Consider the Lorenz system \((13.33)\). (a) Show that the origin \(P_{1}(0,0,0)\) is a critical point and that its stability depends on eigenvalues \(\lambda\) satisfying the cubic $$ (\lambda+\beta)\left[\lambda^{2}+(\sigma+1) \lambda+\sigma(1-\rho)\right]=0 $$ Hence show that \(P_{1}\) is asymptotically stable only when \(0<\rho<1\) (b) Show that there are two further critical points $$ P_{2}, P_{3} \equiv[\pm \sqrt{\beta(\rho-1)}, \pm \sqrt{\beta(\rho-1)},(\rho-1)] $$ when \(\rho>1\), and that their stability depends on eigenvalues \(\lambda\) satisfying the cubic $$ \lambda^{3}+\lambda^{2}(\sigma+\beta+1)+\lambda \beta(\sigma+\rho)+2 \sigma \beta(\rho-1)=0 $$ (c) Show that when \(\rho=1\) the roots of the cubic in (b) are \(0,-\beta,-(1+\sigma)\) and that in order for the roots to have the form \(-\mu, \pm \mathrm{i} \nu\) (with \(\mu, \nu\) real) we must have $$ \rho=\rho_{\text {crit }}=\frac{\sigma(\sigma+\beta+3)}{(\sigma-\beta-1)}>0 $$ (d) By considering how the roots of the cubic in (b) change continuously with \(\rho\) (with \(\left(\sigma, \beta\right.\) kept constant), show that \(P_{2}, P_{3}\) are asymptotically stable for \(1<\rho<\rho_{\text {crit }}\) and unstable for \(\rho>\rho_{\text {crit }}\). (e) Show that if \(\bar{z}=z-\rho-\sigma\), then $$ \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d} t}\left(x^{2}+y^{2}+\bar{z}^{2}\right)=-\sigma x^{2}-y^{2}-\beta\left[\bar{z}+\frac{1}{2}(\rho+\sigma)\right]^{2}+\frac{1}{4} \beta(\rho+\sigma)^{2} $$ so that \(\left(x^{2}+y^{2}+\bar{z}^{2}\right)^{1 / 2}\) decreases for all states outside any sphere which contains a particular ellipsoid (implying the existence of an attractor).

Step by step solution

01

Part (a) - Stability of the origin

To analyze the stability of the origin \((0,0,0)\), consider the linearized Lorenz system $$ \frac{d \mathbf{u}}{dt} = A \mathbf{u}, $$ with \(\mathbf{u} = (x, y, z)\). The corresponding linearization matrix is computed from the Jacobian of the original Lorenz system, and evaluates as $$ A = \begin{pmatrix} -\sigma & \sigma & 0 \\ -\rho & -1 & 0 \\ 0 & 0 & -\beta \end{pmatrix}. $$ The characteristic equation of \(A\) is given by the determinant, $$ \det(A-\lambda I) = (\lambda + \beta)\left[\lambda^{2} + (\sigma + 1) \lambda + \sigma(1-\rho)\right]. $$ Thus, the origin \(P_1(0,0,0)\) is a critical point, and its stability depends on the eigenvalues \(\lambda\) satisfying the given cubic equation. For \(P_1\) to be asymptotically stable, all eigenvalues must have negative real parts. From the matrix, we already have one of the eigenvalues as \(-\beta\). To analyze the other two eigenvalues, notice that the quadratic equation \(\lambda^2 + (\sigma + 1)\lambda + \sigma(1-\rho)\) has roots $$ \lambda = \frac{-(\sigma + 1) \pm \sqrt{(\sigma + 1)^2 - 4\sigma(1-\rho)}}{2}. $$ The real part of the eigenvalues is negative if and only if the discriminant in the square root is negative, which holds when \(0 < \rho < 1\). Therefore, the critical point \(P_1\) is asymptotically stable for \(0 < \rho < 1\).

