From Chaos to Security: A Comparative Study of Lorenz and Rössler Systems in Cryptography

Abstract

Chaotic systems, governed by deterministic nonlinear equations yet exhibiting highly complex and unpredictable behaviors, have emerged as valuable tools at the intersection of mathematics, engineering, and information security. This paper presents a comparative study of the Lorenz and Rössler systems, focusing on their dynamic complexity and statistical independence—two critical properties for applications in chaos-based cryptography. By integrating techniques from nonlinear dynamics (e.g., Lyapunov exponents, KS entropy, Kaplan–Yorke dimension) and statistical testing (e.g., chi-square and Gaussian transformation-based independence tests), we provide a quantitative framework to evaluate the pseudo-randomness potential of chaotic trajectories. Our results show that the Lorenz system offers faster convergence to chaos and superior statistical independence over time, making it more suitable for rapid encryption schemes. In contrast, the Rössler system provides complementary insights due to its simpler attractor and longer memory. These findings contribute to a multidisciplinary methodology for selecting and optimizing chaotic systems in secure communication and signal processing contexts.

1. Introduction

Chaotic systems, known for their deterministic yet seemingly unpredictable behavior, have been extensively studied due to their sensitivity to initial conditions and their potential for modeling complex phenomena [1,2,3,4]. The concept of chaos was first introduced by Henri Poincaré in his work on the three-body problem, which laid the foundation for modern nonlinear dynamics [5]. Beyond their theoretical significance, chaotic systems have found numerous practical applications in the past decades. In particular, their potential in cryptography has been investigated with increasing interest, owing to their capacity to generate complex, unpredictable sequences [6].

Several approaches have been proposed to harness chaos for cryptographic purposes, including techniques based on running-key mechanisms combined with maps like the logistic map [7]. Moreover, studies on specific chaotic maps such as the tent map have focused on evaluating the statistical independence of the generated sequences, which is critical for cryptographic strength [8]. Recent work has further explored the connection between chaos, observability, and independence, applying rigorous tests to evaluate the chaotic output [9,10,11,12].

From a systems engineering perspective, chaotic systems offer opportunities for hardware implementation of secure random number generators, especially when deployed on platforms such as FPGAs [13,14]. These implementations bring together expertise from nonlinear dynamics, digital logic design, and statistical modeling, illustrating the inherently interdisciplinary nature of this research domain.

The theoretical framework underpinning chaotic systems builds on nonlinear differential equations and bifurcation theory, as detailed in classical studies of dynamical systems [15]. In parallel, empirical methods are necessary to validate properties such as statistical independence in the generated sequences [16]. These methods often rely on statistical modeling principles and test theory [17].

The increasing relevance of chaos-based cryptography is also evidenced by comprehensive studies that unify theoretical modeling with algorithmic implementation [18]. Foundational work on synchronization in chaotic systems [19], as well as proposals for practical chaotic cryptosystems [20,21], has laid the groundwork for current developments. Among these, the logistic map has been explored as a core primitive in encryption algorithms due to its simplicity and chaotic behavior [22].

Interdisciplinarity is further exemplified by research that combines mathematical modeling with information security frameworks. For instance, chaos-based public-key cryptography incorporates principles from number theory, dynamical systems, and complexity science [23]. In addition, implementations of pseudo-random number generators based on digitized chaotic signals demonstrate the integration of analog signal theory and cryptographic design [24].

High-dimensional chaotic maps have also been introduced for increasing the entropy of encrypted data streams [25], while chaos-based hash functions have been proposed for ensuring data integrity and authenticity [26]. The utility of chaos extends to video encryption as well, indicating its multidimensional application potential in modern information systems [27]. Nonetheless, the robustness of such systems remains subject to cryptanalytic scrutiny, emphasizing the need for careful system design and vulnerability analysis [28].

Although much of the current literature implements chaos-based cryptography using discretized maps for efficiency, our analysis covers both continuous-time integration (for theoretical completeness) and discrete-time formulations (for modern hardware relevance). The present paper investigates the Lorenz and Rössler chaotic systems with respect to their ability to generate statistically independent sequences suitable for cryptographic use. Through a comparative analysis, we evaluate their performance using a suite of independence and chaos-related metrics.

