Three-dimensional System Of Differential Equations Research Articles

Multi-winged Lorenz attractors due to bifurcations of a periodic orbit with multipliers (−1,i,−i)

Abstract We show that bifurcations of periodic orbits with multipliers ( − 1 , i , − i ) can lead to the birth of pseudohyperbolic (i.e. robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ‘Simó angels’. These three attractors exist in an orientation-reversing, three-dimensional, quadratic Hénon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simó angels. Both for the normal form and the 3D Hénon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent.

Conjoined Lorenz twins-a new pseudohyperbolic attractor in three-dimensional maps and flows.

We describe new types of Lorenz-like attractors for three-dimensional flows and maps with symmetries. We give an example of a three-dimensional system of differential equations, which is centrally symmetric and mirror symmetric. We show that the system has a Lorenz-like attractor, which contains three saddle equilibrium states and consists of two mirror-symmetric components that are adjacent at the symmetry plane. We also found a discrete-time analog of this "conjoined-twins" attractor in a cubic three-dimensional Hénon map with a central symmetry. We show numerically that both attractors are pseudohyperbolic, which guarantees that each orbit of the attractor has a positive maximal Lyapunov exponent, and this property is preserved under small perturbations. We also describe bifurcation scenarios for the emergence of the attractors in one-parameter families of three-dimensional flows and maps possessing the symmetries.

Solution of some three-dimensional boundary value problems for thermoelastic bodies with voids

The paper considers a three-dimensional system of differential equations describing the thermoelastic static equilibrium of homogeneous isotropic materials with voids. In the Cartesian coordinate system the general solution of the mentioned system of equations is constructed using harmonic and metaharmonic functions. On the basis of the constructed general solution using the method of separation of variables boundary value problems for a semi-infinite prism and a rectangular parallelepiped are analytically solved. The corresponding boundary-contact problems for a multilayer rectangular parallelepiped are also considered.

Measuring the Pollutants in a System of Three Interconnecting Lakes by the Semianalytical Method

Pollution has become an intense danger to our environment. The lake pollution model is formulated into the three-dimensional system of differential equations with three instances of input. In the present study, the new iterative method (NIM) was applied to the lake pollution model with three cases called impulse input, step input, and sinusoidal input for a longer time span. The main feature of the NIM is that the procedure is very simple, and it does not need to calculate any special type of polynomial or multipliers such as Adomian polynomials and Lagrange’s multipliers. Comparisons with the Adomian decomposition method (ADM) and the well-known purely numerical fourth-order Runge-Kutta method (RK4) suggest that the NIM is a powerful alternative for differential equations providing more realistic series solutions that converge very rapidly in real physical problems.

A one-locus model describing the evolutionary dynamics of resistance against insecticide in Anopheles mosquitoes

Vector control is regarded an efficient approach for preventing and reducing the spread of malaria. For several decades, insecticides have been used to suppress the population of Anopheles mosquitoes throughout the globe. It turns out that continual usage of a single compounded insecticide has massively contributed to the rise of resistant mosquitoes. Information on the evolution of insecticide resistance is considered an essential aspect in the control of Anopheles mosquitoes. In this study, a mathematical model to describe the dynamics of Anopheles mosquitoes is constructed, highlighting genetic classification based on their resistance status with respect to insecticides. Focusing on a one-locus case related to a specific insecticide class, the model is constructed according to a scenario in which the intermediate form of resistance called heterozygous resistance can be generated from random matings between susceptible and resistant mosquitoes, therefore carrying both genes. The model is governed by a three-dimensional system of differential equations that describes the interaction between homozygous wild, heterozygous, and resistant subpopulations, with three corresponding fitness levels related to their intrinsic growth rates. Criteria for the existence and stability of all existing equilibria are obtained. It is generally concluded from the model that the fitness levels heavily determine to which steady state the model solution converges. Numerical simulations indicate that long-term implementation of high doses of insecticide can increase the proportion of resistant mosquitoes significantly. Accordingly, understanding the fitness levels is very important for selecting proper intervention strategies as well as for predicting the long-term effects of the implementation of certain insecticides.

Integrable Deformations of Three-Dimensional Chaotic Systems

The integrable deformation method for a three-dimensional Hamilton–Poisson system consists in alteration of its constants of motion in order to obtain a new Hamilton–Poisson system. We assume that a three-dimensional system of differential equations has a Hamilton–Poisson part and a nonconservative part. We give integrable deformations of the Hamilton–Poisson part and, adding the nonconservative part, we obtain integrable deformations of the considered three-dimensional system of differential equations. In particular, applying this method to chaotic systems may lead to new systems with chaotic behavior. We use this method to obtain integrable deformations of Lorenz, Chen, and Rössler systems. Using particular deformation functions, we have pointed out some deformations of the above-mentioned attractors.

