Background¶
This page presents the complete theoretical foundations of
chaotic-pfc, covering the physics, mathematics, and engineering
principles behind chaos-based digital communication with FIR-filtered
Hénon maps.
Physical-layer security and chaos-based communication¶
The protection of critical communication infrastructures has become a central element of digital sovereignty and information security in the contemporary landscape, especially in a context shaped by “information warfare” and the constant threat of interception and denial-of-service attacks [OliveiraFilgueiras22] [Sapinski23].
While conventional cryptographic methods operate predominantly at the upper layers of the OSI model (Transport, Application), a strategic advantage of Chaos-Based Communication Systems (CBCS) is providing security directly at the Physical Layer [Baptista21].
This approach exploits the intrinsic properties of the chaotic carrier signal:
Broad bandwidth: the signal occupies a wide portion of the spectrum, resembling white noise.
Noise-like appearance: the waveform shows no discernible pattern to an observer lacking the system parameters.
Sensitivity to initial conditions (SDIC): infinitesimal parameter mismatches cause exponential divergence, preventing unauthorised synchronisation.
Together these properties enable concealed transmission, where the message is not merely encrypted but physically embedded into a chaotic orbit. The scientific foundation for CBCS was established in 1990 when Pecora and Carroll demonstrated that two independent chaotic systems could be synchronised [PecoraCarroll90]. This discovery proved that the apparent unpredictability inherent to chaos could be exploited in a controlled and reproducible manner.
The strategic relevance of CBCS is evidenced by its application in frontier areas such as 5G network standards, ultra-wideband (UWB) communications, and Internet of Things (IoT) systems.
The Hénon map¶
The Hénon map is a two-dimensional discrete dynamical system introduced by Michel Hénon in 1976 as a simplified model of a Poincaré section of continuous systems. Its widespread use as a paradigm for chaotic signal generation stems from the combination of mathematical simplicity with rich dynamical behaviour [Henon76].
Standard form¶
The system is described by the recurrence equations:
where \(x_1[n]\) and \(x_2[n]\) are the state variables at discrete time \(n\), and the parameters \((a, b)\) control the dynamical behaviour.
Canonical chaotic regime¶
For the parameter pair
the map exhibits a strange attractor with:
Fractal dimension \(\approx 1.26\)
Largest Lyapunov exponent \(\lambda_1 \approx 0.42\) (positive, confirming chaos)
SDIC: trajectories with initial separation \(10^{-4}\) diverge beyond recognition within tens of iterations
Phase-space structure resembling a “boomerang” or “seagull wing” shape
Parameter dependence¶
The qualitative behaviour of the Hénon map depends critically on \((a, b)\):
For \(a < 1.06\): the system is periodic (the fixed point is stable).
For \(1.06 \lesssim a \lesssim 1.43\): the system is chaotic, with the canonical chaotic regime at \((1.4, 0.3)\).
For \(a > 1.43\): most initial conditions diverge to infinity, rendering the system inoperable for communication.
The parameter \(b\) controls dissipation: \(|b| < 1\) makes the map area-contracting (dissipative), while \(|b| > 1\) would be area-expanding.
Variants implemented in the package¶
- Generalised Hénon (
henon_generalised()) Uses \(\alpha\) and \(\beta\) as parameters, expressed in a coupled-oscillator form:
\[x_1[n+1] = \alpha - x_1[n]^2 + \beta \, x_2[n]\]This form is convenient for sweeping the parameter space when the dissipative coupling \(\beta\) is the object of study.
- Filtered Hénon (
henon_filtered()) Passes the state through a 2-tap FIR filter before feeding back:
\[x_1[n+1] = \alpha - (c_0 x_1[n] + c_1 x_1[n-1])^2 + \beta \, x_2[n]\]With \(c_0 = 1, c_1 = 0\) this reduces to the generalised map. With nontrivial \((c_0, c_1)\) the filtered term introduces an additional degree of freedom.