In addition to FPGA-based and numerically simulated chaotic systems, an active research direction has emerged in recent decades on the use of optical chaos for secure communications and random number generation. Seminal works in this area have demonstrated high-speed chaotic carriers for encrypted communication [29], optical-feedback-based chaotic lasers for physical random number generation [30], and integrated photonic prototypes capable of generating random bits at hundreds of Gbps rates [31,32]. From a systems engineering perspective, these optical approaches share many of the same fundamental properties as the dynamical systems considered in this paper. Consequently, the independence and chaoticity metrics proposed here could be applied to evaluate and compare optical-chaos-based cryptographic systems, complementing their traditional performance metrics such as bit rate and physical layer robustness.

The structure of this paper is organized as follows:

• Section 2 outlines the theoretical background of chaotic systems and describes the independence metrics employed in our experiments.
• Section 3 presents and interprets the experimental results, focusing on the comparative performance of the Lorenz and Rössler systems.
• Section 4 summarizes our findings and suggests future directions in the context of chaos-based encryption systems.

2. Motivation and Theoretical Background

This section discusses the motivation of the paper, the mathematical description of the Lorenz and Rössler systems, and the statistical independence test used in this study.

2.1. Motivation

One of the motivations for this paper comes from a simple encryption experiment: we used a Vernam cipher, applying a modulo-2 sum between a grayscale image and a binarized chaotic sequence. The chaotic sequences were generated using the Lorenz and Rössler systems by numerically integrating their differential equations in MATLAB R2023b, using the fourth-order Runge–Kutta method with a fixed time step of 0.01. The continuous signals (typically the x-component) were then binarized using a threshold set at the mean value of the series. The result of the encryption is illustrated in Figure 1, and it is visually clear that the image encrypted using Rössler still preserves part of the original content, while the Lorenz-encrypted version appears almost unrecognizable. This observation triggered our desire to better understand why certain systems offer stronger randomness features than others and how this can be measured systematically. The following sections attempt to answer this by proposing a structured, quantitative methodology to compare chaotic systems. This procedure is meant to assist researchers or engineers in making informed decisions when selecting between candidate systems for chaos-based encryption.

2.2. Numerical Formulations for Chaos-Based Cryptography: Continuous-Time Integration vs. Discrete-Time Maps

Chaos-based cryptography can rely on either (i) continuous-time models, where the underlying ODE

$$\dot{x}=f(x;\theta )$$

is integrated numerically at a fixed step h, or (ii) discrete-time maps, where one iterates a native recurrence

$$x_{n+1}=F_h(x_n;\theta ).$$

From a systems point of view, both instantiate a dynamical system on which the same analysis metrics (Lyapunov exponents, KS entropy, Kaplan–Yorke dimension, autocorrelation/correlation time, independence tests) can be consistently applied.

2.2.1. Runge–Kutta (Continuous-Time View)

Given an ODE such as the Lorenz system,

$$\dot{x}=\sigma (y-x),\dot{y}=x(\rho -z)-y,\dot{z}=xy-\beta z,$$

a single RK4 step at step size h defines the one-step flow approximation

$$x_{n+1}=x_n+\frac{h}{6}\left(k_1+2k_2+2k_3+k_4\right),$$

with $k_1=f\left(x_n\right),k_2=f(x_n+\frac{h}{2}{k}_1),k_3=f(x_n+\frac{h}{2}{k}_2),k_4=f(x_n+h{k}_3).$ Although derived as a numerical integrator, this recursion is itself a discrete map $F_h^{\mathrm{RK}4}$ parameterized by h. For sufficiently small h, RK4 preserves the qualitative chaotic features (positive ${\lambda }_1$, strange attractor geometry) while yielding lower local truncation error than first-order schemes.

2.2.2. Explicit Discretization (Discrete-Time View)

A common alternative in practical cryptographic implementations is to work directly with an explicit map, e.g., the forward–Euler discretization:

$$\begin{array}{cc}\hfill x_{n+1}& =x_n+h\sigma (y_n-x_n),\hfill \\ \hfill y_{n+1}& =y_n+h[x_n(\rho -z_n)-y_n],\hfill \\ \hfill z_{n+1}& =z_n+h[x_n{y}_n-\beta z_n],\hfill \end{array}$$

which defines ${\mathbf{F}}_h^{\mathrm{Euler}}$. This approach is appealing for hardware (FPGA/DSP/MCU) because it is algebraically simple, supports fixed-point arithmetic, and avoids runtime ODE solvers. In practice, the discrete map is then digitalized via scaling/quantization and modular arithmetic; a keystream is extracted from selected bits and optionally whitened (e.g., XOR/rotations).

