Application of Chaos Theory in the Differential Equations - Essay Example

?Application of Chaos Theory in the Differential Equations Application of Chaos Theory in the Differential Equations Introduction The chaos theory in the finite dimensional dynamical systems is known to exist and it includes or leads to the development of discrete systems and maps of the ordinary differential equations. This theory has led to profound mathematical theorems that have numerous applications in different fields including chemistry, biology, physics, and engineering among other fields or professions (Wasow, 2002). However, the chaos theory of the partial differential equations has never been well-developed (Jordan and Smith, 2003; p. 55). Therefore, there has been a growing demand for the development for a much stronger theory than for the finite dimensional systems. In mathematics, there are significant challenges in the studies on the infinite dimensional systems (Taylor, 1996; p. 88). For instance, as phase spaces, the Banach spaces have many structures than in Euclidean spaces. In application, the most vital natural phenomena are explained by the partial differential equations, most of important natural phenomena are described by the Yang-Mills equations, partial differential equations, nonlinear wave equations, and Navier-Stokes equations among others. Problem Statement Chaos theory has led to profound mathematical equations and theorems that have numerous applications in different fields including chemistry, biology, physics, and engineering among other fields or professions. Problem Definition The nonlinear wave equations are usually significant class of equations especially natural sciences (Cyganowski, Kloeden, and Ombach, 2002; p. 33). They usually describe a wide spectrum of phenomena including water waves, motion of plasma, vortex motion, and nonlinear optics (laser) among others (Wasow, 2002). Notably, these types of equations often describe differences and varied phenomena; particularly, similar soliton equation that describes several different situations. These types of equations can be described by the nonlinear Schrodinger equation 1 The equation 1 above has a soliton solution 2 Where the variable This leads to 3 The equation leads to the development of the soliton equations whose Cauchy problems that are solved completely through the scattering transformations. The soliton equations are similar to the integrable Hamiltonian equations that are naturally counterparts of the finite dimensionalintegrable differential systems. Setting up the systematic study of the chaos theory in the partial differential equations, there is a need to start with the perturbed soliton equations (Wasow, 2002). The perturbed soliton equations can be classified into three main categories including: 1. Perturbed (1=1) dimensional soliton equations 2. Perturbed soliton lattices 3. Perturbed (1 + n) dimensional soliton equations (n? 2). For each of the above categories, to analyze the chaos theory in the partial differential equations, there is needed to choose a candidate for study. The integrable theories are often parallel for every member within the same category (Taylor, 1996; p. 102). Moreover, members of different categories are often different substantial. Therefore, the theorem that describes the existence of chaos on each candidate can be generalized parallely to other members under the same category (Wasow, 2002). For instance; The candidate in the first category is often described by a perturbed cubic that often focuses on the nonlinear Schrodinger equation 4 Under even and periodic boundary conditions q (x+1) = q (x) and q (x) =q (x), and is a real constant. The candidates in category 2 are often considered as the perturbed discrete cubic that often focus on the nonlinear Schrodinger equation + Perturbations, 5 The above equation is only valid under even and periodic boundary conditions described by +N = The candidates falling under category 3 are perturbed Davey-Stewartson II equations 6 The equation is only satisfied under the even and periodic boundary conditions described as below q (t; x + lx; y) = q(t; x; y) = q(t; x; y + ly); u (t; x + lx; y) = u(t; x; y) = u(t; x; y + ly); And q (t; x; y) = q(t; x; y) = q(t; x; y); u(t; x; y) = u(t; x; y) = u(t; x; y): Numerous programs have established to prove the existence of chaos in the perturbed soliton equations, with certain machineries: 1. Darboux Transformations for the Soliton Equations 2. Isospectral Theory of Soliton Equations when analyzed under Periodic Boundary Conditions 3. Invariant Manifolds and Fenichel Fibers persistences 4. Melnikov Analysis 5. Symbolic Dynamics Construction and Smale Horseshoesof Conley-MoserType The most essential implication of the chaos theory in the differential equation is usually found in the theoretical physics (Wasow, 2002). Understanding this application is often understood by studying a different model; however, for this case, the 2D Navier-Stokes equations will be used under periodic boundary conditions. Moreover, study shall commence with the study of the dynamical system on the 2D turbulence. Notably, the study of turbulence often possesses a Lax pair and the Darboux transformation . 