Lagged Logistic–Sine Map Analysis

• Lagged Logistic–Sine Map is a hybrid dynamical system that blends logistic and sine nonlinearities with modular arithmetic to produce complex bifurcation diagrams.
• Analytical methods like the 0–1 Test and Three-State Test reveal distinct chaotic windows and quasi-periodic intervals across varying parameter ranges.
• The map’s optimal chaotic regions, particularly for specific r values, make it a valuable tool for pseudo‐random generation and cryptographic applications.

A piecewise hybrid map in the context of one-dimensional discrete-time dynamical systems refers to a mathematical function constructed by combining structural elements of different classical maps, typically through additive or multiplicative blending and modular arithmetic. These systems are rigorously studied for their ergodic and chaotic properties, with particular interest in their applications as pseudo-random generators in cryptography. The Logistic–Sine Map (abbreviated as LSM or LSS in the literature) is a representative example of such piecewise hybrid maps, combining logistic and sine map dynamics to yield richer bifurcation and chaos-inducing behavior than either of its constituents alone (Muthu et al., 2020).

1. Mathematical Definition and Structure

The Logistic–Sine Map defines its state evolution on the unit interval $[0,1]$ via the recurrence:

$$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$

Variables and parameters:

• $x_n$: current state in $[0,1]$
• $r$: control parameter (bifurcation parameter), $r\in(0,4]$
• $\bmod 1$ maintains the iterates in the unit interval
• $x_0\in(0,1)$, typically $x_0=0.01$

This construction yields a piecewise-hybrid dynamical system due to the mixture of polynomial (logistic) and transcendental (sine) nonlinearities, and the parameter-dependent contribution of each term controlled by $r$. The map generalizes the logistic map by embedding oscillatory behavior, leading to a multimodal and structurally richer bifurcation diagram.

2. Analytical and Empirical Chaos Detection Methods

Evaluation of chaotic behavior for such hybrid maps employs tools that move beyond the traditional Lyapunov exponent, which has been demonstrated as insufficient for measuring the true chaotic extent in these systems (Muthu et al., 2020).

2.1. The 0–1 Test for Chaos (Gottwald–Melbourne)

Given a time series $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$0 (with $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$1), the test proceeds:

• Select $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$2; in practice, $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$3 is used.
• Compute translation coordinates:

$$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$4

• The mean-square displacement is

$$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$5

• The asymptotic growth rate is summarized by

$$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$6

with $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$7 representing chaos, $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$8 indicating regular or quasi-periodic motion.

2.2. The Three-State Test (3ST)

The 3ST, due to Eyebe & Koepf, characterizes dynamical regimes through local ordinal pattern slopes:

• Divide the time series (length $$x_{n+1} = \left( r\,x_n(1-x_n) + \frac{4-r}{4}\,\sin(\pi x_n) \right) \bmod 1$$9) into overlapping blocks of length $x_n$0 ($x_n$1 used).
• For each block, calculate the empirical slope $x_n$2 and its sample statistics (mean, standard deviation).
• Growth indicator:

$x_n$3

• Interpretation: constant $x_n$4 implies periodicity; saturating $x_n$5 implies quasi-periodicity; unbounded $x_n$6 growth signals chaos.

3. Dynamical Properties Across Parameter Space

Numerical evaluation, systematically performed for $x_n$7, reveals that chaotic strength is not uniform and that the LSM's bifurcation diagram obscures subintervals of quasi-periodicity.

3.1. 0–1 Test Results

$x_n$8	3.15	3.25	3.35	3.45	3.55	3.65	3.75	3.85	3.95
$x_n$9	0.8335	0.7394	0.7692	0.6158	0.6739	0.7112	0.6438	0.5503	0.5137

• Strong chaos: $[0,1]$0 in $[0,1]$1
• Weaker chaos: $[0,1]$2 outside those windows, notably for $[0,1]$3

3.2. 3ST Classification

$[0,1]$4-range	3.10–3.19	3.20–3.29	3.30–3.39	3.40–3.49	3.50–3.59	3.60–3.69	3.70–3.79	3.80–3.89	3.90–3.99
Behavior	chaotic	chaotic	chaotic	quasi-periodic	quasi-periodic	chaotic	quasi-periodic	quasi-periodic	quasi-periodic

• Chaotic windows: $[0,1]$5 and $[0,1]$6
• Quasi-periodic: $[0,1]$7, $[0,1]$8

4. Comparison with Other One-Dimensional Maps

Extensive comparative analysis with the standard logistic map, the Logistic-Tent system (LTS), and the Tent-Sine system (TSS) demonstrates the hybrid map’s distinctive structure:

• The standard logistic map expresses chaos only for $[0,1]$9 and manifests periodic windows.
• The LTS and TSS feature broader chaotic intervals, yet are still interspersed by quasi-periodic subintervals.
• The LSM distinguishes itself by presenting two substantial chaotic windows beginning at lower $r$0 values (starting at $r$1), which is significant for applications requiring chaos at smaller parameter values (Muthu et al., 2020).

5. Cryptographic Applications and Parameter Selection

Piecewise hybrid maps such as the Logistic–Sine Map are desirable for pseudo-random generation and cryptographic primitives due to strong mixing properties and initial condition sensitivity. Practical guidance for parameter selection focuses on sustained regions of strong chaos:

• Recommended $r$2 intervals: $r$3 and $r$4
• Sweet-spot: $r$5 (yielding maximum $r$6 on the 0–1 test)
• Avoided regions: $r$7 or $r$8, where the system defaults to quasi-periodic or weakly chaotic dynamics and the mixing rates are suboptimal
• The combined application of the 0–1 Test and 3ST is vital, as Lyapunov exponents alone have been shown to misrepresent the true chaotic capacity of such systems (Muthu et al., 2020).

6. Practical Illustration and Visualization

The bifurcation diagram for the LSM exhibits what appears to be a full coverage of $r$9 for $r\in(0,4]$0, yet this visual impression conceals the presence of two quasi-periodic bands (around $r\in(0,4]$1 and $r\in(0,4]$2). The tabular results provided by both the 0–1 Test and 3ST afford a more granular and actionable assessment. This enables the delineation of robustly chaotic domains crucial for cryptosystem design, including image and data encryption, where unpredictability and ergodicity must be ensured (Muthu et al., 2020).

7. Broader Implications and Methodological Recommendations

The rigorous analytical framework applied to the LSM underscores the importance of composite diagnostic tools (such as 0–1 Test and 3ST) for genuine chaos certification in nonlinear maps, especially when used in security-critical contexts. The demonstrated non-uniformity in chaotic strength across the parameter space argues against the sole reliance on Lyapunov exponents and advocates for systematic multi-method testing in both theoretical studies and practical cryptographic implementations. A plausible implication is that future research on piecewise hybrid maps should incorporate both spectral and ordinal-pattern-based diagnostics to certify parameter ranges prior to any application—or risk substantial weaknesses due to hidden quasi-periodic intervals (Muthu et al., 2020).