2.2.3. Conceptual Comparison Relevant to Our Metrics

For the same parameters $\theta$ and small h, (i) both ${\mathrm{F}}_h^{\mathrm{RK}4}$ and ${\mathrm{F}}_h^{\mathrm{Euler}}$ preserve the ordering of systems by chaoticity (e.g., Lorenz remains more chaotic than Rössler); (ii) first-order schemes typically introduce extra numerical damping, which leads to slightly smaller maximal Lyapunov exponent ${\lambda }_1$ and KS entropy, a marginally smaller KY dimension, a slower decay of autocorrelation (hence mildly larger correlation time), and a modestly larger temporal separation d required by independence tests; and (iii) Fourier spectra remain consistent qualitatively richer (low-frequency content for Lorenz, simpler for Rössler), with some smoothing under Euler. These differences are quantitative rather than qualitative when h is sufficiently small and no aggressive quantization is applied.

2.2.4. Scope with Respect to Hardware-Oriented Discretizations

When moving to fixed-point and modulo operations, additional phenomena (period-shortening, lattice effects) may arise; these are design issues of the digital PRNG rather than of the continuous dynamics per se. Our methodology—Lyapunov spectrum, KS entropy, KY dimension, autocorrelation/correlation time, and independence tests—remains applicable to the discrete map ${\mathbf{F}}_h$ before or after digitalization and provides a uniform, system-agnostic way to compare chaotic sources intended for cryptographic use.

It is important to emphasize that in many chaos-based cryptographic designs, one starts directly from discrete maps rather than from continuous-time flows. This approach avoids any reference to differential equations and uses native recurrences as the source of pseudo-randomness. In this context, the notion of discrete Lyapunov exponents arises naturally: they can be computed directly from the Jacobian of the iterated map and provide a rigorous characterization of sensitivity to initial conditions in purely discrete systems. Foundational works in this direction include Fridrich’s two-dimensional map cipher [33], the theory of discrete chaos developed by Kocarev and co-authors [34], and Amigó’s exposition on chaos-based cryptography [35]. Our present study focused on continuous-time systems and their numerical discretizations; however, the independence and chaoticity metrics we propose can be consistently applied to purely discrete maps as well. We therefore regard the extension of our methodology to discrete Lyapunov exponents as an important direction for future research.

2.3. Independence Test and Transient Time

Statistical independence tests are used to determine whether two sets of data are independent. In this context, we used two versions of independence tests. The first version of the independence test used [6] involves several steps:

• Sorting the data sets X and Y.
• Calculating the empirical distribution functions $F_{ex}$ and $F_{ey}$.
• Transforming the values into standardized variables U and V using the inverse normal distribution function.
• Computing the correlation coefficient $\rho$ and the test statistic t.
• Comparing the value of t with the critical value $t_{\alpha /2}$ to determine the test result.

The data sets X and Y are transformed into new sets U and V which are normally distributed. The test checks if these two sets conform to a bivariate Gaussian probability law. The hypothesis tested are:

• $H_0$: The two data sets conform to the bivariate Gaussian probability law, indicating that U and V are statistically independent.
• $H_1$: The two data sets do not conform to the bivariate Gaussian probability law.

The improved version of the independence test [16] extends the work from [6] by including an additional analysis:

• Calculating the chi-square distribution to check the uniformity of the distributions of U and V.
• Comparing the value of z with the critical value $z_{\alpha }$ to determine the test result.

The chi-square test is used to classify the data into classes and compute the test value using the formula:

$$z=\sum _{k=1}^M{\displaystyle \frac{{(m_k-N{P}_k)}^2}{N{P}_k}}$$

(1)

where M is the number of classes, $m_k$ is the frequency of class k, N is the data volume, and $P_k$ is the theoretical probability assigned to class k.

The transient time is the period required for a dynamic system to reach a stable or chaotic behavior. In the analysis of the Lorenz system, the transient time is determined based on a visual method and used to select relevant time points for independence analysis (after the transient period ends).

To determine the transient time, the following steps are taken:

• Visualize the distribution of the system states at different time points.
• Identify the time after which the experimental probability density functions stabilize.
• Use this time as the transient time for further analysis.

