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A **limit cycle** is an isolated @closed@trajectory. Isolated means that @neighboring@trajectories are not @closed; they spiral toward or away from the limit cycle.
If all neighboring @trajectories approach the @limit-cycle, we say the @limit-cycle is **stable** or **attracting.**
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:::definition "Unstable limit cycle"
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If a @limit-cycle is not stable, we say it is an **unstable limit cycle**, or in exceptional cases, half-stable.
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:::
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### Non-Existence Criteria
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Sometimes we want to show that a @system does not have a @limit-cycle.
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#### Gradient Systems
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:::definition "Gradient System"
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If a @system can be written in the form $\dot{\vec{x}} = - \nabla V,$ for some @continuously-differentiable, single-values scalar function $V(\vec{x}},$ then it is said to be a **gradient system** with **potential function** $V.$
If we're always moving "downhill" in some direction in a space, it's impossible to come back to where we started. This is another reason why oscillations aren't possible in one dimensional systems.
Consider a system $\dot{\vec{x}} = \vec{f(x)}$ with a fixed point at $\vec{x^*}.$ If it has a function with the following properties:
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1. $V(\vec{x}) > 0$ for all $\vec{x} \neq \vec{x^*},$ and $V(\vec{x^*}) = 0.$ (i.e. $V$ is @positive-definite.)
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2. $\dot{V} < 0$ for all $\vec{x} \neq \vec{x^*}. (All trajectories flow "downhill" toward $\vec{x^*}.$
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Then such a function is called a **Liapunov function.**
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:::
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:::theorem
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If a system has a @liapunov-function, then its fixed point $\vec{x^*}$ is globally asymptotically stable: for all initial conditions, $\vec{x}(t) \to \vec{x^*}$ as $t \to \infty.$ Therefore, the system has no closed orbits.
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::::intuition
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Like gradient systems, we can't get in a loop if we're always moving downhill.
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There is no systematic way to construct @Lianpunov-functions. Strogatz says Divine Inspiration is required.
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#### Dulac's Criterion
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Dulac's Criterion is based on @greens-theorem.
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:::theorem
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Let $\dot{\vec{x}} = \vec{f(x)}$ be a @continuously-differentiable@vector-field defined on a @simply-connected@subset $R$ of the @plane. If there exists a @continuously-differentiable, @real-valued@function $g(\vec{x})$ such that $\nabla \cdot (g\dot{\vec{x}})$ has one sign throughout $R,$ then there are no @closed@orbits lying entirely in $R.$
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The special case where $g(\vec{x}) = 1$ is called **Bendixson's criterion.**
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As with @Liapunov-functions, there is no algorithm for finding $g(\vec{x}).$
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<!-- TODO: definition, isolated closed orbits, stable/unstable/half-stable limit cycles, amplitude and frequency -->
We can do linear stability analysis on this system to understand the local behavior near fixed points. We use the @Jacobian matrix of the system, then plug in $(x^*, y^*).$ For the nonlinear case, we have the same geometric interpretations as the linear case. Topologically, if the real part of any eigenvalue of a fixed point is positive, then the fixed point is unstable. If the real part of all eigenvalues of a fixed point is negative, then the fixed point is asymptotically stable. We call these cases (both eigenvalues have nonzero real parts) hyperbolic @fixed points
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However, if the real part of any eigenvalue is zero, this linearized approached does not tell us about the stability of the fixed point.
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<!-- TODO: converting planar systems to polar form, radial and angular dynamics, using polar coordinates for limit cycle analysis -->
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