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Chapter 9: Problem 8

Carry out the indicated investigations of the Lorenz equations. (a) For \(r=21\) plot \(x\) versus \(t\) for the solutions starting at the initial points \((3,8,0),\) \((5,5,5),\) and \((5,5,10) .\) Use a \(t\) interval of at least \(0 \leq t \leq 30 .\) Compare your graphs with those in Figure \(9.8 .4 .\) (b) Repeat the calculation in part (a) for \(r=22, r=23,\) and \(r=24 .\) Increase the \(t\) interval as necessary so that you can determine when each solution begins to converge to one of the critical points. Record the approximate duration of the chaotic transient in each case. Describe how this quantity depends on the value of \(r\). (c) Repeat the calculations in parts (a) and (b) for values of \(r\) slightly greater than 24 . Try to estimate the value of \(r\) for which the duration of the chaotic transient approaches infinity.

Short Answer

Expert verified
#Summary# This exercise aims to analyze the behavior of the Lorenz equations for various initial conditions and values of the parameter \(r\). The steps involve setting up a Lorenz equation solver, plotting the solutions for different values of \(r\) and various initial conditions, and determining the dependency of the chaotic transient duration on \(r\). Finally, estimate the value of \(r\) for which the chaotic transient duration approaches infinity by analyzing the convergence behavior seen in the graphs and the solutions obtained.

Step by step solution

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01

Define a Lorenz equation solver

Use a numerical integration method (such as Runge-Kutta) to determine the trajectories of the Lorenz equations for varying initial conditions and parameter values. Implement this as a function which takes as input the initial conditions, parameter values, and the time interval, and returns the trajectories of x, y, and z over time. ##Step 2. Plot Solutions for r = 21##
02

Solve and plot solutions for r = 21

Use the Lorenz equation solver defined in Step 1 to calculate the solutions for the initial conditions \((3,8,0)\), \((5,5,5)\), and \((5,5,10)\) with \(r = 21\). Then, plot the solutions of \(x\) versus \(t\) over the time interval \(0 \leq t \leq 30\). Compare them with the graphs in Figure 9.8.4. ##Step 3. Repeat for r = 22, 23, 24##
03

Solve and plot solutions for r=22, r=23, and r=24

Repeat Step 2 for \(r\) values of 22, 23, and 24. Additionally, increase the time interval as necessary to determine when each solution begins to converge to one of the critical points. Record the approximate duration of the chaotic transient in each case. ##Step 4. Analyze Chaotic Transient Duration Dependency##
04

Chaotic transient dependency on r

Describe how the duration of the chaotic transient depends on the value of \(r\). Based on the solutions and graphs obtained in Step 3, write your observations on the relationship between \(r\) and the chaotic transient duration. ##Step 5. Repeat for r > 24##
05

Solve and estimate for r > 24

Repeat the calculations from Steps 2 and 3 for values of \(r\) slightly greater than 24. Estimate the value of \(r\) for which the duration of the chaotic transient approaches infinity, based on the convergence behavior seen in the graphs and the solutions obtained.

Key Concepts

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

Chaotic Transient Duration

In the study of dynamical systems, such as the Lorenz equations, the concept of chaotic transient duration refers to the length of time a system's behavior appears chaotic before it settles into a stable pattern or a periodic orbit. During the transient period, the system's evolution is sensitive to initial conditions, meaning that small changes can lead to vastly different outcomes over time. This sensitivity is a hallmark of chaos.

  • The length of the chaotic transient can vary depending on parameters within the system, such as the Rayleigh number r in Lorenz equations.
  • As r is altered, the duration of the chaotic transient period can either increase or decrease, eventually reaching a point where the system may never settle into a stable pattern, signifying an infinite transient.

To investigate this characteristic within the Lorenz system, one would observe how the trajectory of a solution starting from given initial conditions changes over time for different values of r. As the value of r approaches a critical threshold, the chaotic transient duration typically increases, providing insight into the system's stability and behavior under varying conditions.

Numerical Integration Methods

When exploring complex systems like the Lorenz equations, numerical integration methods are indispensable tools. These methods allow mathematicians and scientists to approximate the solutions to differential equations that cannot be solved analytically.

  • These techniques transform differential equations into a series of algebraic equations that can be solved with a computer.
  • Numerical methods include Euler's method, the Runge-Kutta methods, and others, each with varying degrees of accuracy and computational complexity.

An essential aspect of using numerical methods is selecting an appropriate step size. Too large a step size can result in significant errors, while too small a step size can lead to long computation times without a substantial increase in accuracy. When analyzing chaotic systems, where precision is crucial to capture the sensitive dependence on initial conditions, choosing an appropriate numerical method and step size is of paramount importance.