The transient time is crucial for ensuring that the system has reached a stable state before performing any statistical analysis. This helps in obtaining accurate and reliable results.

2.4. Chaoticity Metrics

2.4.1. Lyapunov Exponents

The Lyapunov coefficients/exponents were the first metric used to compare the two systems of interest in this paper. Lyapunov exponents measure the rates at which nearby trajectories in a dynamical system diverge or converge. They provide insight into the system’s stability and chaotic behavior. A positive Lyapunov exponent indicates chaos, as it signifies exponential divergence of nearby trajectories. A zero exponent corresponds to neutral stability, while a negative exponent indicates convergence and stability.

The calculation involves integrating the system’s equations to generate trajectories, tracking the evolution of small perturbations (tangent vectors), and periodically orthogonalizing these vectors to maintain numerical accuracy. The growth rates of the tangent vectors are averaged over time to obtain the Lyapunov exponents. In essence, Lyapunov exponents quantify the sensitivity of a system to initial conditions, revealing its long-term behavior and stability characteristics.

2.4.2. Kolomogorov–Sinai Entropy and Kaplan–Yorke Dimension

The Kolmogorov–Sinai (KS) entropy measures the rate of information growth in a dynamical system and serves as an indicator of chaos. Essentially, KS entropy quantifies how unpredictable a system becomes over time. A system with high KS entropy is more chaotic and sensitive to initial conditions.

The Kaplan–Yorke (KY) dimension is a measure of the complexity of a dynamical system and is related to the fractal dimension of the system’s attractor. It provides an estimate of the number of independent variables needed to fully describe the system’s behavior.

Calculation method for KS and KY metrics:

• Calculation of Lyapunov Exponents: First, the Lyapunov exponents of the system are calculated, as described previously.
• KS Entropy: It is calculated by summing the positive Lyapunov exponents. This represents the total rate of information growth in the system.
• KY Dimension: It is calculated using the Lyapunov exponents and their cumulative sum. The largest index j for which the cumulative sum is positive is identified, and the KY dimension is given by Formula (2):

  $$D_{KY}=j+{\displaystyle \frac{{\sum }_{i=1}^j{\lambda }_i}{|{\lambda }_{j+1}|}}$$

  (2)

  where ${\lambda }_i$ are the Lyapunov exponents ordered in descending order.

These two metrics provide valuable insights into the complexity and chaos of a dynamical system, helping to understand its long-term behavior.

2.5. The Autocorrelation Function and the Correlation Time

The next metric set is represented by the autocorrelation function and correlation time. The autocorrelation function measures the correlation of a signal with a delayed version of itself over varying time lags. It provides insight into the periodicity and memory of the system. The autocorrelation function is defined as Equation (3):

$$R\left(\tau \right)={\displaystyle \frac{1}{N{\sigma }^2}}\sum _{t=1}^{N-\tau }(x_t-\mu )(x_{t+\tau }-\mu )$$

(3)

where $x_t$ is the signal at time t, $\mu$ is the mean of the signal, $\tau$ is the time lag, and N is the total number of data points.

Equation (3) includes the normalization factor ${\sigma }^2$, ensuring that the autocorrelation function is normalized, as is customary in statistical practice.

The correlation time is a measure of the time over which the autocorrelation function decays significantly. It provides an estimate of how long the system retains memory of its past states. The correlation time ${\tau }_c$ is typically defined as the time lag at which the autocorrelation function falls to $1/e$ of its initial value.

Calculation method:

• The autocorrelation function is computed by correlating the signal with itself over a range of time lags. This involves calculating the mean of the signal, subtracting it from the signal, and then computing the correlation for each time lag.
• The correlation time is determined by finding the time lag at which the autocorrelation function decays to $1/e$ of its initial value.

These metrics provide valuable insights into the periodicity and memory of a dynamical system, helping to understand its temporal behavior.

Fourier Spectrum

The Fourier spectrum is a representation of a signal in the frequency domain. It shows how the signal’s energy is distributed across different frequencies. This metric is useful for identifying periodic components and understanding the frequency characteristics of a system.

Calculation method:

• The Fourier transform is applied to the time-series data of the system to convert them from the time domain to the frequency domain. This is typically done using the Fast Fourier Transform (FFT) algorithm.
• The power spectrum is obtained by taking the square of the magnitude of the Fourier transform. This represents the energy of the signal at each frequency.
• The frequencies and their corresponding power values are plotted to visualize the distribution of energy across frequencies.

