Limit cycles are isolated closed trajectories in the phase plane of a dynamical system. They represent self-sustained oscillations, meaning the system will repeatedly cycle through the same states. The Poincaré-Bendixson Theorem provides a powerful tool for proving the existence of limit cycles in two-dimensional dynamical systems. Specifically, it states that if a trajectory remains confined within a bounded region of the plane that contains no fixed points, then the trajectory must approach a limit cycle.
Understanding limit cycles and the Poincaré-Bendixson Theorem is crucial for analyzing the long-term behavior of many systems, from electrical circuits to ecological models. These concepts help us predict and understand stable oscillatory behavior.
Match the term with its correct definition:
| Term | Definition |
|---|---|
| 1. Limit Cycle | A. A point where the vector field is zero. |
| 2. Trajectory | B. A region in the phase plane where all trajectories move inward. |
| 3. Fixed Point | C. An isolated closed trajectory in phase space. |
| 4. Phase Plane | D. The path of a system's state through time. |
| 5. Invariant Region | E. A two-dimensional space where the axes represent the state variables. |
Complete the following paragraph with the correct terms:
The Poincaré-Bendixson Theorem applies to _______________ dynamical systems. It states that if a _______________ remains within a bounded region containing no _______________, then it must approach a _______________. This theorem helps prove the _______________ of self-sustained oscillations.
Consider a system described by the following differential equations:
$\frac{dx}{dt} = y + x(1 - x^2 - y^2)$
$\frac{dy}{dt} = -x + y(1 - x^2 - y^2)$
Explain how you might use the Poincaré-Bendixson Theorem to show that this system has a limit cycle. What steps would you take, and what conditions would need to be satisfied?
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