nonlinear dynamics homework solutions

MATH 412: Nonlinear Dynamics and Chaos (Spring 2015)

Prof. matthew pennybacker.

This first course in nonlinear dynamics and chaos is aimed at upper-level undergraduate and graduate students. We will use analytical methods, concrete examples, and geometric intuition to develop the basic theory of dynamical systems, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles, and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.

Your grade will be determined by nine homework assignments and a final exam. Homework will comprise seventy percent of your grade, and the final exam will comprise the remaining thirty percent. A combined total of ninety percent or higher will guarantee an A, eighty percent a B, seventy percent a C, and sixty percent a D.

You have the option of completing a project to replace your two lowest homework grades. For the project, you need to read a research paper and complete the following tasks:

• Explain in detail the mathematical model and the techniques used to analyze it.
• Summarize the results of the paper and, if possible, attempt to reproduce them.
• Propose one or more potential extensions to the model and discuss how the results may change.
• Why are more people right handed?
• Chaos and Scheduling Buses.
• Synchronization and the Millenial Bridge.
• Cats, rats, and seabirds.
• Water mites and mosquitoes.
• Coexistence and chaos in complex ecologies.
• Tuberculosis in badgers and cows.
• Modeling the decline of religous affliation.
• Modeling language death.
• Ultimate fate of constrained voters.
• Rock, paper, scissors and the evolution of lizards.
• Modeling the love story in "Gone with the Wind"
• Dynamical Characteristics Common to Neuronal Competition Models.
• The power of true believers.
• Sychrony in frogs.
• Predator prey food chains with adaptation.

Each assignment is due in class on the date indicated. For some of the exercises, you will need the PPLANE utility for Matlab, which can be downloaded here for version R2014a or older and here for version R2014b or newer.

Advanced Dynamics

Hw_03 - nonlinear solutions, hw_03 - nonlinear solutions ¶.

Creating a solution for the hanging chain, we reached a point where the constants required a nonlinear solution to an algebraic equation,

\(y(x) = \cosh \frac{\rho g a}{2c} - \cosh\frac{\rho g x}{c}\)

\(L = \int_{-a/2}^{a/2}\cosh\frac{\rho g x}{c}dx \rightarrow L = \frac{c}{\rho g} \sinh\frac{\rho g a}{c}\)

The second equation does not have an “analytical” solution. Where “analytical” refers to an equation with seperable input/output. What you need is a “numerical” solution to equation 2:

what \(c\) will satisfy this equation?

\(f(c)=L - \frac{c}{\rho g} \sinh\frac{\rho g a}{c}=0\)

These problems often come up when engineering systems have large displacements or large rotations that cannot be ignored. One way to approach this problem is to guess the solution. You could try:

If you happen to guess numbers that change the sign of \(f(c)\) , then you know one interval where \(f(c_{solution})=0\) must have been true. I find it helps to plot the function to see where the solution may exist

Numerical solution ¶

We can use fsolve to automate the guess-and-check method. You need 2 things:

a function f(c) that returns the result \(f(c)=L - \frac{c}{\rho g} \sinh\frac{\rho g a}{c}=0\)

an initial guess, c_0

Numerical solutions always require an initial guess for the solution and they will iterate until your function f(c_sol) \(\approx0\) .

Note : fsolve has more advanced features than ‘guess-and-check’, but at its core it uses algorithms to reduce the number of guesses and checks.

Define f(c) with lambda ¶

In Python, you can use the lambda function to create functions in one line. The other way to create a function is using def .

Note : def is a much richer way to create functions in Python. We will use it later when we want more involved functions.

Here, you define the function f(c) with lambda :

Solve f(c_sol)=0 with fsolve ¶

The numerical solver, fsolve , is part of the scipy.optimize library. Import the function with the from … import -command.

Now, you can solve for the value of c_sol that creates a solution to f(c_sol)=0 . Use the function, f and an initial guess, c0=40 .

Plug into catenary equation ¶

Now, you have a solution for \(c\) that describes the hanging chain. Plug it into the original equation

and plot the final shape.

Problem 1 ¶

Plot the solution for two hanging chains, the same as we did above:

\(g = 9.81~m/s/s\) \(L = 1~m\) \(rho = 5~kg/m\)

\(a = 0.9~m\)

\(a = 0.7~m\)

Problem 2 ¶

In the four-bar linkage show above there are 3 bodies moving in 2D (9 DOF) and 4 pins (8 constraints). The linkage configuration is constrained by the two nonlinear equations

\(l_1\sin\theta_1+l_2\sin\theta_2-l_3\sin\theta_3 -d_y = 0\)

\(l_1\cos\theta_1+l_2\cos\theta_2-l_3\cos\theta_3 -d_x = 0\)

If you have one of the angles, \(\theta_1\) , you can use equations 1 and 2 to solve for the other two angles, \(\theta_2~and\theta_3\) using fsolve only now the input is a vector with two values and the output is a vector with two values.

\(\bar{f}(\bar{x})= \left[\begin{array}{c} f_1(\theta_2,~\theta_3) \\ f_2(\theta_2,~\theta_3)\end{array}\right]=\left[\begin{array}{c} l_1\sin\theta_1+l_2\sin\theta_2-l_3\sin\theta_3 -d_y\\ l_1\cos\theta_1+l_2\cos\theta_2-l_3\cos\theta_3 -d_x \end{array}\right]\)

The linkage system has the following properties:

link 1: \(l_1 = 0.5~m\)

link 2: \(l_2 = 1~m\)

link 3: \(l_3 = 1~m\)

when \(\theta_1=90^o\) , \(\theta_2=0^o\) , and \(\theta_3=90^o\) . So the two grounded pins have a fixed relative position, \(r_{3/1} = d_x\hat{i}+d_y\hat{j} = 1\hat{i}-0.5\hat{j}\) .

Below, the definition of Fbar is defined for \(\bar{f}(\bar{x})\) and the function is satisfied for \(\theta_1=\theta_3=90^o\) and \(\theta_2=0^o\) . Then, the links are plotted with rx and ry , where

\(rx = \left[\begin{array}~0\\l_1\cos(\theta_1)\\l_1\cos(\theta_1)+l_2\cos(\theta_2)\\ l_1\cos(\theta_1) + l_2\cos(\theta_2)-l_3\cos(\theta_3)\end{array}\right]\)

\(ry = \left[\begin{array}~0\\l_1\sin(\theta_1)\\l_1\sin(\theta_1)+l_2\sin(\theta_2)\\ l_1\sin(\theta_1)+l_2\sin(\theta_2)-l_3\sin(\theta_3)\end{array}\right]\)

Your goal: ¶

Change the angle to \(\theta_1=45^o,~135^o,~and~180^o\) . Plot the three configurations like above. Use fsolve to find \(\theta_2~and~\theta_3\) .

Example: Pendulum support by spring has two stable positions

Homework #4

Snapsolve any problem by taking a picture. Try it in the Numerade app?

Solutions for Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering 1st

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Bifurcations, flows on the circle, linear systems, phase plane, limit cycles, bifurcations revisited, lorenz equations, one-dimensional maps, strange attractors.

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