The Fourier spectrum provides valuable insights into the periodicity and dominant frequencies of a dynamical system, helping to understand its oscillatory behavior.

3. Experimental Results

Here, we present the experimental comparison of the Lorenz and Rössler systems, highlighting their differences in transient time, statistical independence, and the metrics described in detail in Section 2.

The Lorenz system is a set of three coupled, nonlinear differential equations originally developed by Edward Lorenz in 1963 to model atmospheric convection. The equations are given by Equation (4):

$$\left\{\begin{array}{c}{\displaystyle \frac{dx}{dt}}=\sigma (y-x)\hfill \\ {\displaystyle \frac{dy}{dt}}=x(\rho -z)-y\hfill \\ {\displaystyle \frac{dz}{dt}}=xy-\beta z\hfill \end{array}\right.$$

(4)

where x, y, and z represent the system’s state variables, and $\sigma$, $\rho$, and $\beta$ are parameters. Typically, $\sigma =10$, $\rho =28$, and $\beta ={\displaystyle \frac{8}{3}}$. The Lorenz system is known for its chaotic behavior, which is highly sensitive to initial conditions.

The Rössler system, introduced by Otto Rössler in 1976, is another example of a chaotic system. It consists of three nonlinear differential equations shown in Equation (5):

Although the Rössler system is less common in chaos-based cryptographic algorithms compared to maps such as Logistic, Tent, or Henon, its inclusion in our study is deliberate. First, Rössler’s attractor exhibits fundamentally different dynamical features from Lorenz, enabling us to test the robustness and generality of our chaoticity and independence metrics across distinct classes of chaotic flows. Second, despite its lower prevalence, recent works have demonstrated successful applications of the Rössler system in cryptographic contexts, including S-box generation [36], image scrambling and diffusion [37], and hybrid encryption schemes [38,39]. By evaluating Rössler alongside Lorenz, we ensure that our methodology is not tailored to a single, widely adopted chaotic system but is broadly applicable to both well-established and less conventional chaotic sources, thereby strengthening its relevance for diverse cryptographic designs.

$$\left\{\begin{array}{c}{\displaystyle \frac{dx}{dt}}=-y-z\hfill \\ {\displaystyle \frac{dy}{dt}}=x+ay\hfill \\ {\displaystyle \frac{dz}{dt}}=b+z(x-c)\hfill \end{array}\right.$$

(5)

where x, y, and z are the state variables, and a, b, and c are parameters. The values $a=0.1$, $b=0.1$, and $c=14$ are valid and frequently used in several studies. However, they are not the most widely cited in the literature. Another well-known parameter set, often reported in the literature, is $a=0.2$, $b=0.2$, and $c=5.7$. Both choices are valid, and our results are not limited to one specific configuration. The Rössler system exhibits chaotic dynamics and is often used to study the properties of chaotic attractors.

Figure 2 shows the evolution of the probability density function in time for the Lorenz system. The plots were obtained by iterating over time 1000 initial conditions (x, y, and z), normally distributed (mean = 0 and variance = 1), and analyzing the histograms in order to identify the moment when the distribution stabilizes (the transient time is surpassed). It can be noticed that after 20–30 s, the distribution of the values of x remains almost unchanged, and this is valid for the other state variables too (y and z) and later in time. We can say with a high degree of confidence based on this visual method that the transient time for the Lorenz system is 30 s [10], so any independence assessment for the Lorenz system should be conducted after this moment in time.

A similar plot can be obtained for the second system analyzed, the Rössler system, in Figure 3. It can be noticed that the behavior is the same as that of the Lorenz system, i.e., the probability density functions for x vary at the beginning of the iterations and stabilize in time. The difference is that the transient time for the Rössler system is higher than for the Lorenz system, 100 s being a reasonable estimate (compared to 30 s for the Lorenz system).

The results from Figure 2 and Figure 3 already show a difference between the two chaotic systems analyzed: Rössler seems to be “slower” in getting over the transient time window. Let us visualize the two systems from the perspectives of the next metrics proposed and check if any general conclusion emerges.

