How Hyperbolic Sinusoidal Function is used in simple chaotic function and in Voice Encryption

Chaotic System: A chaotic system is highly sensitive to initial conditions, meaning small differences in the starting point can lead to vastly different outcomes over time.

  

Dynamic Behavior: The system's dynamic properties are studied using various mathematical techniques, such as Lyapunov exponents, bifurcation diagrams, and Poincaré maps. These tools help analyze the stability and chaotic nature of the system.

 

Random Number Generation (RNG): The chaotic system is used to design a random number generator (RNG). RNGs are critical for encryption because they ensure the randomness required for secure communication.

Small changes in the starting point lead to vastly different outcomes. This unpredictability makes them excellent sources for generating random numbers.

RNG generates random bit sequences that are used to encrypt voice data. The voice data is first converted into a binary format, and then an XOR operation is applied between the voice data and the random bit sequence. This XOR-based encryption ensures that the encrypted voice data.

• Variables: x, y, and z represent the system’s states, where each changes over time.

• Coefficients: a1,a2,…,a16​ are constants controlling how different terms in the equations contribute to the behavior of the system.

• Nonlinearities: Terms like xy, xz, and the hyperbolic sine function sinh⁡(y) introduce nonlinearity into the system, which is crucial for creating chaotic behavior.

• b and c are scalar values that further modify the system.

This system models a three-dimensional chaotic flow.

The Jacobian matrix is used to analyze the local stability of the system near equilibrium points. It contains the partial derivatives of the system’s functions with respect to the variables x, y, and z. Here:

cosh⁡(y) is the hyperbolic cosine function, which reflects the nonlinearity introduced by the y-equation.

The matrix helps to determine the system’s behavior around equilibrium points by calculating eigenvalues.

This equation is derived from the Jacobian and is used to find the eigenvalues λ, which help in understanding the stability and type of equilibrium points (e.g., stable, unstable, saddle).

Lyapunov exponent measures how quickly an infinitesimally small distance between two initially close states grows over time.

If Lyapunov's exponent (lamda in below equation) is positive, then the behavior of dynamic system is chaotic. If Lyapunov's exponent is negative, then behavior of dynamic system is non chaotic. Bifurcation Diagrams show how system transitions between different types of behavior(chaotic, periodic etc) as a parameter changes.