Dynamical analysis of a two prey-one predator system with quadratic self interaction

In this paper we investigate the dynamical properties of a two prey-one predator system with quadratic self interaction represented by a three-dimensional system of differential equations by using tools of computer algebra. We first investigate the stability of the singular points. We show that the trajectories of the solutions approach to stable singular points under given conditions by numerical simulation. Then, we determine the conditions for the existence of the invariant algebraic surfaces of the system and we give the invariant algebraic surfaces to study the flow on the algebraic invariants which is a useful approach to check if Hopf bifurcation exists.

A hybrid symbolic-numerical approach to the center-focus problem

We propose a new hybrid symbolic-numerical approach to the center-focus problem. The method allowed us to obtain center conditions for a three-dimensional system of differential equations, which was previously not possible using traditional, purely symbolic computational techniques.

Elastic Equilibrium of Porous Cosserat Media with Double Porosity

The static equilibrium of porous elastic materials with double porosity is considered in the case of an elastic Cosserat medium. The corresponding three-dimensional system of differential equations is derived. Detailed consideration is given to the case of plane deformation. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and two solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with double porosity. A concrete boundary value problem for a concentric ring is solved.

Solution of some boundary value thermoelasticity problems for a rectangular parallelepiped taking into account micro-thermal effects

A three-dimensional system of differential equations is considered that describes a thermoelastic equilibrium of homogeneous isotropic elastic materials, microelements of which, in addition to classical displacements and thermal fields, are also characterized by microtemperatures. In the Cartesian system of coordinates the general solution of this system of equations is constructed using harmonic and metaharmonic functions. Some boundary value micro-thermoelasticity problems are stated for the rectangular parallelepiped. An analytical solution of this class of boundary value problems is constructed using the above-mentioned general solution. When the coefficients characterizing microthermal effects are zero, the obtained solutions lead to the solutions of corresponding classical boundary value thermoelasticity problems, the majority of which have been solved for the first time. It should be noted that the aim of the given work is to construct an effective (analytical) solution for a class of boundary vale problems rather than to investigate the validity or applicability of the involved theory.

AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR

A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple “canard” orbits if one of the parameters is small. The presence of chaotic attractors is also detected.

Easy-to-implement method to target nonlinear systems.

In this work we present a method to rapidly direct a chaotic system, to an aimed state or target, through a sequence of control perturbations, with few different amplitudes chosen according to the allowed control-parameter changes. We applied this procedure to the one-dimensional Logistic map, to the two-dimensional Henon map, and to the Double Scroll circuit described by a three-dimensional system of differential equations. Furthermore, for the Logistic map, we show numerically that the resulting trajectory (from the starting point to the target) goes along a stable manifold of the target. Moreover, using the Henon map, we create and stabilize unstable periodic orbits, and also verify the procedure robustness in the presence of noise. We apply our method to the Double Scroll circuit, without using any low-dimensional mapping to represent its dynamics, an improvement with respect to previous targeting methods only applied for experimental systems that are mapping-modeled. (c) 1998 American Institute of Physics.

Three-dimensional motion of a material point

The two-dimensional motion of a material point over the active portion of its trajectory can be generalized in a natural way to three dimensions. Corresponding to the traditional flight plane, to which the trajectory of motion is confined in three dimensions, we have a set of flight surfaces obtained from it by bending. The three-dimensional system of differential equations governing the motion of a material point splits into a two-dimensional system, which describes the motion in the flight surface, and a system of ordinary differential equations, which describes the bending of the surface. By solving this system of equations one can determine by analytical means how the velocity and coordinate vectors over the active portion of the trajectory depend on its three-dimensional distortion. The results obtained may be used to analyse the three-dimensional motion of a material point, to select trajectories in space and to control the three-dimensional motion of the centre of mass over the active portions. In some cases one can actually derive analytical expressions for solutions to boundary-value and extremal problems associated with the three-dimensional motion of a material point.

Macromodeling of nonlinear systems with piecewise-constant inputs

A representation theorem for ground stable nonlinear systems with piecewise-constant inputs is presented. It is a three-dimensional system of differential equations augmented by an algebraic output equation. The representation is characterized by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N + 6</tex> parameters, where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</tex> is the number of harmonics used to represent transients. Five of the parameters are positive constants, and the remaining <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N + 1</tex> parameters are functions of either <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</tex> or <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</tex> variables. Applications involving the macromodeling of higher dimensional linear and nonlinear systems are presented.