Engr 231 Linear Engineering Systems Lab 2 In-Class Assignmen

Engr 231 Linear Engineering Systemslab 2 In Class Assignment Summ

In this assignment, students will compare the critical points and their stability for two related autonomous differential equations. The second equation is adjusted so that its largest root becomes a double root. Students are instructed to use MATLAB to find all critical points of the first differential equation, plot phase plots with stability markers, overlay phase lines, and then repeat the analysis for the second differential equation with the modified root structure. The task emphasizes proper visualization, marking stability types with colors, and correctly demonstrating phase lines indicating regions of positive and negative derivatives. A complete, well-labelled plot with all the critical points and phase line markers is required, as well as a thorough discussion of stability and root significance.

Paper For Above instruction

The analysis of stability and equilibrium points in autonomous differential equations is fundamental in understanding the long-term behavior of dynamical systems. In this study, we focus on two related equations, starting with an initial polynomial differential equation and proceeding to a modified version where the root structure is intentionally altered to examine the effects on stability and phase portrait visualization.

Finding Critical Points of the First Differential Equation

Using MATLAB, the first step involves defining the polynomial representing the differential equation, then solving for its roots to identify the critical points where the derivative goes to zero. These roots are the equilibrium solutions of the system. The process employs the MATLAB functions ‘sym’, ‘poly2sym’, and ‘roots’ to convert symbolic representations into polynomial form and extract roots.

Given the roots: -2, -0.5, 0, 0.5, 2, 3, the roots should be sorted in ascending order for clarity. The stability of each equilibrium point depends on the sign of the derivative of the polynomial at those points: if the derivative changes from positive to negative, the point is stable; if from negative to positive, unstable; and if the derivative touches zero without changing sign, the point may be semi-stable.

Plotting Phase Portraits and Marking Critical Points

The phase plot is generated by plotting the function \(f(y)\) versus \(y\) across a suitable range that includes all critical points, such as from -5 to 5. Using MATLAB’s ‘plot’ function, the roots are marked with circles whose colors depend on their stability: green for stable, red for unstable, and yellow for semi-stable. Adding grid lines, labels, and annotations enhances interpretability. The ‘hold on’ command ensures the phase plot and markers are combined seamlessly.

Additionally, phase lines are overlaid by plotting arrows or symbols (e.g., ‘>’ or ‘’ for positive derivatives and red ‘

Analysis of the Modified Equation with a Double Root

For the second differential equation, the root structure is modified so that the largest root becomes a double root. In MATLAB, this can be realized by multiplying the right-hand side of the differential equation by a factor that introduces multiplicity, effectively creating a repeated root without changing the roots’ locations. This adjustment results in a ‘semi-stable’ equilibrium at the double root, as typical in differential systems where multiple roots lead to neutral stability.

The same process of root finding, phase plotting, and phase line overlaying is repeated. The challenge is to correctly display the double root, with the stability classification reflecting its semi-stable nature. The plot must include all the roots, appropriately marked, with comprehensive labels, axes, and a phase line that accurately shows the flow influence around each equilibrium.

Visualization and Interpretation of Results

Visual clarity and accuracy are essential. The axes should be appropriately scaled to include all critical points with some margin. Each marker must be correctly colored to reflect the stability, and the phase line symbols must correspond with the derivative sign at each equilibrium. The final plots demonstrate how the change in root multiplicity impacts system stability, highlighting the quasi-stable nature at the double root, which appears as a ‘semi-stable’ point with flow lines approaching but not diverging away.

The comprehensive phase portraits with annotated critical points and phase lines enable a detailed qualitative understanding of the system's dynamics under the given differential equations. Such visual tools are vital in analyzing nonlinear systems, contributing to broader insights in control theory, ecology, or any field where the stability of equilibria dictates system evolution.

Conclusion

This assignment emphasizes the importance of visualizing critical points, understanding their stability, and recognizing the effects of root multiplicity on system dynamics. MATLAB proves to be an invaluable tool for symbolic and numerical analysis, enabling clear graphical representation and detailed stability analysis. Mastery of these techniques enhances comprehension of nonlinear systems and supports their application across scientific disciplines.

References

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