---
title: 'Logistic-Sine Map: A Hybrid Chaos Model'
url: https://www.emergentmind.com/topics/logistic-sine-map-lsm
type: topic
---

# Logistic-Sine Map: A Hybrid Chaos Model

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 [2012.04156].

## 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 $r$.

## 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 $\{\phi(n)\}$, derive translation variables: $p(n)=\sum_{j=1}^n \phi(j)\cos(cj)$, $q(n)=\sum_{j=1}^n \phi(j)\sin(cj)$, for a constant $c\in(0,2\pi)$ ($c=0.8$ in the cited work).
- Compute the mean-square displacement $M(n)$ and the growth rate $K = \lim_{n\to\infty}\frac{\log M(n)}{\log n}$.
- $K\approx 1$ indicates chaos; $K\approx 0$ indicates regular dynamics.

**B. Three-State Test (3ST):**

- Partition the series into overlapping blocks (block size $n=50$, total $N=5000$).
- For each block, compute state-transition slopes and their statistics.
- Average slope $S$ and its standard deviation $\sigma_S(N,n)$ are used to define a growth indicator $u(N,n)$, from which $K_{\rm 3ST}$ is derived.
- Saturated $S$ indicates quasi-periodicity, unbounded growth of $S$ indicates chaos.

Both methods provide parametric diagnostics of chaoticity beyond Lyapunov exponents, relevant for rigorous cryptosystem assessment [2012.04156].

## 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 ($K$ values for selected $r$):**

| $r$    | 3.15   | 3.25   | 3.35   | 3.45   | 3.55   | 3.65   | 3.75   | 3.85   | 3.95   |
|--------|--------|--------|--------|--------|--------|--------|--------|--------|--------|
| $K$    | 0.8335 | 0.7394 | 0.7692 | 0.6158 | 0.6739 | 0.7112 | 0.6438 | 0.5503 | 0.5137 |

**3ST Intervals:**

| $r$-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:

- $r\in[3.10,\,3.39]$ (contiguous window)
- $r\in[3.60,\,3.69]$ (secondary window)

Outside these, especially for $3.40\le r\le3.59$ and $r\ge3.70$, 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 $r \gtrsim 3.57$, with periodic windows; 3ST detects strong chaos only in $r$ ranges $3.70$–$3.79$ and $3.90$–$3.99$.
- **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-$r$ robust chaos seen in the LSM.
- **Significance:** The LSM’s strong chaos begins at $r\approx3.10$, offering a broader parameter spectrum for practical applications where robust, persistent chaos is required [2012.04156].

## 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 $r$:

- **Strongest-chaotic windows:** $3.10\leq r\leq 3.39$ and $3.60\leq r\leq 3.69$.
- **Parametric “sweet spot”:** $r\approx 3.15$, exhibiting maximal $K=0.8335$ per the 0–1 test.
- **Avoidance zones:** $3.40\le r\le 3.59$ and $r\ge 3.70$; 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 [2012.04156].

## 6. Visualization and Diagnostic Overview

Empirical diagnostics reveal the nuanced dynamical landscape of the LSM:

- **Bifurcation diagrams** (e.g., Figure 1(c) in [2012.04156]) show full coverage of $[0,1]$ as $r$ varies, but do not immediately reveal the quasi-periodic bands near $r\approx 3.45$ or $r\gtrsim 3.7$.
- **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.

Source: https://www.emergentmind.com/topics/logistic-sine-map-lsm