- Order-N Hénon (
henon_order_n()) Generalises to \(N_c\) filter taps, yielding an \(N_c\)-dimensional state vector. This is the main workhorse of the parameter-sweep pipeline. For \(N_c = 2\) with \(c_0 = 1, c_1 = 0\) it reproduces the standard map.
FIR filtering and band-limited chaos¶
The bandwidth problem¶
Chaotic signals are inherently broadband: their continuous power spectrum extends over a wide frequency range. However, physical transmission channels are invariably band-limited: every real-world communication medium (copper wire, optical fibre, radio-frequency spectrum) imposes a finite bandwidth constraint.
Transmitting a broadband chaotic signal through a band-limited channel without adaptation results in severe distortion, compromising both synchronisation quality and message recovery.
The Fontes-Eisencraft solution¶
To reconcile bandwidth constraints with chaotic signal generation, Fontes and Eisencraft (2016) proposed inserting finite impulse response (FIR) filters directly into the feedback loop of the chaotic generator [FontesEisencraft16]. The filter design relies on classical signal-processing techniques: windowing methods (Hamming, Blackman, Kaiser), as established by Oppenheim and Schafer [OppenheimSchafer09].
In this architecture, the variable \(x_1[n]\) in the nonlinearity \(x_1[n]^2\) is replaced by the filtered version \(x_3[n] = \sum_{k=0}^{N_z-1} c_k \, x_1[n-k]\), where \(c_k\) are the FIR coefficients and \(N_z\) is the filter order (number of taps).
The result is a band-limited chaotic signal whose spectrum is shaped by the FIR filter, making it compatible with practical channel bandwidth constraints without sacrificing chaotic behaviour, provided the filter parameters are chosen within regions that preserve the chaotic regime.
Dimensionality expansion¶
Inserting an FIR filter into the feedback loop expands the system dimensionality. For a map of order \(K\) and a filter with \(N_s\) coefficients, the resulting filtered system has dimension:
For the 2-D Hénon (\(K = 2\)) with an \(N_s\)-tap filter, this yields a \((N_s + 1)\)-dimensional system. The stability analysis then requires the Jacobian matrix of the expanded \(K'\)-dimensional system.
Shrimps and bifurcation cascades¶
Investigations by Borges and Eisencraft (2022) revealed that FIR filtering introduces unexpected dynamical complexity [BorgesEisencraft22]. The linear filter interacts non-trivially with the map’s nonlinearity, potentially inducing:
Bifurcation cascades: sequences of period-doubling transitions
that accumulate toward chaos from a periodic state. * “Shrimps”: islands of periodicity immersed within chaotic
regions of the parameter space. Named for their characteristic shape in parameter-space diagrams.
The presence of shrimps is critical for communication security: if the system enters a periodic regime due to quantisation errors or hardware variations, the signal becomes predictable and vulnerable to spectral analysis.
This interaction motivates the central research question of the project:
Under what conditions does FIR filtering preserve the chaotic regime necessary for physical-layer security under bandwidth constraints?
Lyapunov exponents¶
Definition¶
The largest Lyapunov exponent \(\lambda_{\max}\) quantifies the average rate of exponential divergence (or convergence) of infinitesimally close trajectories in phase space:
where \(\delta \mathbf{x}_k\) is the perturbation vector at iteration \(k\), evolved through the tangent map of the system.
Classification¶
Condition |
Meaning |
Regime |
|---|---|---|
\(\lambda_{\max} > 0\) |
Exponential divergence of nearby trajectories; SDIC present |
Chaotic |
\(\lambda_{\max} \leq 0\) |
Trajectories converge or remain bounded without exponential growth |
Periodic / quasiperiodic |
\(\lambda_{\max} = \text{NaN}\) |
Trajectory diverged to infinity within the simulation window |
Divergent (unbounded) |
Important
The condition \(\lambda_{\max} > 0\) is the mathematical imperative that guarantees SDIC and, consequently, the security of concealed transmission. This is the metric used throughout the sweep pipeline to classify each (order, cutoff) grid point.
