4-dimensional Systems Research Articles

Application of Ant Colony Programming Approach for Solving Systems of Stochastic Differential Equations

Stochastic differential equations (SDE) have wide applications in natural phenomena, engineering, finance, and biological models. Obtaining analytic solutions for an SDE is often complex, and the complexity increases for an SDE system. The paper introduces ant colony programming (ACP) as a novel approach for solving SDE system. Ant colony programming was developed in two directions, the first is to add the Wiener process as a variable to the terminals and functions, and the second is to construct the appropriate fitness function . ACP constructs mathematical expressions and evaluates them using the fitness function . The ACP proposed effectiveness has been demonstrated by applying to 2,3 and 4-dimensional SDE systems. The most important finding of this work is that ACP generates symbolic stochastic processes that represent solutions for SDE system. Methods for solving SDE systems are important tools for study phenomena that involve noise or randomness.

Packing Densities of Delzant and Semitoric Polygons

Exploiting the relationship between 4-dimensional toric and semitoric integrable systems with Delzant and semitoric polygons, respectively, we develop techniques to compute certain equivariant packing densities and equivariant capacities of these systems by working exclusively with the polygons. This expands on results of Pelayo and Pelayo-Schmidt. We compute the densities of several important examples and we also use our techniques to solve the equivariant semitoric perfect packing problem, i.e., we list all semitoric polygons for which the associated semitoric system admits an equivariant packing which fills all but a set of measure zero of the manifold. This paper also serves as a concise and accessible introduction to Delzant and semitoric polygons in dimension four.

Path integration method based on a decoupling probability mapping for fast solving the stochastic response of dynamical systems

An efficient path integration method is proposed for obtaining the stochastic response of dynamical systems. This method benefits from a new short-time transition probability density function (TPDF) matrix. First, a decoupling probability mapping is revealed, which achieves the decoupling of the short-time TPDF from stochastic processes that the system suffers. Then, the short-time transition paths carrying known probabilities can be determined by solving the probability mapping. Thus, the new TPDF matrix can be obtained simply and efficiently. The transient and steady responses of two 4-dimensional systems for different noise parameters and system parameters are explored by using the efficient path integration method. The results confirm that our method requires a cheap time cost to achieve good results that fit well with the Monte Carlo simulations.

Synchronization and Anti-synchronization for a 4-dimensional Hyperchaotic System

Between two 4-dimensional hyper-chaotic systems with unknown parameters, using Lyapunov stability theory, adaptive controller and parameter estimation law are designed to achieve their globally asymptotical synchronization and anti-synchronization. By numerical calculation with MATLAB, synchronization or anti-synchronization is arrived theoretically but synchronous or anti-synchronous errors of state variables cannot all converge to zero simultaneously during a limited time. The comparative and contrastive studies about system without cubic term indicates that the primary cause is existence of cubic term. It is verified by numerical simulation examples consistently.

On the Moduli Space of Wigner Quasiprobability Distributions for N-Dimensional Quantum Systems

A mapping between operators on the Hilbert space of an N-dimensional quantum system and Wigner quasiprobability distributions defined on the symplectic flag manifold is discussed. The Wigner quasiprobability distribution is constructed as a dual pairing between the density matrix and the Stratonovich–Weyl kernel. It is shown that the moduli space of Stratonovich–Weyl kernels is given by the intersection of the coadjoint orbit space of the group SU(N) and a unit (N − 2)-dimensional sphere. The general considerations are exemplified by a detailed description of the moduli space of 2, 3, and 4-dimensional systems.

On Hamiltonian Systems Integrable in Elliptic Functions That Describe Waves over Underwater Banks and Ridges

We discuss the 4-dimensional Hamiltonian systems that describe waves over underwater banks and ridges. The systems are exactly integrable in terms of elliptic functions and of solutions to nontrivial transcendental equations involving the elliptic integrals (Weierstrass' $\zeta$-function).

Research on 4-dimensional Systems without Equilibria with Application

Recently chaos-based encryption has been obtained more and more attention. Chaotic systems without equilibria may be suitable to be used to design pseudorandom number generators (PRNGs) because there does not exist corresponding chaos criterion theorem on such systems. This paper proposes two propositions on 4-dimensional systems without equilibria. Using one of the propositions introduces a chaotic system without equilibria. Using this system and the generalized chaos synchronization (GCS) theorem constructs an 8-dimensional discrete generalized chaos synchronization (8DBDGCS) system. Using the 8DBDGCS system designs a 216-word chaotic PRNG. Simulation results show that there are no significant correlations between the key stream and the perturbed key streams generated via the 216-word chaotic PRNG. The key space of the chaotic PRNG is larger than 21275. As an application, the chaotic PRNG is used with an avalanche-encryption scheme to encrypt an RGB image. The results demonstrate that the chaotic PRNG is able to generate the avalanche effects which are similar to those generated via ideal chaotic PRNGs.