From a Lyapunov coefficients point of view, the Lorenz system is more “chaotic”, which can be noticed by the high positive value of ${\lambda }_1$ in Figure 4. A positive ${\lambda }_1$ indicates that trajectories diverge exponentially in the primary direction, which is a hallmark of chaotic behavior. Both the Lorenz and Rössler systems exhibit this characteristic, with the Lorenz system showing much stronger chaos due to its higher ${\lambda }_1$ value. Negative ${\lambda }_2$ and ${\lambda }_3$ indicate that trajectories converge exponentially in the secondary and tertiary directions, providing stability in those directions. This is common in chaotic systems, where there is a mix of instability and stability. In summary, both the Lorenz and Rössler systems exhibit chaotic behavior, but the Lorenz system has much stronger chaos (higher ${\lambda }_1$) compared to the Rössler system.

Another approach is to compare the Lorenz and Rössler chaotic systems based on the KS entropy and KY dimension introduced in Section 2. Figure 5 depicts the comparison between the two metrics for both systems.

KS entropy measures the rate of information production in a dynamical system. Higher values indicate a higher degree of chaos, meaning the system is more unpredictable and complex. In Figure 5, we see the following:

• The Lorenz system has a KS entropy of 10.8929, indicating a high level of chaos.
• The Rössler system has a KS entropy of 0.0160, indicating a much lower level of chaos compared to the Lorenz system.

The KY dimension estimates the fractal dimension of the attractor in the system, which reflects the complexity of the system’s behavior. Higher values indicate a more complex and higher-dimensional attractor. In Figure 5, we see the following:

• The Lorenz system has a KY dimension of 1.3207, suggesting a moderately complex attractor.
• The Rössler system has a KY dimension of 1.0030, indicating a simpler attractor compared to the Lorenz system.

To sum up, the Lorenz system exhibits higher chaos and complexity compared to the Rössler system, as indicated by the higher KS entropy and KY dimension values. These metrics help quantify the chaotic behavior and complexity of dynamical systems.

The autocorrelation function and the correlation time lead to a similar conclusion. As it can be noticed in Figure 6, the autocorrelation function for the Lorenz system falls much faster than that of the Rössler system, and it is more oscillatory in nature. The correlation time is smaller for Lorenz’s compared to Rössler’s system (the point in time where the autocorrelation function becomes 1/e).

The autocorrelation function for the Lorenz system shows a more complex, oscillatory pattern. This indicates that the Lorenz system has intricate temporal dependencies and exhibits periodic behavior. The oscillations suggest that the system’s state at one point in time is influenced by its state at previous time points, leading to recurring patterns. The autocorrelation function for the Rössler system displays a smooth, parabolic shape peaking at lag zero. This indicates that the Rössler system has simpler temporal dependencies compared to the Lorenz system. The smooth curve suggests that the system’s state is less influenced by its past states, leading to a more straightforward and predictable behavior.

The correlation time for the Lorenz system is 33. This indicates a faster decay of correlations in the system, suggesting that the Lorenz system loses its memory of initial conditions relatively quickly. From a chaos theory perspective, this means that the Lorenz system exhibits more chaotic behavior, with rapid divergence of trajectories and sensitivity to initial conditions. The correlation time for the Rössler system is 108. This indicates a slower decay of correlations in the system, suggesting that the Rössler system retains its memory of initial conditions for a longer period compared to the Lorenz system. From a chaos theory perspective, this means that the Rössler system exhibits less chaotic behavior, with slower divergence of trajectories and less sensitivity to initial conditions.

The comparison of the Fourier spectra for the Lorenz and Rössler (Figure 7) systems reveals distinct characteristics that can be interpreted from a chaos theory perspective. The Fourier spectrum of the Lorenz system shows a series of peaks at lower frequencies with diminishing amplitude as frequency increases. This indicates that the Lorenz system has multiple dominant frequencies at which it oscillates. From a chaos theory perspective, the presence of these peaks suggests that the Lorenz system exhibits complex or chaotic behavior, with a rich frequency content and rapid divergence of trajectories. The system’s sensitivity to initial conditions is reflected in the varied frequency components. The Fourier spectrum of the Rössler system has a single dominant peak close to 0 Hz frequency and quickly drops to near zero for other frequencies. This indicates that the Rössler system has a simpler oscillatory behavior compared to the Lorenz system. From a chaos theory perspective, the presence of a single dominant peak suggests that the Rössler system exhibits less chaotic behavior, with slower divergence of trajectories and lower sensitivity to initial conditions. The system retains its memory of initial conditions for a longer period, as indicated by the slower decay of correlations. In summary, the Lorenz system exhibits more chaotic behavior with a richer frequency content, while the Rössler system exhibits less chaotic behavior with a simpler frequency spectrum. This comparison highlights the differences in their oscillatory behavior and dominant frequencies, providing insights into their chaotic dynamics.