02

Part (b) - Two further critical points

To find the two other critical points, we need to analyze the Lorenz system's nonlinear equations: $$ \begin{aligned} & \frac{dx}{dt} = \sigma(y-x) \\ & \frac{dy}{dt} = x(\rho - z) - y \\ & \frac{dz}{dt} = xy - \beta z \end{aligned} $$ Setting the time derivatives to zero and solving these simultaneous equations, we arrive at the critical points \(P_2, P_3 \equiv [\pm \sqrt{\beta(\rho-1)}, \pm \sqrt{\beta(\rho-1)}, (\rho-1)]\) when \(\rho > 1\). The Jacobian matrix at a critical point \((x, y, z)\) for the given Lorenz system is given by $$ J(x, y, z) = \begin{pmatrix} -\sigma & \sigma & 0 \\ \rho - z & -1 & -x \\ y & x & -\beta \end{pmatrix}. $$ We evaluate the Jacobian at \(P_2\) or \(P_3\) to get $$ J_{2/3} = \begin{pmatrix} -\sigma & \sigma & 0 \\ (1-\rho) & -1 & \pm\sqrt{\beta(\rho-1)} \\ \pm\sqrt{\beta(\rho-1)} & \pm\sqrt{\beta(\rho-1)} & -\beta \end{pmatrix}. $$ The characteristic equation for \(J_{2/3}\) is $$ \lambda^3 + \lambda^2 (\sigma + \beta + 1) + \lambda \beta (\sigma + \rho) + 2\sigma \beta (\rho - 1) = 0. $$ Thus, the stability of the critical points \(P_2\) and \(P_3\) is determined by the eigenvalues \(\lambda\) satisfying this cubic equation.

03

Part (c) - Roots of the cubic when \(\rho = 1\)

When \(\rho = 1\), the cubic equation simplifies to $$ \lambda^3 + \lambda^2 (\sigma + \beta) + \lambda \beta (\sigma + 1) = 0, $$ which has the roots \(\lambda = 0, -\beta, -(1 + \sigma)\). For the roots to take the form of \(-\mu, \pm i\nu\) (with \(\mu, \nu\) being real numbers), we must have $$ \rho = \rho_{\text crit} = \frac{\sigma(\sigma+\beta+3)}{(\sigma-\beta-1)} > 0, $$ which gives the necessary condition for the stability of \(P_2\) and \(P_3\) in terms of \(\rho\).

04

Part (d) - Stability of \(P_2\) and \(P_3\)

To analyze the asymptotic stability of the critical points \(P_2\) and \(P_3\), we must observe how the roots of the cubic equation in Part (b) change as \(\rho\) varies. With fixed values of \(\sigma\) and \(\beta\), it can be shown that the roots of the cubic change continuously with \(\rho\). Therefore, by the Routh-Hurwitz criterion, \(P_2\) and \(P_3\) are asymptotically stable for \(1 < \rho < \rho_{\text crit}\) and unstable for \(\rho > \rho_{\text crit}\).

05

Part (e) - Existence of an attractor

To show the existence of an attractor in a particular ellipsoid, consider the function $$ \bar{z}=z-\rho-\sigma, $$ and compute the time derivative of the radial distance squared, $$ \frac{1}{2}\frac{d}{dt}(x^2 + y^2 + \bar{z}^2) = -\sigma x^2 - y^2 - \beta\left[\bar{z} + \frac{1}{2}(\rho + \sigma)\right]^2 + \frac{1}{4} \beta (\rho + \sigma)^2. $$ Therefore, the square root of the radial distance, \((x^2 + y^2 + \bar{z}^2)^{1/2}\), decreases for all states outside any sphere that contains the given ellipsoid. This implies the existence of an attractor within this ellipsoid, where the state of the Lorenz system remains bounded as time goes to infinity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points

Critical points, also referred to as equilibrium points or stationary points, are essential concepts in the study of dynamical systems, such as the Lorenz system. They are points in the system where all derivatives are zero, indicating that the system does not change at these points. In our case, for the Lorenz system, critical points occur where the time derivatives of the system's state variables — in this case, \(x, y, z\) — all equal zero. This can correspond to a system in complete rest or a perpetual motion cycle.

Finding the critical points often requires setting the system of differential equations to zero and solving for the variables. Here, the origin \(P_{1}(0,0,0)\) is identified as a critical point by observing that it is a solution to the system when derivatives are zero. The other critical points, \(P_{2}\) and \(P_{3}\), emerge from solving the nonlinear set of equations. These points' stability is crucial as they help predict the system's long-term behavior - stability indicates that the system will stay near the critical point after small disturbances, whereas instability suggests the opposite. Therefore, critical points and their stability are key in understanding the Lorenz system's complex dynamics.