Adaptive Synchronization of 4-Dimensional Energy Resource Unknown Time-Varying Delay Systems

This paper proposed a fuzzy adaptive chaos synchronization method for a 4-D energy resource system with time-varying delays. The unknown time-varying delays are compensated based on Lyapunov–Krasovskii functional. Fuzzy logic systems are adopted to approximate the unknown nonlinear functions. Under the condition that all the system parameters are useless, an adaptive synchronization scheme is presented based on robust adaptive theory and Lyapunov stability to ensure that the two chaotic systems asymptotically synchronized. It is shown that all the signals in the closed-loop system are proved to be bounded, and the system output could track the given reference signal. The simulation example is given to illustrate the effectiveness of the design proposed schemes.

Superrelativity

Did any physics experts expect SUPERRELATIVITY paper, a physics revolution producing the EINSTEIN-RODGERS RELATIVITY EQUATION, producing the HAWKING-RODGERS BLACK HOLE RADIUS, and producing the STEFAN-BOLTZMANN-SCHWARZSCHILD-HAWKING-RODGERS BLACK HOLE RADIATION POWER LAW, as the author gave a solution to The Clay Mathematics Institute’s very difficult problem about the Navier-Stokes Equations? The Clay Mathematics Institute in May 2000 offered that great $million prize to the first person providing a solution for a specific statement of the problem: “Prove or give a counter-example of the following statement: In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier-Stokes Equations.” Did I, the creator of this paper, expect SUPERRELATIVITY to become a sophisticated conversion of my unified field theory ideas and mathematics into a precious fluid dynamics paper to help mathematicians, engineers and astro-physicists? [1]. Yes, but I did not expect such superb equations that can be used in medicine or in outer space! In this paper, complicated equations for multi-massed systems become simpler equations for fluid dynamic systems. That simplicity is what is great about the Navier-Stokes Equations. Can I delve deeply into adding novel formulae into the famous Schwarzschild’s equation? Surprisingly, yes I do! Questioning the concept of reversibility of events with time, I suggest possible 3-dimensional and 4-dimensional co-ordinate systems that seem better than what Albert Einstein used, and I suggest possible modifications to Maxwell’s Equations. In SUPERRELATIVITY, I propose that an error exists in Albert Einstein’s Special Relativity equations, and that error is significant because it leads to turbulence in the universe’s fluids including those in our human bodies. Further, in SUPERRELATIVITY, after I create Schwarzschild-based equations that enable easy derivation of the Navier-Stokes Equations, I suddenly create very interesting exponential energy equations that simplify physics equations, give a mathematical reason for turbulence in fluids, give a mathematical reason for irreversibility of events with time, and enable easy derivation of the Navier-Stokes Equations. Importantly, my new exponential Navier-Stokes Equations are actually wave equations as should be used in Fluid Dynamics. Thrilled by my success, I challenge famous equations by Albert Einstein and Stephen Hawking [2] [3].

The exact traveling wave solutions to two integrable KdV6 equations

The exact explicit traveling solutions to the two completely integrable sixth-order nonlinear equations KdV6 are given by using the method of dynamical systems and Cosgrove’s work. It is proved that these traveling wave solutions correspond to some orbits in the 4-dimensional phase space of two 4-dimensional dynamical systems. These orbits lie in the intersection of two level sets defined by two first integrals.

Hybrid Chaos Synchronization of Hyperchaotic LIU And Hyperchaotic CHEN Systems by Active Nonlinear Control

This paper investigates the hybrid chaos synchronization of identical 4-D hyperchaotic Liu systems (2006), 4-D identical hyperchaotic Chen systems (2005) and hybrid synchronization of 4-D hyperchaotic Liu and hyperchaotic Chen systems. The hyperchaotic Liu system (Wang and Liu, 2005) and hyperchaotic Chen system (Li, Tang and Chen, 2006) are important models of new hyperchaotic systems. Hybrid synchronization of the 4-dimensional hyperchaotic systems addressed in this paper is achieved through complete synchronization of two pairs of states and anti-synchronization of the other two pairs of states of the underlying systems. Active nonlinear control is the method used for the hybrid synchronization of identical and different hyperchaotic Liu and hyperchaotic Chen and the stability results have been established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the proposed nonlinear control method is effective and convenient to achieve hybrid synchronization of the hyperchaotic Liu and hyperchaotic Chen systems. Numerical simulations are shown to demonstrate the effectiveness of the proposed chaos synchronization schemes.