Combinatorial validation protocol¶
The sweep pipeline estimates \(\lambda_{\max}\) at every point of a 2-D grid of (filter order \(N_z\), cutoff frequency \(\omega_c\)) pairs, building a classification map of the entire parameter space. For each grid point:
1. FIR bank precomputation: all filter coefficient sets are
computed once using SciPy’s firwin().
2. Initial condition generation: \(N_{\text{IC}}\) initial conditions are sampled from a uniform box around the fixed point.
3. Transient discard: the first \(N_{\text{discard}}\) iterations are discarded to eliminate transient dynamics.
4. Lyapunov estimation: \(N_{\text{itera}}\) iterations are performed, tracking the evolution of \(N_s\) tangent vectors via modified Gram-Schmidt orthogonalisation.
Ensemble aggregation: \(\lambda_{\max}\) is reported as the mean over all \(N_{\text{IC}}\) initial conditions.
Tangent-map method and Modified Gram-Schmidt¶
The tangent-map method propagates both the main orbit and a set of linearised perturbation vectors simultaneously. However, the exponential growth of these vectors would quickly cause numerical overflow or collapse (all vectors aligning with the dominant eigen-direction).
The Modified Gram-Schmidt (MGS) procedure is applied at every iteration to re-orthonormalise the perturbation vectors, with the logarithmic scaling factors accumulated for the final estimate of \(\lambda_{\max}\). The implementation uses QR factorisation compiled via Numba JIT for performance.
Fixed-point stability¶
For the filtered Hénon map, the stability analysis also requires the Jacobian matrix of the expanded \(K'\)-dimensional system. Convergent results from Borges, Silva, and Eisencraft (2024) demonstrate that:
The fixed-point locations depend primarily on the total filter gain \(G\).
The stability of these points is highly sensitive to the individual coefficient distribution and the zero locations of the filter in the complex plane [BorgesSilvaEisencraft24].
This dependence motivates the need for a systematic (rather than pointwise) characterisation of the filters employed.
Numba JIT acceleration¶
The inner loop of the Lyapunov computation is \(\mathcal{O}(N_s^3)\) due to the MGS re-orthonormalisation of an \(N_s\)-dimensional perturbation ensemble. Numba JIT compilation reduces this to practical execution times, enabling sweeps with up to \(N_s = 41\) filter taps, \(N_{\text{IC}} = 25\) initial conditions per grid point, and \(N_{\text{itera}} = 3\,000\) iterations per IC, totalling \(40 \times 100 = 4\,000\) grid points, each with \(25 \times 3\,000 = 75\,000\) orbit evaluations. Without Numba, this would require tens of hours; with JIT, it completes in minutes on a modern multicore processor.
Pecora-Carroll synchronisation¶
Principle¶
The receiver runs a copy of the transmitter’s chaotic oscillator driven by the received signal. This is formalised as a conditional response subsystem: given a master chaotic system with state vector \(\mathbf{x}(t)\), a slave subsystem is synchronised when:
where \(\hat{\mathbf{x}}(t)\) is the receiver state.
The necessary condition for convergence is that all conditional Lyapunov exponents of the slave subsystem are negative, ensuring that perturbations in the state difference are attenuated over iterations [PecoraCarroll90] [Williams01].
Receiver equations for the 2-D Hénon¶
For the standard 2-D Hénon, the receiver state evolves as:
where \(r[n]\) is the received (possibly channel-distorted) carrier. The receiver does not use its own \(y_1[n]\) in the nonlinearity; instead, it is driven by \(r[n]\), which carries the transmitter state plus the embedded message.
After a short transient (typically 200–500 iterations), the receiver state converges: \(y_1[n] \approx x_1[n]\). The message is then recovered by subtracting the local estimate:
where \(\mu\) is the modulation depth (a small positive constant, typically 0.01).