7 Additionally, it should be noted that these the understanding of these equations and the 2D Euler equations are the fundamentals of the analytical study of the chaos theory application in the partial differential equations. The high Reynolds number describes the 2D Navier-Stokes equations are usually regarded as the singular perturbation of the 2D described by the Euler equations (Jordan and Smith, 2003; p. 119). This understanding is only valid under the perturbation parameter that describes the inverse of the Reynolds number (Taylor, 2005; p. 121). These mathematical relations are often related to their corresponding singular perturbations that is a description of the nonlinear Schrodinger equation (Wasow, 2002). In other words, these equations leads to the identification of the unstable eigenvalues that give absolute or an approximate hyperbolic structures presentation that well related the application of the chaos theory in partial differential equation relating to geometric designs. Application of the Chaos Theory of Partial Differential Equations 1. The Two-Dimensional Euler Equation The profound implication of the chaos theory in the partial differential equations is often in the theoretical physics particularly in the study of the turbulence (Wasow, 2002). For the goals, this essay has picked on the 2D Navier – Strokes equations that are defined within certain periodic boundary conditions. To begin with, this analysis there must be a study of dynamical systems. 1.1 X and y define the periodic boundary conditions; moreover, they also give the direction upon which the analysis can be directed. 2 defines the period while gives the velocity, u and v are the velocity components along x and y directions respectively. and f describes the forces acting on the body under investigation. When the 2D Euler equations become, 1.2 1.3 The relation between stream function vorticity is given as in the equation below, 1.4 The stream function is defined by 1.5 2. Lax Pair and Darboux Transformation The main advantage of the application of the chaos theory in the applied differential equation is its advancement to the discovery of the Lax pair founded in the 2D Euler equation. The 2D Euler equation has philosophical significance in the Lax (Wasow, 2002). This significant goes beyond the chaos in the differential equations. If the integrability of an equation is defined by the availability of the Lax pair, then the 2D Euler equation will automatically be intergratable. Furthermore, the 2D Navier – Stokes equation at the high Reynolds number is often near the integrable numbers or system (Cyganowski, Kloeden, and Ombach, 2002; p. 24). Notably, this point automatically changes the old ideology on Navier – Strokes and the Euler equations. From the Lax pairs, it is clear that homoclinic structures can be constructed from the Darboux transformation. Moreover, this transformation has led to the invention of the Lair pair of the 2D Euler equation (Jordan and Smith, 2003; p. 169). Therefore, the initial step is to identify a structured figure eight that emerges or evident of the 2D Euler equation thereby studying their consequences in the two dimension Navier – Strokes equations (Wasow, 2002). It has been viewed that the high Reynolds number two dimension Navier – Strokes equation often forms a singular perturbation in the two dimensions Euler equation. This understanding often comes through the perturbation that had ealier been described by that in turn forms the inverse of the Reynolds Number (Toma?s - Rodriguez, and Banks, 2010; p. 91). The above-mentioned singular perturbation has been analyzed through investigation for the nonlinear Schrodinger equations. For the analysis, we must consider the 2D Euler equation, 2.1 The bracket in the above equation is usually defined as, and Theorem in this equation (Wasow, 2002) can be defined by the Lax pair of the two dimension Euler equation (2.1) that leads to the equation 2.3 below 2.3 Where and the ? is a complex constant and defines the complex valued function. It should be noted that a Backlund – Darboux transformation usually forms part of the above Lax pair. However, considering the Lax pair 2.3 above at ? = 0, then the equation become {?, p} = 0 2.4 + {?,} = 0, 2.5 In the equations, the notation has been replaced by p (Jordan and Smith, 2003; p. 189). The same Lax Pair and Darboux Transformation can be analyzed using a different theorem. For instance, let f = f (t, x, y) be a fixed solution for the above systems defined in the