Question

Consider the ellipsoid $$ V(x, y, z)=r x^{2}+\sigma y^{2}+\sigma(z-2 r)^{2}=c>0 $$ (a) Calculate \(d V / d t\) along trajectories of the Lorenz equations \((1) .\) (b) Determine a sufficient condition on \(c\) so that every trajectory crossing \(V(x, y, z)=c\) is directed inward. (c) Evaluate the condition found in part (b) for the case \(\sigma=10, b=8 / 3, r=28\)

Short answer

Question: Determine the sufficient condition on \(c\) for every trajectory crossing the ellipsoid \(V(x, y, z) = rx^{2} + \sigma y^{2} + \sigma(z-2r)^{2} = c\) to be directed inward, given the Lorenz equations. Evaluate this condition for \(\sigma = 10, b = 8 / 3, r = 28\). Answer: The sufficient condition on \(c\) for inward trajectories crossing the ellipsoid is \(c > \frac{1}{\sigma}\left[\frac{8}{3}z^{2}-2r(xy+xz)+8r^{2}\right]\). Specifically, for \(\sigma = 10, b = 8 / 3, r = 28\), the condition becomes \(c > \frac{1}{10}\left[\frac{8}{3}(z^{2})-56(x(y+z))+8(28^{2})\right]\).

Step-by-step solution

(a) Calculating the derivative dV/dt along the trajectories of the Lorenz equations

First, we find the gradient of V with respect to x, y, and z: $$\begin{aligned} \frac{\partial V}{\partial x} &= 2r x \\ \frac{\partial V}{\partial y} &= 2\sigma y \\ \frac{\partial V}{\partial z} &= 2\sigma(z-2r) \end{aligned}$$ Now, we can calculate \(\frac{dV}{dt}\) by using the chain rule: $$\frac{dV}{dt} = \frac{\partial V}{\partial x}\frac{dx}{dt} + \frac{\partial V}{\partial y}\frac{dy}{dt} + \frac{\partial V}{\partial z}\frac{dz}{dt}$$ Plug in the expressions for the gradient of V and the Lorenz equations: $$\frac{dV}{dt} = 2rx(\sigma(y-x)) + 2\sigma y(rx-y-xz) + 2\sigma(z-2r)(-\frac{8}{3}z + xy)$$

(b) Finding a sufficient condition on c for inward trajectories

For every trajectory crossing the ellipsoid \(V(x,y,z)=c\) to be directed inward, we require \(\frac{dV}{dt}<0\). From the expression we calculated in (a), we want to find a condition on \(c\) such that: $$2rx(\sigma(y-x)) + 2\sigma y(rx-y-xz) + 2\sigma(z-2r)(-\frac{8}{3}z + xy) < 0$$ Rearrange the inequality to get: $$c > \frac{1}{\sigma}\left[\frac{8}{3}z^{2}-2r(xy+xz)+8r^{2}\right]$$ This inequality provides a sufficient condition on \(c\) for inward trajectories crossing the ellipsoid.

(c) Evaluating the condition found in part (b) for specific values of sigma, b, and r

For \(\sigma = 10, b = 8 / 3, r = 28\), the sufficient condition on \(c\) becomes: $$c > \frac{1}{10}\left[\frac{8}{3}(z^{2})-56(x(y+z))+8(28^{2})\right]$$ To have inward trajectories for all points on the ellipsoid, \(c\) must be greater than this expression. In essence, the ellipsoid with the given parameters must satisfy this inequality for all inward trajectories crossing its surface.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They are central to understanding a wide array of phenomena in physics, biology, economics, and engineering, such as pattern formation, population dynamics, and mechanical systems. The Lorenz equations, featured in this exercise, are a set of three coupled, non-linear differential equations that model convection rolls in the atmosphere. They are a prime example of a system that can exhibit chaotic behavior, making them a fascinating subject of study.

In the exercise, the derivative of a potential function V with respect to time, i.e., \( \frac{dV}{dt} \), along the trajectories of the Lorenz equations, illustrates the application of differential equations to analyze the system's evolution which is central to predicting its behavior.

Ellipsoid

An ellipsoid is a three-dimensional surface, all plane sections of which are ellipses or circles. It can be represented by the equation \( V(x, y, z) = rx^2 + \sigma y^2 + \sigma(z - 2r)^2 = c \). In the context of our problem, the ellipsoid functions as a boundary, where the dynamics of the Lorenz system are analyzed in terms of entering and exiting trajectories. Key to understanding this concept is visualizing the ellipsoid as an enclosure that influences how the system behaves when trajectories intersect with it. The characteristics and shape of the ellipsoid, defined by parameters like \( r \), \( \sigma \), and the constant \( c \), will directly affect the behavior of trajectories that approach and cross it.

Boundary Value Problems

Boundary value problems (BVPs) are a type of differential equation problem with conditions specified at the boundaries of the range in question. Solving a BVP involves finding a solution to the differential equations that not only satisfies the specified conditions at the boundaries but also within the entire domain. In our exercise, the boundary in question is the ellipsoid \( V(x, y, z) \), and we seek to define conditions (in this case, values of \( c \)) so that the Lorenz system's trajectories will behave in a desirable way (here, directed inward) when they intersect with this boundary. This underscores the importance of BVPs in predicting the long-term behavior of dynamical systems and effectively encapsulates the interplay between diverse mathematical concepts.