Chaotic Differential Equation Research Articles

A New Dynamical Circuit Based on CCII+, Physical Implementation and Synchronization

In this study, a second-generation positive current conveyor (CCII+)-based analog circuit is proposed for the electronic implementation of a different dynamical system which is an adaptation of the chaotic Lorenz differential equation set. The proposed circuit is more cost-effective and contains less active and passive elements than the circuit obtained by applying the classical parallel synthesis method with opamps. Mathematical analyses and SPICE simulations are performed for chaotic phase portraits and bifurcation diagrams. The proposed dynamical circuit is implemented on the board by using commercially available active and passive elements on the market and an experimental study is conducted. In order to demonstrate the usability of this proposed circuit in secure communication studies, three different synchronization methods are applied and one of them is implemented. The obtained experimental results are in good agreement with the mathematical analysis and simulation results.

Lightweight secure image encryption scheme based on chaotic differential equation

The ultimate secure choice for block cryptosystem until now is advanced encryption standard (AES). It is very difficult to implement AES for the constrained situations such as sensor networks, image encryption and RFID tags. In this article, a chaotic oscillator generated by a second order differential equation is used to produce confusion and diffusion in the plaintext message to achieve the desired secrecy. The produced chaotic sequence of random numbers from dynamical system is utilized to scramble the pixels of an image to obtain an encrypted image. Chaos based encryption technique is found secure enough to tackle chosen plaintext attacks and brute force attacks. The specific attributes of chaotic system like, sensitivity to initial conditions, randomness and uncertainty make it suitable for the design of cryptosystem. The dominance of the proposed scheme is acknowledged due to the fact of better cryptographic properties when compared with the algorithms already developed in the literature.

Electronic Chaotic Oscillator Realization with Potential Uses in Communication Systems

Chaotic systems have some unique properties that can be taken advantage of in some practical systems. These systems have characteristics such as long-term aperiodicity, continuous power spectral density, topological mixing, and sensitivity to initial conditions, all while still having a clearly defined deterministic structure. The property of continuous power spectral density is of particular interest in spread spectrum communication applications. This work looks to maintain these complex properties in a practical custom electronic realization through careful layout and device selection. Included are simulation results demonstrating the system's sensitivity to initial conditions and topological mixing. In addition to this, the electronic simulation maintains a continuous spectral power density up the fundamental frequency of the oscillator. These simulation results are used design the chaotic oscillator in a hardware demonstration. The hardware results exhibit similar dynamics to the original motivation system. Presented here is a relatively simple electronic implementation that closely maintains the complex properties of an ideal chaotic differential equation.

A Class of Integer Order and Fractional Order Hyperchaotic Systems via the Chen System

In this article, we investigate the generation of a class of hyperchaotic systems via the Chen chaotic system using both integer order and fractional order differential equation systems. Based on the Chen chaotic system, we designed a system with four nonlinear ordinary differential equations. For different parameter sets, the trajectory of the system may diverge or display a hyperchaotic attractor with double wings. By linearizing the ordinary differential equation system with divergent trajectory and designing proper switching controls, we obtain a chaotic attractor. Similar phenomenon has also been observed in linearizing the hyperchaotic system. The corresponding fractional order systems are also considered. Our investigation indicates that, switching control can be applied to either linearized chaotic or nonchaotic differential equation systems to create chaotic attractor.

An efficient approach to design confusion creating component and its analyses using majority logic criterion

In 1949, the concept of confusion creating component was proposed by some cryptographers for the sake of good security, so, from then to now, every block cipher encryption system has this component as an essential part. The aim of this work is to propose a scheme for the assembly of 8×8 nonlinear confusion component for encryption algorithms based on 'fractional-order hyper chaotic Lorentz differential equation system' (FOHCLS). The security analyses of the proposed confusion component and its utility in image encryption application show that it is invulnerable against renowned attacks.

Continuous-Time Limit of a Shift Map Yielding an Exactly Solvable Chaotic Differential Equation

A chaotic nonlinear differential equation possessing an exact analytic solution is derived from a continuous-time limit of an iterated shift map. The resulting equation consists of an unstable harmonic oscillator forced by a nonlinear piecewise-constant feedback. The continuous time solution exhibits a shift dynamic at integer return times, and the waveform is described completely in terms of the symbols of the contained shift map. Significantly, this exactly solvable nonlinear system provides the first example of a chaotic ordinary differential equation possessing an exact symbolic dynamics.

ANALYTIC PATTERNS FOR CHAOTIC EQUATIONS

Chaotic evolution equations sometimes display regular patterns (fronts, kinks, holes), which often correspond to closed form analytic solutions. In the dissipative-dispersive Kuramoto and Sivashinsky travelling wave reduction u(x − ct), νu‴ + bu″ + μu′ + u2/2 + A = 0, ν ≠ 0, with (ν,b,μ,A) constants, such analytic solutions are known for heteroclinic solutions, but one has also observed (Toh, 1987) homoclinic solutions without corresponding analytic solutions yet. We review the search for the most general analytic solution admissible by this chaotic differential equation. Several investigations, both analytic by the Painlevé test (Thual and Frisch, 1986) and numerical by Padé approximants (Yee, Conte, Musette, 2003) indicate its quite probable single valuedness for any (ν,b,μ,A). Moreover, Nevanlinna theory on the growth of solutions near infinity rules out (Eremenko, 2005) the possibility for the singularities of this unknown closed form single valued expression to be only poles.We propose a qualitative closed form involving a deformed elliptic function.

REVIEW OF CHAOS COMMUNICATION BY FEEDBACK CONTROL OF SYMBOLIC DYNAMICS

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.

Chaos in a class of impulsive differential equation

In this paper, the chaotic differential equation with impulse effect is introduced including its fundamental theory. For illustrating the chaotic characteristic of this kind of system, the theory of the dynamics is applied and the computer simulation is given.

Using chaos to shadow the quadratic map for all time

We present a new way of proving that a computer-generated orbit for the chaotic attractor outside the periodic windows of the quadratic map can be shadowed for all time (i.e., there exist true orbits which stay near a numerical orbit for all time). This is done by computing a numerical orbit for a particular value of a and show that . The true orbits are found using slightly different maps . This technique can therefore be applied to other chaotic differential equation and discrete systems.