In our comparative study of the Lorenz and Rössler systems up to this moment, we investigated several key metrics to evaluate their chaotic behavior and suitability for cryptographic applications. The Lorenz system demonstrated a shorter transient time to chaos, stabilizing after approximately 30 s, compared to the Rössler system, which required around 100 s. Lyapunov exponents revealed that the Lorenz system exhibited stronger chaos, with higher positive values indicating rapid divergence of trajectories. The Kolmogorov–Sinai entropy and Kaplan–Yorke dimension further confirmed the Lorenz system’s higher complexity and unpredictability. Autocorrelation functions showed that the Lorenz system had a faster decay of correlations, suggesting quicker loss of memory of initial conditions. The Fourier spectra analysis highlighted the richer frequency content of the Lorenz system, indicating more intricate oscillatory behavior. Overall, the Lorenz system’s higher sensitivity to initial conditions and rapid transition to chaos make it more suitable for generating pseudo-random sequences and enhancing cryptographic security. In contrast, the Rössler system, with its simpler dynamics and slower transition to chaos, offers complementary insights but is less optimal for rapid encryption schemes.

It is worth visualizing the statistical independence results for the two chaotic systems. In order to obtain the statistical independence results, the following steps were followed:

• The data were generated as mentioned at the beginning of Section 3.
• A thousand random initial conditions ($x, y, z$) were generated over time by solving the differential set of equations with a sampling frequency of 100 Hz.
• After the transient time elapsed (more than 30 s for Lorenz and 100 s for Rössler), a data set was selected at moment $k_1$ and a second data set was selected at moment $k_2=k_1+d$. We used state variable x for this purpose, but similar conclusions can be obtained for y and z.
• These yielded the X and Y variables described in Section 2, which were further transformed into U and V, on which the independence tests were applied (the two versions of the Badea–Vlad test [6,16]).

Figure 8 shows the results of the execution of the independence algorithm for the Lorenz system and the scatter plots obtained for different distances tested. The test passed, and this can be seen from the scatter plot, which looks close to the scatter plot of two jointly Gaussian random variables when the distance is higher or equal to 10 s. For $d<10$ s, there is still a degree of dependency between the variables. Note: a more accurate analysis can be conducted to find the exact statistical independence distance, but for the purpose of this study, this result is sufficient, as the Rössler statistical independence distance is much higher, by at least an order of magnitude.

Figure 9 shows the results of the execution of the independence algorithm for the Rössler system and the scatter plots obtained for different distances tested. The test failed for d up to 4800 s, when the test passed, but the scatter plot was not as expected. This statistical independence figure was much higher than the one obtained for the Lorenz system, a result which aligned with the previous ones obtained when comparing the two systems from the point of view of the different chaotic metrics. The conclusion based on this metric, the statistical independence distance, is that the Lorenz system is much more suitable from a computational point of view for generating pseudo-random data compared with the Rössler system.

Comparison of Results: RK4 Integration vs. Explicit Discrete Map

To address the relevance of discrete-time formulations in modern chaos-based cryptography, we repeated the entire analysis pipeline of Section 3 using a simple forward–Euler discretization of the Lorenz and Rössler systems, with the same parameters, initial conditions, and step size $h=0.01$. The discrete map was defined by:

$$\begin{array}{c}{x}_{n+1}=x_n+h\sigma (y_n-x_n),\hfill \\ y_{n+1}=y_n+h[x_n(\rho -z_n)-y_n],\hfill \\ z_{n+1}=z_n+h[x_n{y}_n-\beta z_n],\hfill \end{array}$$

and analogously for the Rössler equations. No fixed-point quantization or modulo operations were applied in order to isolate the numerical–integration effect.

Table 1. Relative changes (Euler vs. RK4) for the metrics in Section 3 ($h=0.01$). Positive $\Delta$ means an increase compared to RK4.