System Stability Analysis

System stability analysis is a core component of understanding dynamical systems like those described by the Lorenz equations. Stability analysis allows us to determine the behavior of a system over time and whether perturbations will grow or dampen, leading to different types of equilibria or steady states.

  • By examining the eigenvalues of the Jacobian matrix at particular points, one can infer the local stability of those points.
  • A stable system will return to an equilibrium state after a disturbance, whereas an unstable system may diverge away from its original state.
  • In chaotic systems, this analysis is complicated by the presence of strange attractors and sensitive dependence on initial conditions, which may lead to a diverging trajectory even if the system is locally stable.

For the Lorenz equations, this type of analysis reveals critical points and helps understand the emergence of chaos as a result of parameter changes, like variations in the value of r. This is crucial to predict how the system might behave over time and assess the potential for chaotic dynamics.

Runge-Kutta Method

The Runge-Kutta method is a powerful family of iterative numerical integration methods used to solve ordinary differential equations (ODEs). One of the most commonly used variants is the fourth-order Runge-Kutta method (often abbreviated as RK4), due to its balance between accuracy and computational effort.

  • RK4 provides an approximate solution to differential equations by calculating multiple slopes at strategically chosen points within an interval and then combining these slopes to produce a better estimate for the next value.
  • The method is particularly effective for problems where higher precision is required, such as the Lorenz equations, where small differences in initial conditions can lead to significant divergence in the system's behavior over time.
  • To apply the Runge-Kutta method to the Lorenz equations, one would systematically apply the RK4 algorithm across the chosen time interval, continuously updating the solution at each step based on the calculated slopes.

While more computationally intensive than simpler methods like Euler's method, the Runge-Kutta method provides a good compromise between complexity and precision, making it ideal for studying systems where chaos and complex dynamics are present.

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Most popular questions from this chapter

Assuming that the trajectory corresponding to a solution \(x=\phi(t), y=\psi(t),-\infty0\) such that \(\phi\left(t_{0}+T\right)=x_{0}, \psi\left(t_{0}+T\right)=y_{0} .\) Show that \(x=\Phi(t)=\phi(t+T)\) and \(y=\Psi(t)=\psi(t+T)\) is a solution and then use the existence and uniqueness theorem to show that \(\Phi(t)=\phi(t)\) and \(\Psi(t)=\psi(t)\) for all \(t .\)

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=x-x^{2}-x y, \quad d y / d t=\frac{1}{2} y-\frac{1}{4} y^{2}-\frac{3}{4} x y $$

(a) Find all the critical points (equilibrium solutions). (b) Use a computer to draw a direction field and portrait for the system. (c) From the plot(s) in part (b) determine whether each critical point is asymptotically stable, stable, or unstable, and classify it as to type. $$ d x / d t=(2+x)(y-x), \quad d y / d t=y\left(2+x-x^{2}\right) $$

We will prove part of Theorem 9.3 .2 : If the critical point \((0,0)\) of the almost linear system $$ d x / d t=a_{11} x+a_{12} y+F_{1}(x, y), \quad d y / d t=a_{21} x+a_{22} y+G_{1}(x, y) $$ is an asymptotically stable critical point of the corresponding linear system $$ d x / d t=a_{11} x+a_{12} y, \quad d y / d t=a_{21} x+a_{22} y $$ then it is an asymptotically stable critical point of the almost linear system (i). Problem 12 deals with the corresponding result for instability. In this problem we show that the Liapunov function constructed in the preceding problem is also a Liapunov function for the almost linear system (i). We must show that there is some region containing the origin for which \(\hat{V}\) is negative definite. (a) Show that $$ \hat{V}(x, y)=-\left(x^{2}+y^{2}\right)+(2 A x+B y) F_{1}(x, y)+(B x+2 C y) G_{1}(x, y) $$ (b) Recall that \(F_{1}(x, y) / r \rightarrow 0\) and \(G_{1}(x, y) / r \rightarrow 0\) as \(r=\left(x^{2}+y^{2}\right)^{1 / 2} \rightarrow 0 .\) This means that given any \(\epsilon>0\) there exists a circle \(r=R\) about the origin such that for \(0

This problem extends Problem 22 to a damped pendulum . The equations of motion are $$ d x / d t=y, \quad d y / d t=-4 \sin x-\gamma y . $$ where \(\gamma\) is the damping coefficient, with the initial conditions \(x(0)=0, y(0)=v\) (a) For \(\gamma=1 / 4\) plot \(x\) versus \(t\) for \(v=2\) and for \(v=5 .\) Explain these plots in terms of the motions of the pendulum that they represent. Also explain how they relate to the corresponding graphs in Problem 22 (a). (b) Estimate the critical value \(v_{c}\) of the initial velocity where the transition from one type of motion to the other occurs. (c) Repeat part (b) for other values of \(\gamma\) and determine how \(v_{c}\) depends on \(\gamma\).
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