Hybrid Chaos Synchronization of 4-D Hyperchaotic QI And JIA Systems By Active Nonlinear Control

This paper investigates the hybrid chaos synchronization of identical 4-D hyperchaotic Qi systems, 4-D identical hyperchaotic Jia systems and hybrid synchronization of 4-D hyperchaotic Qi and Jia systems. The hyperchaotic Qi system (2008) and hyperchaotic Jia system (2007) are important models of hyperchaotic systems. Hybrid synchronization of the 4-dimensional hyperchaotic systems addressed in this paper is achieved through complete synchronization of two pairs of states and anti-synchronization of the other two pairs of states of the underlying systems. Active nonlinear control is the method used for the hybrid synchronization of identical and different hyperchaotic Qi and Jia systems and the stability results have been established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the proposed nonlinear control method is effective and convenient to achieve hybrid synchronization of the hyperchaotic Qi and Jia systems. Numerical simulations are presented to demonstrate the effectiveness of the proposed chaos synchronization schemes.

The use of the minimum projection mode in 4-dimensional examination of the fetal heart with spatiotemporal image correlation.

The minimum projection mode (MPM) is a rendering algorithm available in some 3- and 4-dimensional ultrasonography systems that, in 1 image, allows the visualization of vessels and cystic anatomic structures located in different scanning planes. The objective of this study was to compare the information displayed in images obtained with the MPM with their corresponding 2-dimensional ultrasonographic images from fetuses with and without structural heart defects. Thirty-two volume data sets acquired with the spatiotemporal image correlation technique from fetuses with (n = 15) and without (n = 17) structural heart defects were analyzed. Rendered images of the upper abdomen, 4-chamber view, and 3-vessel view were compared with the corresponding 2-dimensional images. Two independent observers with experience in fetal echocardiography reviewed these volume data sets using the MPM. The visualization rate of specific anatomic structures seen from a transverse sweep of the heart was recorded for each observer, and the interobserver agreement was assessed by statistical indices of agreement (kappa statistic). Images obtained by the MPM displayed more vascular structures than those of corresponding 2-dimensional images at the level of the 3-vessel view. For conotruncal anomalies, the minimum mode projected the aorta and pulmonary arteries in 1 single plane, facilitating the understanding of their spatial relationships. The interobserver agreement score was moderate to "almost perfect" for assessment of most anatomic structures in the upper abdomen, 4-chamber view, and 3-vessel view sections. However, interobserver agreement ranged from fair to poor for visualization of the left outflow tract, atrial septum, and flap of the foramen ovale. The MPM is an alternative rendering modality that facilitates visualization of normal and abnormal vascular connections to the fetal heart at the level of the 3-vessel view. This technique may be useful in prenatal diagnosis of conotruncal anomalies and in assessment of the spatial relationships of abnormal vascular connections in the upper mediastinum.

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.

High-speed electronic-analogue computing techniques

The paper describes circuits and techniques of measurement developed for use in an electronic-analogue differential analyser running at high repetition rates. The object has been to explore the upper limits to the speed conveniently attainable in various basic operations.After a general description of the problem and the computer which was ultimately developed, these basic operations are treated in turn. Precision methods have been developed for resetting and releasing integrators and altering the coefficients electronically at repetition rates up to 25 000 per second, making possible extremely rapid automatic search processes of trial and error. A multiplier on the “quarter squares” principle has been developed, using biased diodes to control the feedback characteristic of an amplifier, giving an output which shows negligible phase-shifts at 50 kc/s.Arbitrary functions are synthesized by approximation in diode-controlled networks.Essential to the efficient use of such high operating speeds has been the development of projective multidimensional displays with an adequate frequency response. Three-and 4-dimensional systems using resistive resolvers are described, and the principles and techniques developed for accurate measurement at high speeds are discussed in detail.The final Section describes measurements made on each of the basic units to evaluate their performance.It is concluded that the potential information capacity of electronic-analogue computing methods is much higher than can be realized by conventional techniques producing single solutions on a 2-dimensional display. With multi-dimensional displays and electronic programming, a hundred-or even thousand-fold increase in information capacity is quite readily attainable with problems requiring systematic search processes for their solution.

The cgs Units of Magnetic Susceptibility and Specific Magnetization

The units of susceptibility and magnetization can be chosen so that they are identical in the (unrationalized) 3- and 4-dimensional cgs systems. These units and the defining equations are given, and inconsistencies in the literature are pointed out. Adoption of the present definitions would eliminate much existing confusion and satisfy the various schools of thought.