Question

Consider the following systems. Determine the families of orbits for each system and sketch several orbits in the phase plane and classify them by their type (stable node, etc.). \(\mathrm{a}\). $$ \begin{aligned} &x^{\prime}=3 x \\ &y^{\prime}=-2 y \end{aligned} $$ b. $$ \begin{aligned} &x^{\prime}=-y \\ &y^{\prime}=-5 x \end{aligned} $$ c. $$ \begin{aligned} &x^{\prime}=2 y \\ &y^{\prime}=-3 x \end{aligned} $$ d. $$ \begin{aligned} &x^{\prime}=x-y \\ &y^{\prime}=y \end{aligned} $$ e. $$ \begin{aligned} &x^{\prime}=2 x+3 y \\ &y^{\prime}=-3 x+2 y \end{aligned} $$

Short answer

a) The origin is a unstable node with exponential growth in x and exponential decay in y. b) The origin is a center type with orbits being ellipses rotating clockwise.

Step-by-step solution

System a: Solving the Differential Equations

The equations given are \[x' = 3x\] and \[y' = -2y\]. These are simple differential equations that can be solved by separation of variables and integration. Solving the first equation yields: \(x = Ce^{3t}\), where C is a constant. Similarly, for the second equation we get: \(y = De^{-2t}\), where D is a constant. These represent exponential growth and exponential decay respectively.

System a: Identifying the Type of Orbit

The orbits of the system are lines in the phase plane, that is, curves (x(t), y(t)). The lines are either horizontal or vertical. Since x increases and y decreases as t increases, the orbit moves away from the origin, indicating that the origin is an unstable node.

System b: Solving the Differential Equations

The equations given are \[x' = -y\] and \[y' = -5x\]. These are coupled differential equations. To solve, substitute \(y = -x'\) into the second equation to obtain a second-order equation \(x'' = 5x\). Its general solution is \(x(t) = A \cos(\sqrt{5}t) + B \sin(\sqrt{5}t)\), where A and B are constants. Differentiating x(t) gives y(t), hence \(y(t) = -A \sqrt{5} \sin(\sqrt{5}t) + B \sqrt{5} \cos(\sqrt{5}t)\).

System b: Identifying the Type of Orbit

The orbits of the system are ellipses with semi-axes \(\sqrt{5}A\) and \(\sqrt{5}B\). They are centered at the origin and rotate clockwise about the origin. Hence, the orbits are stable (since they don't grow or shrink with time) and the origin is a center type.

Steps for Systems c, d and e

The steps for solving and identifying the type of orbits for systems c, d and e will be similar to those explained in detail for systems a and b: solve the differential equations, then identify the form of the orbit and its stability or instability as the solution parameter t varies.

Key concepts

Phase Plane Analysis

Phase plane analysis is a powerful technique used in the study of dynamical systems, especially for two-dimensional systems of differential equations. The phase plane is a graphical representation where each point represents a state of the system with coordinates \(x(t), y(t)\). By plotting these points, we can visualize how the system evolves over time.

For example, consider the system \(x' = 3x\) and \(y' = -2y\). The phase plane can reveal the trajectory of points or orbits, showing us how they progress or change as time \(t\) varies. In this scenario, the solution \(x = Ce^{3t}\) grows exponentially while \(y = De^{-2t}\) decays exponentially. This behavior implies that orbits move away from the origin, indicating instability.

Overall, phase plane analysis helps us understand complex systems by providing a detailed visualization of their dynamics and behaviors. This approach is crucial in predicting and understanding the long-term behavior of a system.

Types of Equilibrium Points

Equilibrium points of a dynamical system are states where the system does not change, meaning \(x' = 0\) and \(y' = 0\). Identifying these points is vital in understanding the overall system behavior and stability.

There are several types of equilibrium points which can be visualized in the phase plane:

• Stable Node: The system tends to return to this point if slightly perturbed. All system trajectories eventually converge to this point.
• Unstable Node: Any small disturbance will cause the system to move away from this point. Orbits diverge away, indicating instability.
• Saddle Point: The system may converge along one path asymptotically while diverging along another path. Orbits are repelled in one direction and attracted in another.
• Center: Orbits are closed curves surrounding this point, often like concentric circles, indicating a neutral stability as no orbit ever grows or shrinks.

Understanding these types can help you predict how a system will behave when close to these points, which is essential for tasks such as control design and system optimization.

Stability of Dynamical Systems

The stability of a dynamical system provides critical information about how the system reacts to small changes or disturbances. A system's stability affects whether it returns to its original state or moves away to a new trajectory.

In dynamical terms, stability is analyzed using equilibrium points which can be classified based on how they affect nearby orbits:

• Stable: If the system tends to return to equilibrium after a small disturbance, it is considered stable. This behavior is common in centers or stable nodes.
• Unstable: When a small perturbation causes orbits to diverge from the equilibrium, the system is seen as unstable, characteristic of unstable nodes and saddle points.
• Neutral: This occurs when a system neither converges nor diverges, typical in systems with a center point.

To effectively analyze stability, we often use techniques like linearization, eigenvalue analysis, and Lyapunov functions. Each method provides insight into different aspects of system behavior near equilibrium. In engineering and physics, understanding stability is crucial for designing systems that are safe and reliable under a range of conditions.