Logistic-Sine Map: A Hybrid Chaos Model

• LSM is a one-dimensional piecewise hybrid system that blends logistic and sine functions to yield robust chaotic behavior, with key chaotic windows identified at r in [3.10, 3.39] and [3.60, 3.69].
• Advanced chaos quantifiers, including the 0–1 test (with maximal K ≈ 0.8335 at r ≈ 3.15) and the three-state test, rigorously differentiate chaotic from quasi-periodic regimes.
• Compared to classical maps, LSM offers broader, lower-parameter rich chaos regimes, making it especially valuable for cryptographic systems and randomness-based applications.

A piecewise hybrid map, with the Logistic–Sine Map (LSM) as a key exemplar, is a one-dimensional dynamical system designed by combining classical map structures such as the logistic and sine maps in a nonlinear, piecewise fashion. The intent is to modulate the chaotic characteristics of the underlying system—specifically, to optimize the system’s strong chaos intervals for applications in chaos-based cryptography and related areas. Rigorous comparative analyses using advanced chaos quantifiers, such as the 0–1 test and the three-state test (3ST), reveal non-uniformity in the chaotic strength of such maps. The combination in the LSM yields unique parametric intervals with robust chaotic behavior at lower control-parameter values than competing classical maps (Muthu et al., 2020).

1. Mathematical Formulation

The LSM is defined for real-valued iterates $x_n \in [0,1]$ and a single bifurcation parameter $r\in(0,4]$:

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

where:

• $x_0\in(0,1)$ (typically $x_0 = 0.01$),
• The logistic map $r x_n (1 - x_n)$ and the sine map term $\sin(\pi x_n)$ are combined with a weighted sum that depends on $r$,
• $\bmod 1$ enforces confinement to the unit interval.

This formulation subsumes the classical logistic and sine maps as particular cases and provides tunability by $r$, where the contribution of each term varies continuously. The mapping is known to exhibit both strong and weak chaos, as well as quasi-periodic behavior, depending on the parameter $r\in(0,4]$0.

2. Chaos Quantification Methodologies

Characterizing the chaoticity of hybrid piecewise maps necessitates more than the Lyapunov exponent due to nonuniform chaos strength. The leading methodologies include:

A. 0–1 Test for Chaos (Gottwald–Melbourne):

• Given a scalar time series $r\in(0,4]$1, derive translation variables: $r\in(0,4]$2, $r\in(0,4]$3, for a constant $r\in(0,4]$4 ($r\in(0,4]$5 in the cited work).
• Compute the mean-square displacement $r\in(0,4]$6 and the growth rate $r\in(0,4]$7.
• $r\in(0,4]$8 indicates chaos; $r\in(0,4]$9 indicates regular dynamics.

B. Three-State Test (3ST):

• Partition the series into overlapping blocks (block size $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$0, total $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$1).
• For each block, compute state-transition slopes and their statistics.
• Average slope $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$2 and its standard deviation $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$3 are used to define a growth indicator $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$4, from which $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$5 is derived.
• Saturated $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$6 indicates quasi-periodicity, unbounded growth of $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$7 indicates chaos.

Both methods provide parametric diagnostics of chaoticity beyond Lyapunov exponents, relevant for rigorous cryptosystem assessment (Muthu et al., 2020).

3. Numerical Characterization of Chaotic Intervals

Quantitative analysis via the 0–1 test and 3ST reveals distinct chaotic and quasi-periodic intervals for the LSM. Summary results are as follows:

0–1 Test Results ($$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$8 values for selected $$x_{n+1} = \Bigl(r\,x_n(1 - x_n) + \frac{4 - r}{4}\,\sin(\pi x_n)\Bigr) \bmod 1$$9):

$x_0\in(0,1)$0	3.15	3.25	3.35	3.45	3.55	3.65	3.75	3.85	3.95
$x_0\in(0,1)$1	0.8335	0.7394	0.7692	0.6158	0.6739	0.7112	0.6438	0.5503	0.5137

3ST Intervals:

$x_0\in(0,1)$2-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

There are two robustly chaotic intervals:

• $x_0\in(0,1)$3 (contiguous window)
• $x_0\in(0,1)$4 (secondary window)

Outside these, especially for $x_0\in(0,1)$5 and $x_0\in(0,1)$6, the system exhibits predominantly quasi-periodic behavior.

4. Comparative Analysis with Other One-Dimensional Maps

The behavior of the LSM diverges significantly from classical benchmarks:

• Standard Logistic Map: Chaoticity largely restricted to $x_0\in(0,1)$7, with periodic windows; 3ST detects strong chaos only in $x_0\in(0,1)$8 ranges $x_0\in(0,1)$9–$x_0 = 0.01$0 and $x_0 = 0.01$1–$x_0 = 0.01$2.
• Logistic-Tent System (LTS) and Tent-Sine System (TSS): Both offer distinct patterns of alternating chaos and quasi-periodicity, yet neither achieves the extent of low-$x_0 = 0.01$3 robust chaos seen in the LSM.
• Significance: The LSM’s strong chaos begins at $x_0 = 0.01$4, offering a broader parameter spectrum for practical applications where robust, persistent chaos is required (Muthu et al., 2020).

5. Practical Guidance for Parameter Selection

The optimal utilization of the LSM for cryptographic or other randomization-critical applications depends on conscientious choice of the control parameter $x_0 = 0.01$5:

• Strongest-chaotic windows: $x_0 = 0.01$6 and $x_0 = 0.01$7.
• Parametric “sweet spot”: $x_0 = 0.01$8, exhibiting maximal $x_0 = 0.01$9 per the 0–1 test.
• Avoidance zones: $r x_n (1 - x_n)$0 and $r x_n (1 - x_n)$1; these are quasi-periodic and show significantly reduced sensitivity to initial conditions and mixing.

These recommendations are actionable for image and data-encryption system design, where uniform unpredictability and sensitivity are critical design constraints (Muthu et al., 2020).

6. Visualization and Diagnostic Overview

Empirical diagnostics reveal the nuanced dynamical landscape of the LSM:

• Bifurcation diagrams (e.g., Figure 1(c) in (Muthu et al., 2020)) show full coverage of $r x_n (1 - x_n)$2 as $r x_n (1 - x_n)$3 varies, but do not immediately reveal the quasi-periodic bands near $r x_n (1 - x_n)$4 or $r x_n (1 - x_n)$5.
• 0–1 test and 3ST classification tables provide granular subdivision into strong/weak chaotic and quasi-periodic domains, directly informing parameter selection for application domains.

A plausible implication is that LSM-based hybrid maps, subject to careful parametric analysis using advanced chaos-detection tools, offer more flexible and robust chaos regimes than canonical one-dimensional maps—particularly valuable in cryptographic contexts requiring uniform key-space unpredictability.