Analysis and computation for a class of semilinear elliptic boundary value problems

AbstractIn this paper, with the help of super-solutions and sub-solutions, we set up a general framework and get a positive threshold Λ for solution existence and non-existence of a class of semilinear elliptic Dirichlet boundary value problems. Moreover, a result on multiplicity is obtained when λ is large enough. We also give a numerical method to solve and visualize the positive solutions of the problem. Theoretical results are illustrated by numerical simulation

Computational Intelligence and Complexity Measures for Chaotic Information Processing

This dissertation investigates the application of computational intelligence methods in the analysis of nonlinear chaotic systems in the framework of many known and newly designed complex systems. Parallel comparisons are made between these methods. This provides insight into the difficult challenges facing nonlinear systems characterization and aids in developing a generalized algorithm in computing algorithmic complexity measures, Lyapunov exponents, information dimension and topological entropy. These metrics are implemented to characterize the dynamic patterns of discrete and continuous systems. These metrics make it possible to distinguish order from disorder in these systems. Steps required for computing Lyapunov exponents with a reorthonormalization method and a group theory approach are formalized. Procedures for implementing computational algorithms are designed and numerical results for each system are presented. The advance-time sampling technique is designed to overcome the scarcity of phase space samples and the buffer overflow problem in algorithmic complexity measure estimation in slow dynamics feedback-controlled systems. It is proved analytically and tested numerically that for a quasiperiodic system like a Fibonacci map, complexity grows logarithmically with the evolutionary length of the data block. It is concluded that a normalized algorithmic complexity measure can be used as a system classifier. This quantity turns out to be one for random sequences and a non-zero value less than one for chaotic sequences. For periodic and quasi-periodic responses, as data strings grow their normalized complexity approaches zero, while a faster deceasing rate is observed for periodic responses. Algorithmic complexity analysis is performed on a class of certain rate convolutional encoders. The degree of diffusion in random-like patterns is measured. Simulation evidence indicates that algorithmic complexity associated with a particular class of 1/n-rate code increases with the increase of the encoder constraint length. This occurs in parallel with the increase of error correcting capacity of the decoder. Comparing groups of rate-1/n convolutional encoders, it is observed that as the encoder rate decreases from 1/2 to 1/7, the encoded data sequence manifests smaller algorithmic complexity with a larger free distance value

Research on digital image watermark encryption based on hyperchaos

The digital watermarking technique embeds meaningful information into one or more watermark images hidden in one image, in which it is known as a secret carrier. It is difficult for a hacker to extract or remove any hidden watermark from an image, and especially to crack so called digital watermark. The combination of digital watermarking technique and traditional image encryption technique is able to greatly improve anti-hacking capability, which suggests it is a good method for keeping the integrity of the original image. The research works contained in this thesis include: (1)A literature review the hyperchaotic watermarking technique is relatively more advantageous, and becomes the main subject in this programme. (2)The theoretical foundation of watermarking technologies, including the human visual system (HVS), the colour space transform, discrete wavelet transform (DWT), the main watermark embedding algorithms, and the mainstream methods for improving watermark robustness and for evaluating watermark embedding performance. (3) The devised hyperchaotic scrambling technique it has been applied to colour image watermark that helps to improve the image encryption and anti-cracking capabilities. The experiments in this research prove the robustness and some other advantages of the invented technique. This thesis focuses on combining the chaotic scrambling and wavelet watermark embedding to achieve a hyperchaotic digital watermark to encrypt digital products, with the human visual system (HVS) and other factors taken into account. This research is of significant importance and has industrial application value

Investigation of chaos in biological systems

Chaos is the seemingly irregular behavior arising from a deterministic system. Chaos is observed in many real-world systems. Edward Lorenz’s seminal discovery of chaotic behavior in a weather model has prompted researchers to develop tools that distinguish chaos from non-chaotic behavior. In the first chapter of this thesis, I survey the tools for detecting chaos namely, Poincaré maps, Lyapunov exponents, surrogate data analysis, recurrence plots and correlation integral plots. In chapter two, I investigate blood pressure fluctuations for chaotic signatures. Though my analysis reveals interesting evidence in support of chaos, the utility such an analysis lies in a different direction that I point to the reader. In chapter three, I investigate a simulation of predator-prey interactions. My analysis casts doubt on some of the claims laid by past researchers, and I prompt future researchers to probe some specific questions that I have outlined in this thesis