Modulation schemes¶
The package implements three modulation techniques:
CSK: Chaos Shift Keying. Sistematised by Williams (2001) [Williams01]. Each binary symbol corresponds to a distinct set of parameters or initial conditions of the chaotic generator: the message is encoded in the selection between two different chaotic attractors. The receiver performs detection by correlation or energy comparison, identifying which attractor the received signal belongs to. CSK requires a synchronisation period before each symbol, reducing spectral efficiency.
DCSK: Differential Chaos Shift Keying. Introduced by Kolumban et al. (1996) [Kolumban96]. Each bit period is split into two equal halves: a reference slot transmitting a chaotic reference sequence, followed by a data slot transmitting the same reference (bit 0) or its negated version (bit 1). The receiver correlates the two slots to recover the bit. DCSK does not require chaos synchronisation, making it inherently robust to channel variations. The cost is a 50% reduction in bit rate (one information bit per two transmitted chaotic sequences).
EF-DCSK: Efficient DCSK. Proposed by Kaddoum et al. (2013) [Kaddoum13]. Improves the data rate by using a single slot per bit: the reference and its time-reversed copy are superposed in the same slot. The decoder correlates the received signal with its own time-reverse to recover the bit. This doubles the throughput of classical DCSK with minimal BER degradation.
Performance metrics¶
Quantitative evaluation of digital communication performance relies on two standard metrics [Haykin01] [LathiDing09]:
BER: Bit Error Rate. The ratio of incorrectly received bits to the total number of transmitted bits. In chaotic systems, BER integrates the combined effects of channel AWGN (additive white Gaussian noise), synchronisation imperfections, quantisation errors, and transient settling before receiver convergence.
SNR: Signal-to-Noise Ratio. The ratio of average signal power to noise power, expressed in dB:
The BER-vs-SNR curve is the primary comparison tool between modulation schemes, identifying the SNR threshold required for acceptable operation (commonly \(\text{BER} \leq 10^{-3}\)).
The chaotic-pfc run dcsk command produces exactly this curve for
Pecora-Carroll synchronisation, classical DCSK, and EF-DCSK over an
FIR-filtered Hénon map with AWGN.
References¶
M. S. Baptista. “Chaos-based communication systems: Current trends and challenges.” Springer, 2021.
V. S. Borges, M. Eisencraft. “A filtered Hénon map.” Chaos, Solitons and Fractals, vol. 165, 2022.
V. S. Borges, R. Silva, M. Eisencraft. “Stability analysis of the filtered Hénon map.” (preprint), 2024.
R. Fontes, M. Eisencraft. “A digital bandlimited chaotic communication system.” Commun. Nonlinear Sci. Numer. Simul., vol. 37, pp. 374–385, 2016.
S. Haykin. “Communication Systems.” 4th ed., Wiley, 2001.
M. Hénon. “A two-dimensional mapping with a strange attractor.” Commun. Math. Phys., vol. 50, pp. 69–77, 1976.
B. P. Lathi, Z. Ding. “Modern Digital and Analog Communication Systems.” 4th ed., Oxford University Press, 2009.
G. Kolumban, M. P. Kennedy, G. Kis. “Performance evaluation of FM-DCSK.” Proc. IEEE ISSSTA, pp. 686–690, 1996.
G. Kaddoum, E. Soujeri, C. Arcila, K. Eshteiwi. “Design and performance analysis of a multiuser OFDM based differential chaos shift keying communication system.” IEEE Trans. Commun., v. 61, n. 12, pp. 4908–4920, 2013.
E. V. Oliveira, R. Filgueiras. “A importância da segurança da informação para as organizações.” Revista Alomorfia, v. 6, n. 1, pp. 438–447, 2022.
A. V. Oppenheim, R. W. Schafer. “Discrete-Time Signal Processing.” 3rd ed., Pearson, 2009.
L. M. Pecora, T. L. Carroll. “Synchronization in chaotic systems.” Physical Review Letters, v. 64, n. 8, p. 821, 1990.
A. Sapinski. “The Importance and Challenges of Information Security in the Digital Age.” Scientific Journal of Bielsko-Biala School of Finance and Law, v. 1, pp. 52–55, 2023.