Question

The ODE system given by $$ \begin{aligned} &y_{1}^{\prime}=\alpha-y_{1}-\frac{4 y_{1} y_{2}}{1+y_{1}^{2}} \\ &y_{2}^{\prime}=\beta y_{1}\left(1-\frac{y_{2}}{1+y_{1}^{2}}\right) \end{aligned} $$ where \(\alpha\) and \(\beta\) are parameters, represents a simplified approximation to a chemical reaction. There is a parameter value \(\beta_{c}=\frac{3 \alpha}{5}-\frac{25}{\alpha}\) such that for \(\beta>\beta_{c}\) solution trajectories decay in amplitude and spiral in phase space into a stable fixed point, whereas for \(\beta<\beta_{c}\) trajectories oscillate without damping and are attracted to a stable limit cycle. (This is called a Hopf bifurcation.) (a) Set \(\alpha=10\) and use any of the discretization methods introduced in this chapter with a fixed step size \(h=0.01\) to approximate the solution starting at \(y_{1}(0)=0, y_{2}(0)=2\), for \(0 \leq t \leq 20\). Do this for the parameter values \(\beta=2\) and \(\beta=4\). For each case plot \(y_{1}\) vs. \(t\) and \(y_{2}\) vs. \(y_{1} .\) Describe your observations. (b) Investigate the situation closer to the critical value \(\beta_{c}=3.5\). (You may have to increase the length of the integration interval \(b\) to get a better look.)

Short answer

Question: Analyze the behavior of a simplified chemical reaction system for different values of the parameter \(\beta\), comparing it to its critical value \(\beta_c\), using Euler's method for numerical integration. Describe the observed behavior by plotting \(y_1\) vs. \(t\) and \(y_2\) vs. \(y_1\). Answer: When \(\beta\) is below the critical value of 3.5, the reaction reaches a stable equilibrium, visible as horizontal lines on \(y_1\) vs. \(t\) plot. When \(\beta\) is above the critical value, the system exhibits oscillatory behavior, with peaks and troughs in the \(y_1\) vs. \(t\) plot and closed loops in the \(y_2\) vs. \(y_1\) plot. Near the critical value of 3.5, the system shows a transition between stable equilibrium and oscillatory behavior, which is more apparent when using a longer integration interval.

Step-by-step solution

Initialize Parameters and Functions

First, let's set the given parameter values, \(\alpha=10\), the step size \(h=0.01\), and the initial conditions \(y_1(0)=0, y_2(0)=2\). We also need to define the system of differential equations as functions: $$ \begin{aligned} f_1(y_1, y_2) &= \alpha-y_{1}-\frac{4 y_{1} y_{2}}{1+y_{1}^{2}} \\ f_2(y_1, y_2) &= \beta y_{1}\left(1-\frac{y_{2}}{1+y_{1}^{2}}\right) \end{aligned} $$ where \(y_1'\) and \(y_2'\) represent the first derivatives of \(y_|RFPLC\) with respect to time.

Choose a Discretization Method

We will use the Euler's method to discretize the problem. For each differential equation, we will iteratively approximate the solution using the formula: $$ y_{n+1}=y_{n}+hf(y_{n}, t_{n}) $$ where \(h\) is the step size and \(f\) represents \(f_1\) or \(f_2\) depending on \(y_1\) or \(y_2\).

Implement Euler's Method

For each value of \(\beta\) in the list \([2, 4]\), perform the following steps: 1. Initialize lists for \(y_1\), \(y_2\), and \(t\) with their initial values. 2. Iterate through time steps from 0 to 20 using a step size of 0.01. 3. At each time step, use the Euler's method to calculate the next values of \(y_1\) and \(y_2\). 4. Add the calculated values of \(y_1\), \(y_2\), and the current time step to their respective lists. 5. After completing all the iterations, plot \(y_1\) vs. \(t\), and \(y_2\) vs. \(y_1\) and describe your observations.

Investigate the Critical Value

To investigate the situation for \(\beta\) closer to the critical value \(\beta_c=3.5\), repeat steps 1-3 from the previous step with \(\beta = 3.5\) and a longer integration interval, such as \(b = 50\). Plot the results and describe your observations.

Key concepts

Euler's Method

Euler's Method is a straightforward way to approximate solutions to ordinary differential equations (ODEs). It's ideal for solving initial value problems where values at the starting point are known. Here's how it works:
The method updates solutions by stepping a fixed size, called the step size (\(h\)), from a known point using the derivative's value at that point. The update rule for Euler's Method is \(y_{n+1}=y_{n}+hf(y_{n}, t_{n})\). This means we take the current value and add the product of the step size and the function's slope (or derivative) at the current point.

For our system of ODEs, the method calculates successive points for \(y_1\) and \(y_2\) over time. Using the initial conditions \(y_1(0)=0\) and \(y_2(0)=2\), we compute new values iteratively for \(t\) ranging from 0 to 20.

Though simple, this approach is not always precise for all types of ODEs. However, it's a useful introduction to numerical methods as it shows how solutions can be built step-by-step.

Hopf Bifurcation

A Hopf bifurcation in a dynamic system occurs when a parameter value shifts, changing the behavior of a system's solution from a stable state to periodic oscillations.
When computing, one often sees a fixed point (steady state) become unstable, leading to sustained, repeating oscillations termed a 'limit cycle.'

In our ODE system, we observe this phenomenon based on changes in \(\beta\). As \(\beta\) exceeds the critical value \(\beta_c\), solutions spiral inward, stabilizing to a fixed point. However, for \(\beta\) less than \(\beta_c\), solutions demonstrate oscillations, settling into a limit cycle.

This bifurcation exemplifies intricate dynamics that can arise in nonlinear systems, showing how seemingly simple changes in parameters lead to drastically different system behavior. Grasping the concept of Hopf bifurcation is essential for understanding complex systems such as ecological models and chemical reactions.

Numerical Methods

Numerical methods play a crucial role in solving ODEs when analytical solutions are difficult to derive. These methods transform continuous problems into discrete ones that computers can solve.
Examples include Euler's Method, Runge-Kutta methods, and Multi-step methods like Adams-Bashforth.

Numerical methods allow for simulations across diverse fields like physics and engineering where modeling exact solutions is impractical. They ensure we can predict and evaluate future system states based on current conditions, albeit with some approximation.
These methods require considerations of accuracy and stability to choose the right one for the problem at hand.

Overall, numerical methods expand our ability to handle intricate systems and build powerful simulations, making them invaluable in both educational and professional contexts.

Stability Analysis

Stability analysis examines how solutions to equations behave when influenced by small disturbances.
It involves determining if a solution will eventually return to a steady state or diverge over time.

In the context of our exercise, stability analysis highlights whether the system will stabilize to a fixed point or oscillate into a limit cycle. By examining critical parameters like \(\beta_c\), we assess where stability transitions to instability, and vice versa.

Analyzing stability is paramount in systems where maintaining control is critical, such as in control theory and engineering designs. These analyses help predict responses to environmental changes, ensuring safety and efficiency in applications like climate models and mechanical systems.

By understanding how to apply stability analysis, we can better manage systems, predicting their evolution under varying conditions, which is essential for effective decision-making.