The Forced van der Pol Equation II: Canards in the reduced system

This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. The first paper described the reduced system, a two dimensional flow with jumps that reflect fast trajectory segments in this vector field with two time scales. This paper extends the reduced system to account for canards, trajectory segments that follow the unstable portion of the slow manifold in the forced van der Pol oscillator. This extension of the reduced system serves as a template for approximating the full nonwandering set of the forced van der Pol oscillator for large sets of parameter values, including parameters for which the system is chaotic. We analyze some bifurcations in the extension of the reduced system, building upon our previous work in [J. Guckenheimer, K. Hoffman, and W. Weckesser, SIAM J. Appl. Dyn. Syst., 2 (2003), pp. 1–35]. We conclude with computations of return maps and periodic orbits in the full three dimensional flow that are compared with the computations and analysis of the reduced system. These comparisons demonstrate numerically the validity of results we derive from the study of canards in the reduced system

On period doubling bifurcations and on compact analytic semigroups

SIGLEAvailable from British Library Document Supply Centre- DSC:D80447 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

The effects of ball interactions on the dynamic behavior of a ball balancer rotating at speeds above the translational resonant frequency of the system.

Many applications are inherently rotational in nature. These applications range from industrial products to consumer products. As seen in a traditional frequency response plot, the motion of a rotating imbalance does not approach zero as the forcing frequency is increased. Unlike a traditional forced mass spring oscillator, the motion of the rotating imbalance approaches some non-zero value. To account for this residual motion, some systems utilize a balancing device to reduce this motion. These balancing devices can be passive or active, depending on the design considerations. This paper will focus on the traditional, passive ball-type balancer due to its simplicity and extensive use in application. This paper derives the equations of motion for a vertically oriented ball-type balancing system. Due to the high non-linearity of these equations, a fourth order Runge-Kutta numerical integration method is used. The ball balancer equations of motion contain the proper physics needed for full operation such that the ball balancer can translate horizontally, vertically and rotate angularly in the presence of gravity. Acceleration terms are included such that a wide range of operating conditions can be tested. Additionally, n number of balls are present, which are affected by rolling friction and viscous fluid drag. Unlike many numerical models published in the past, the ball-to-ball interactions are not neglected within this model. These interactions include collisions, and train formations and separations. An application of the method presented by (Henon 1982) is utilized where the equations of motion are altered such that an exact integration step can be solved. This is based on the need for a displacement step (collision) or a force step (separation). Although the model presented can accommodate n number of balls, only a maximum ball count of two is considered. It is shown how the behavior of the balls affect the motion response of the ball balancer at rotational velocities above the translational resonance of the system. It is seen that a critical transition is reached; the operating point at which the ball balancer becomes effective at offsetting an eccentric mass. It is also seen that ball balancer displacement decreases until a point of saturation, after which ball balancer displacement increases. Also for the two ball case, it is shown that the spatial characteristics of the balls do affect steady state motion. The angle that separates two contacting balls alters the center of gravity of the train of balls such that the balancing capacity of the system is reduced. Although this effect is shown to be small for a two ball case, the balancing capacity is further reduced as the angle between two contacting balls becomes larger

Chaotic-Based Encryption Algorithm using Henon and Logistic Maps for Fingerprint Template Protection

Fingerprint is a reliable user authentication method as it is unique to individual users that makes it efficient for authenticating users. In a fingerprint authentication system, user fingerprint information is stored in databases in an image format known as a fingerprint template. Although fingerprint is reliable, the templates stored in the database are exposed to security threats either during the data transmission process over the network or in storage. Therefore, there is a need to protect the fingerprint template, especially in unsecured networks to maintain data privacy and confidentiality. Many past studies proposed fingerprint template protection (FTP) using chaotic-based encryption algorithms that are more suitable to secure images than conventional encryption such as DES, AES, and RSA. The chaotic-based encryption algorithms have been improved a lot in terms of their robustness. However, the robustness of the algorithm caused a trade-off to encryption speed where it remains an issue in FTP.  Hence, this study aims to improve the limitations found in the existing chaotic-based encryption algorithms for FTP by improving its encryption speed using Henon and Logistic map. A series of simulations were conducted using MATLAB to evaluate the performance of the proposed chaotic-based encryption algorithm for FTP through different analyses covering key sensitivity, histogram, correlations, differential, information entropy, and encryption/decryption speed. The performance proposed encryption algorithm was promising which could be a starting point for detailed analysis and implementation in real application domains