Chaotic Behavior Of The Logistic Equation

Abstract chaotic system is a nonlinear dynamic system with random and unpredictable behavior. The trajectory of the chaotic dynamic system is sensitive to the initial condition, and in a sense, the initial condition is slightly different, so the bias is seen in the trajectory. In order to study chaos, the solution behavior of the logical equation is taken into account. In this paper, the solution of the logical equation was analyzed for different parameters. At some point, the solution points to multiple equilibrium points, and the periodicity increases as the parameter increases.

Logical mapping is the most fundamental recursive expression and shows various levels of chaos based on its parameters. It is used in demographics to simulate chaotic behavior. Here we explore this model in the context of probabilistic simulation and revisit the singular aperiodic random number generator found 70 years ago based on the logical mapping equation. Next, I will explain the pitfalls and benefits of widely used random number generators, and how to reverse engineer these algorithms. Finally, we discuss the quantum algorithm as it applies to our context.

Let's see an example of a well-known process determined by the so-called "logical expression". In chaos theory, logical equations are studied as equations that produce deterministic processes of random observations. The deterministic process is completely determined by previous conditions. Based on the given situation, the evolution of the future is fully determined in advance. The logical expression is x (t + 1) = r * x (t) * (1 - x (t)) and describes the process of giving the known value of variable x at time t. You can see the next state x (t). + 1) at time t + 1. x is a variable in the interval, and r is a fixed parameter that can be obtained from the range. Interestingly, depending on the value of parameter r, the value chain looks very random, but it is completely deterministic. This discovery is a major advance in the field of power systems (1).

It is a simple system with several variables, but it still shows unpredictable, sometimes confusing behavior ... Libchaber made a series of breakthrough experiments. He made a small system to study cubic millimeter convection (chaotic behavior) in his laboratory. By gradually warming up from the bottom, he can create a controlled turbulence condition. Even in this strictly controlled environment we do chaotic behavior. It is a complex and unpredictable confusion that is inconsistent with "regular" rules.