equations 2.4 and 2.5 above (Taylor, 2005; p. 156). Using the Gauge transformation, the above systems can be defined differently, Gauge transform is defined as 2.6 On the other hand, the transformation of the potential is given by 2.7 Where defines the subject of constraints {?, ?F} = 0, {?, F} = 0 2.8 From the above understanding, it is apparent that p” solves the equation defined by the systems 2.4 and 2.5 above and at (?”, ?”). Therefore, equation 2.6 and 2.7 above forms the Darboux transformation for the two dimension Euler equation 2.3 and its respective Lax pair 2.4 and 2.5. Existence of Chaos theory in the Perturbed Equations The existence of the chaos in an equation often means that there is Smale horseshoe and the dynamics shift in the Bernoulli Poincare maps (Cyganowski, Kloeden, and Ombach, 2002; p. 82). Therefore, for the lower dimensional systems, many theorems exist for chaos. Under the dissipative perturbations, initial step is to determine the existence of a Silnikov homoclinic orbit in such system or equation thereafter the Poincare section is defined (Wasow, 2002). The Poincare is usually transversal to the Silnikov homoclinic orbit and it often maps Poincare onto the Poincare section that in turns induces the flow (Taylor, 2005; p. 171). The final step aims in understanding how to construct the Smale horseshoe that defines the Poincare map. To establish the existence of the Silnikov homoclinic orbit, there is need to build a Melnikov by analyzing the Darboux transformations that aims at generating an explicit representation for the unperturbed (Wasow, 2002) heteroclinic orbit. In most cases, application of the chaos theory cannot evade the understanding of soliton equations or the perturbed soliton equations; therefore, before analyzing the application of the chaos theory and its subsequent theorems, it is important to understand the connectivity or the existence of chaos theory in the perturbed soliton equations (Jordan and Smith, 2003; p. 189). Additionally, the understanding of the soliton equations isospectral theory is vital since it helps in generating the Melnikov vectors as well as the persistence of Fenichel fibers and invariant manifolds (Toma?s - Rodriguez, and Banks, 2010; p. 91). Moreover, there is need to utilize Fenichel fibers properties in building a second measurement within a slow manifold. The joint application of these two techniques helps in analyzing differential equations from the normal techniques or understandings. The second measurement and the Melnikov measurement leads to the establishment of the existence of a smooth linearization within the peripheries of the saddle point or the Silnikov homoclinic orbit’s asymptotic point. The neighborhood dynamics are usually governed by the linear partial differential equations that are explicitly solvable (Taylor, 2005; p. 171). Globally, the tubular dynamics of the neighborhood in the Silnikov homoclinic orbit often helps in approximating the linear flow that are defined by region that are far from the nonlinear neighborhoods. Through such approximation, it often becomes simple to approximate or rather determine semi explicit presentations that map the differential equations on entire region of application (Wasow, 2002). Furthermore, the application of the chaos theory often helps in establishing fixed points under given except one-point conditions (Cyganowski, Kloeden, and Ombach, 2002; p. 142). Therefore, effective understanding and analysis of the applications of the chaos theory in the differential equations often require the understanding different and varied differential principles and applications since the theory is widely applied in different fields of study. The application of the chaos theory in the differential equations also engulfs different equations that must be well interrelated to understand their application as a unit towards solving a single application. The best example in this case is the application of chaos theory in solving the linearized two dimensions Euler equation. For all applications of the chaos theory, boundaries must be defined; otherwise, the intended calculation may lead to nothing. References Cyganowski, S., Kloeden, P., & Ombach, J. (2002). From elementary probability to stochastic differential equations with MAPLE. Berlin [u.a.], Springer. Jordan, D. W., & Smith, P. (2003). Nonlinear ordinary differential equations: an introduction to dynamical systems. [Oxford], Oxford University Press. Taylor, J. R. (2005). Classical mechanics. Sausalito, Calif, Univ. Science Books. Taylor, M. E. (1996). Partial differential equations 1 Basic theory. New York [u.a.], Springer. Toma?s-Rodriguez, M., & Banks, S. P. (2010). Linear, time-varying approximations to nonlinear dynamical systems. Berlin, Springer. Wasow, W. (2002). Asymptotic expansions for ordinary differential equations. Mineola, Dover. Read More