Piecewise Hybrid Map (AHYB) Dynamics

• Piecewise Hybrid Map (AHYB) is a one-dimensional discrete system that hybridizes logistic and sine maps to create tunable chaotic behavior.
• It employs advanced chaos-detection methodologies, such as the 0–1 test and Three-State Test, to classify its nonuniform dynamical regimes.
• Its robust and variable chaotic dynamics, especially in specific parameter intervals, make it highly applicable for cryptography and random number generation.

A Piecewise Hybrid Map, exemplified by the Logistic–Sine Map (LSM), is a class of one-dimensional discrete dynamical systems that combine distinct nonlinear behaviors—typically from well-established chaotic maps such as the logistic, tent, and sine maps—into a single piecewise or hybridized functional form. These maps are studied for their complex bifurcation structures and nonuniform chaotic dynamics, which have important implications for applications such as cryptography and random number generation. The LSM, studied in detail in (Muthu et al., 2020), illustrates how piecewise hybridization yields broad, but non-uniform, intervals of strong and weak chaoticity, requiring nuanced parameter selection for robust applications.

1. Mathematical Structure and Definition

The canonical piecewise hybrid map described in (Muthu et al., 2020) is the Logistic–Sine Map (LSM), formally given by:

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

where:

• $x_n \in [0,1]$ is the state at iteration $n$,
• $r \in (0,4]$ is the control (bifurcation) parameter,
• The initial condition $x_0$ is typically chosen in $(0,1)$, often $0.01$,
• The modular operation ensures state confinement to $[0,1]$.

This construction blends the canonical logistic map $r x_n(1-x_n)$ and a normalized sine map $(4-r)/4\,\sin(\pi x_n)$, with weights adjusting continuously with $x_n \in [0,1]$0. The resulting dynamics represent a nontrivial hybridization, not reducible to either individual map for intermediate $x_n \in [0,1]$1, providing a tunable and piecewise diverse chaotic regime.

2. Chaos Detection Methodologies

The nonuniform and intricate structure of chaos in hybrid maps such as the LSM necessitates robust diagnostic tools beyond standard Lyapunov exponent analysis, which may fail to characterize the true complexity of the system’s dynamics. Two contemporary chaos-detection methods are pivotal:

2.1 0–1 Test for Chaos

Given a time series $x_n \in [0,1]$2 ($x_n \in [0,1]$3), and a randomly chosen $x_n \in [0,1]$4 (typically $x_n \in [0,1]$5), the test constructs translation variables:

$x_n \in [0,1]$6

The mean-square displacement is defined by:

$x_n \in [0,1]$7

The scaling exponent $x_n \in [0,1]$8 is then:

$x_n \in [0,1]$9

Interpretation: $n$0 indicates regular (periodic or quasiperiodic) dynamics, $n$1 signals chaos. In (Muthu et al., 2020), $n$2, $n$3, $n$4.

2.2 Three-State Test (3ST)

This test employs ordinal-pattern analysis:

1. Divide the trajectory into overlapping blocks of length $n$5.
2. For each block, compute state-transition slopes $n$6.
3. Compute block-averaged slope $n$7 and standard deviation $n$8.
4. Define a growth indicator:

$n$9

and $r \in (0,4]$0.

Behavioral classes: constant $r \in (0,4]$1 (periodic), saturating $r \in (0,4]$2 (quasiperiodic), unbounded $r \in (0,4]$3 (chaotic). Implementation: $r \in (0,4]$4, $r \in (0,4]$5, $r \in (0,4]$6 blocks, time-lag $r \in (0,4]$7.

3. Characterization of Chaotic and Nonchaotic Regimes

Numerical investigation reveals that the strength and presence of chaos in the LSM are nonuniform across $r \in (0,4]$8. Representative quantitative results for the 0–1 test are tabulated below.

$r \in (0,4]$9	3.15	3.25	3.35	3.45	3.55	3.65	3.75	3.85	3.95
$x_0$0	0.8335	0.7394	0.7692	0.6158	0.6739	0.7112	0.6438	0.5503	0.5137

Strong chaos ($x_0$1) is observed for $x_0$2. Weaker chaos arises elsewhere, notably for $x_0$3.

The 3ST test further classifies subintervals:

$x_0$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	quasiperiodic	quasiperiodic	chaotic	quasiperiodic	quasiperiodic	quasiperiodic

The map exhibits true chaos only in two windows: $x_0$5 and $x_0$6. Outside these intervals, the dynamics are predominantly quasiperiodic.

4. Comparative Analysis with Other Chaotic Maps

The LSM’s hybrid structure leads to substantial differences from standard maps:

• The standard logistic map, $x_0$7, exhibits chaos only for $x_0$8, with extensive periodic windows. 3ST detects chaos in $x_0$9 and $(0,1)$0.
• The Logistic–Tent System (LTS) offers broader, but still interrupted, chaotic coverage.
• The Tent–Sine System (TSS) displays another distribution of chaotic and quasiperiodic intervals.

A key distinction is that the LSM’s hybridization yields relatively broad intervals of sustained chaos beginning at lower $(0,1)$1 values ($(0,1)$2). This control-parameter flexibility makes it attractive in contexts demanding lower parameter ranges for security or unpredictability.

5. Practical Implications for Cryptographic Applications

Piecewise hybrid maps—specifically, the LSM—are of particular practical interest in cryptosystem design because they enable tunable, strong unpredictability over large parameter intervals. The bifurcation diagram of the LSM (Figure 1(c) in (Muthu et al., 2020)) appears to span the full unit interval for $(0,1)$3, but detailed tests expose hidden nonchaotic windows.

Practical guidance for parameter selection includes:

• Select $(0,1)$4 in the robustly chaotic intervals: $(0,1)$5 or $(0,1)$6.
• The strongest unpredictability is obtained near $(0,1)$7, where $(0,1)$8.
• Avoid $(0,1)$9 in $0.01$0 or $0.01$1, where mixing and sensitivity degrade significantly.

The empirically validated tests (0–1 test and 3ST) supersede sole reliance on Lyapunov exponents in this context.

6. Limitations, Testing Practices, and Outlook

The analysis in (Muthu et al., 2020) demonstrates that neither Lyapunov exponent analysis nor superficial inspection of bifurcation diagrams fully characterize the nonuniform and intricate structure of chaos in hybrid maps. Piecewise hybrid maps demand advanced diagnostics that reveal both strong and weak chaoticity, as well as cryptographically weak, quasiperiodic parameter regimes potentially masked in traditional analyses.

A plausible implication is that standard cryptosystem key selection practices may require reevaluation, favoring hybrid maps only with empirically confirmed, strongly chaotic parameter windows. This underscores the need for adopting stringent empirical chaos-detection protocols when deploying such maps in sensitive applications.