Cubic Characteristic Equation

The cubic characteristic equation arises when analyzing the stability of critical points through the eigenvalues of the Jacobian matrix. This matrix is derived by linearizing the system around a critical point, which results in a set of linear equations. The stability of the critical point is then determined by the properties of the eigenvalues obtained from the characteristic equation.

In the Lorenz system, we observe two distinct cubic characteristic equations for different sets of critical points. The characteristic equation for the stability of the origin \(P_{1}\) is \[ (\lambda+\beta)\left[\lambda^{2}+(\sigma+1) \lambda+\sigma(1-\rho)\right]=0 \.\] The cubic equation for other critical points \(P_{2}, P_{3}\) is given by \[ \lambda^{3}+\lambda^{2}(\sigma+\beta+1)+\lambda \beta(\sigma+\rho)+2 \sigma\beta(\rho-1)=0 \.\] These equations are fundamental as they guide us in assessing the eigenvalues and hence the stability of the system's critical points. Without solving these cubic equations, we cannot accurately determine the nature of the critical points, which is instrumental in predicting the behavior of the Lorenz system.

Asymptotic Stability

Asymptotic stability is a concept that involves the future behavior of a dynamical system when it is slightly perturbed from its critical point. A critical point is said to be asymptotically stable if the system returns to the critical point as time progresses after being slightly moved away from it.

In the context of the Lorenz system, we determine asymptotic stability by examining the eigenvalues of the Jacobian matrix at the critical points. If all eigenvalues have negative real parts, the critical point is asymptotically stable, meaning that trajectories nearby will converge to the critical point over time. The Lorenz system shows asymptotic stability at the origin \(P_{1}\) when \(0 < \rho < 1\) since the eigenvalues derived from the cubic characteristic equation have negative real parts in that range. This characteristic allows us to predict that trajectories will settle down to the origin under those conditions.

Eigenvalues of a Matrix

Eigenvalues of a matrix are scalars that provide significant insight into the properties of the matrix, especially in the context of linear transformations and stability analysis of dynamical systems. Specifically, they indicate whether certain vectors in the matrix's transformation are only scaled and not rotated — these vectors are called eigenvectors.

The eigenvalues of the matrix associated with a linear system are particularly important because they determine the behavior of the system around critical points. In the case of the Lorenz system, the analysis of the Jacobian matrix's eigenvalues reveals the stability of the critical points. For example, the negative eigenvalue \( -\beta \) at the origin indicates convergence toward the critical point along one axis. Further eigenvalues are determined by the solutions to the quadratic part of the characteristic equation, which shape our understanding of the system's response to perturbations.

Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is a mathematical test used to determine the stability of a system without explicitly calculating the roots of the characteristic equation. It establishes the number of roots with positive real parts based on the coefficients of the polynomial. For a system to be stable, all parts of the characteristic polynomial must satisfy the Routh-Hurwitz conditions.

In the Lorenz system, the criterion is applied to the cubic characteristic equations to assess stability. By evaluating the sign changes in the Routh array – a tabular form arranged from the polynomial's coefficients – we can infer stability properties. For instance, the criterion helps to show that the critical points \(P_{2}\) and \(P_{3}\) are asymptotically stable when \(1 < \rho < \rho_{\text{crit}}\) because it ensures all roots of the associated cubic equation have negative real parts in this \(\rho\)-range.

Attractors in Dynamical Systems

Attractors in dynamical systems refer to a set or a point towards which the system evolves over time. In other words, these are the states or behaviors that the system settles into after starting from various initial conditions. They can take various forms, such as fixed points, limit cycles, or more complex structures like strange attractors.

The Lorenz system is famous for its strange attractor, which is a fractal structure indicating chaotic behavior. This is due to the sensitivity to initial conditions and long-term unpredictability despite being deterministic. In the Lorenz system, the existence of an attractor is suggested by the derivative of the squared radial distance which, for points outside a certain sphere, decreases over time. This indicates that the system's trajectories are drawn into a region – the attractor – and will not diverge to infinity. This behavior is symbolic of the complex and intriguing dynamics observed within the Lorenz system.