One construction rationalizes successive Turing-machine tape configurations into $30$8 and proves that bounded, aperiodic, non-Cauchy orbits are Devaney-chaotic on an invariant subset; with a counting-and-measure argument, random universal computation is then “almost surely” chaotic (Mondal et al., 2011). Another construction gives a reversible one-dimensional cellular automaton $30$9 that is Devaney-chaotic and whose finite-time trace-prediction problem is $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$0-complete, while the infinite-time version is $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$1-complete, thereby showing that Devaney chaos and maximal trace-complexity universality coexist in the same reversible CA (Kari et al., 2014).

Symbolic and categorial formulations push the term still further. In binary category computation, every infinite binary string is said to be morphically compressible under the fixed $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$2-bit code

$$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$3

which implements the local XOR rule in the category $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$4 and yields “chaotic categorial dynamics” underlying algorithmically random patterns (Gonçalves, 2010). In quantum computing structures built from coherent states of bosonic Fock space, bounded nonlinear maps on $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$5 emerge through path-dependent unitary gates; for a version of Chirikov’s standard map, the associated quantum ensemble displays three phases of dynamics: regular, chaotic, and an intermediate “complex quantum stochastic phase” analogous to edge-of-chaos behavior (Gonçalves, 2012).

6. Applications, limitations, and research trajectory

Across the literature, ChaosComp is associated with a wide application envelope. Refined composite multiscale entropy is explicitly linked to physiology, heart-rate and gait analysis, materials science, finance, and traffic signals, and it is presented as a way to compare Lyapunov-exponent-style, fractal-dimension-style, and information-theoretic perspectives within one multiscale measure; the same work emphasizes, however, that the choice of $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$6, $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$7, and $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$8 affects absolute values and that long data series are required for stable statistics (Brechtl et al., 2017). The compression-based classifier explicitly warns that backward iteration can suffer floating-point underflow, that the current formulation is limited to one-dimensional maps, and that threshold and block-length sensitivity necessitate cross-validation; it also stresses that the goal is reinterpretation rather than state-of-the-art accuracy chasing (Naik et al., 4 Aug 2025).

In physical systems, applications include secure communications and sound synthesis in Chua-based processors, true random number generation and photonic reservoir computing in comb-injected lasers, secure communication, LiDAR, and reinforcement learning in chaos-on-comb photonics, and metrology, sensing, and communication in magnonic chaotic combs. The same papers also delimit the engineering constraints: synchronization can be fragile to parameter mismatch in analog chaos processors; comb-based photonic systems remain limited by seed bandwidth, modulator $$W(t)=\sum_{k=1}^{20} y^{-(2-D)k}\cos(y^k t+\phi_k),$$9, residual cross-correlations, and waveshaper reconfiguration speed; chaos-assisted spectrometers require pre-calibration on the order of $y=2$0 and currently operate on end-to-end timescales of several seconds; magnonic systems face thermal-noise and bias-field control issues (Li, 2017, Malica et al., 2024, Shi et al., 4 Jan 2026, Zhang et al., 18 Jun 2025, Sun et al., 29 May 2025).

This suggests that ChaosComp is best understood not as a single method but as a research program. Its recurring structure is the operationalization of chaos: as a class-conditional generative prior, as a multiscale entropy profile, as a random-yet-programmable analog encoder, or as a complexity growth law that diagnoses scrambling and instability. The common thread is not merely that chaos is present, but that chaos is made algorithmically or physically useful.