Metric	Lorenz $\mathbf{\Delta }$	Rössler $\mathbf{\Delta }$	Interpretation	Qualitative Impact
${\lambda }_1$	$-3.8\%$	$-4.5\%$	Slight reduction in maximal divergence rate	Minor
KS entropy	$-3.9\%$	$-4.7\%$	Directly tied to ${\lambda }_1$	Minor
KY dimension	$-1.5\%$	$-1.8\%$	Attractor complexity slightly lower	Negligible
Correlation time ${\tau }_c$	$+4.1\%$	$+5.5\%$	Slower decorrelation	Minor
Independence distance d	$+0.5\mathrm{s}$	$+1.2\mathrm{s}$	Requires slightly larger separation	Minor
Transient time	≈	≈	No perceptible change	None

summarizes the quantitative differences between RK4 and Euler for the main metrics considered. All values were averaged over the same simulation duration and number of initial conditions; differences are reported relative to the RK4 baseline.

Qualitatively, Fourier spectra remained consistent in their envelope and dominant frequencies, with a slight smoothing in the Euler case. The probability density functions were visually indistinguishable at the binning resolution used in Figure 1.

Overall, while the discrete map introduces mild numerical damping (slightly lower ${\lambda }_1$, KS, KY; slightly higher ${\tau }_c$ and d), the key conclusions of Section 3 remain unaffected: the Lorenz system attains useful statistical properties for cryptography with a much shorter independence distance and transient time than the Rössler system, regardless of whether the underlying model is integrated via RK4 or iterated as a discrete map.

4. Conclusions

This section summarizes the key findings, discusses their implications, and suggests potential future research directions.

In our comparative study of the Lorenz and Rössler systems, we investigated several key metrics to evaluate their chaotic behavior and suitability for cryptographic applications. The Lorenz system demonstrated a shorter transient time to chaos, stabilizing after approximately 30 s, compared to the Rössler system, which required around 100 s. This indicates that the Lorenz system reaches a chaotic state more rapidly, making it more efficient for applications requiring quick transitions to chaos.

Lyapunov exponents revealed that the Lorenz system exhibited stronger chaos, with higher positive values indicating rapid divergence of trajectories. This suggests that the Lorenz system is more sensitive to initial conditions, a desirable property for generating pseudo-random sequences in cryptographic applications. In contrast, the Rössler system, while still chaotic, showed less sensitivity to initial conditions, as indicated by its lower Lyapunov exponents.

The Kolmogorov–Sinai entropy and Kaplan–Yorke dimension further confirmed the Lorenz system’s higher complexity and unpredictability. The higher KS entropy value for the Lorenz system indicates a greater rate of information production, which is beneficial for creating secure cryptographic keys. The KY dimension also suggests that the Lorenz system has a more complex attractor, providing a richer source of pseudo-randomness.

Autocorrelation functions showed that the Lorenz system had a faster decay of correlations, suggesting quicker loss of memory of initial conditions. This rapid decay is advantageous for cryptographic applications, as it ensures that the generated sequences are less predictable. The Rössler system, with its slower decay of correlations, retains memory of initial conditions for a longer period, which may be less desirable for certain cryptographic uses.

The Fourier spectra analysis highlighted the richer frequency content of the Lorenz system, indicating more intricate oscillatory behavior. The presence of multiple dominant frequencies in the Lorenz system suggests a higher degree of complexity and chaos, which is beneficial for generating diverse and unpredictable sequences. The Rössler system, with its simpler frequency spectrum, exhibits less chaotic behavior and may be less suitable for applications requiring high levels of pseudo-randomness.

Overall, the Lorenz system’s higher sensitivity to initial conditions and rapid transition to chaos make it more suitable for generating pseudo-random sequences and enhancing cryptographic security. In contrast, the Rössler system, with its simpler dynamics and slower transition to chaos, offers complementary insights but is less optimal for rapid encryption schemes. Future research could explore the potential of combining these systems to leverage their respective strengths and further enhance cryptographic security.

Furthermore, this study opens the path to a structured and clear methodology for selecting chaotic systems for various applications. By systematically evaluating key metrics such as transient time, Lyapunov exponents, KS entropy, KY dimension, autocorrelation functions, and Fourier spectra, researchers can make informed decisions about the most suitable chaotic systems for specific purposes. This approach provides a robust framework for optimizing chaos-based applications, ensuring that the chosen systems meet the desired criteria for complexity, unpredictability, and efficiency.