Polar Coordinate Methods
Given a system $\dot{x} = f(x,y), \dot{y} = g(x,y),$ it's sometimes useful, especially for the analysis of limit cycles, to convert to polar coordinates. This can be done the manual way from the start, which leads to needing to solve a system of equations for $\dot{r}$ and $\dot{\theta},$ or we can use these handy formulas, which are equivalent:
$$ \dot{r} = \frac{x \dot{x} + y \dot{y}}{r}, \quad \dot{\theta} = \frac{x \dot{y} - y \dot{x}}{r^2}. $$
Conservative Systems
Given a system $\dot{\vec{x}} = \vec{f(x)},$ a conserved quantity is a @real-valued continuous function $E(\vec{x})$ that is constant on trajectories, i.e. $dE/dt = 0.$ We also require that $E(\vec{x})$ is nonconstant on every open set, to avoid trivial examples such as $E(\vec{x}) = 0.$
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As an example, consider a particle of mass $m$ moving along the $x$-axis, subject to a nonlinear force $F(x):$
$$ m\ddot{x} = F(x). $$
Note that $F(x)$ is not dependent on $\dot{x}$ or $t,$ so there is no damping or friction, and the driving force doesn't vary with time. Under these conditions, we can show that energy is conserved. Let $V(x)$ denote the potential energy, defined by $F(x) = -dV/dx.$ Then
$$ m\ddot{x} + \frac{dV}{dx} = 0. $$
Now, if we multiply both sides times $\dot{x}$ we get
$$ m \dot{x} \ddot{x} + \frac{dV}{dx} \dot{x} = 0 \implies \frac{d}{dt} \left [ \frac{1}{2} m \dot{x}^2 + V(x) \right ] = 0, $$
which means that the LHS is an @exact time derivative. We get here by applying the chain rule in reverse, i.e.
$$ \frac{d}{dt} V(x(t)) = \frac{dV}{dx} \frac{dx}{dt}. $$
Therefore, for a given solution $x(t),$ the total energy
$$ E = \frac{1}{2} m \dot{x}^2 + V(x) $$
is a constant function of time. The energy is called a conserved quantity, a constant of motion, or a first integral.
Systems for which a conserved quantity exist are called conservative systems.
Conservative systems have no attracting fixed points.
Suppose $\vec{x^*}$ were an attracting fixed point in a conservative system. Then, note that all trajectories in the basin of attraction approach $\vec{x^*}$ as $t \to \infty.$ Since $E(\vec{x})$ is continuous, the energy in the limit as trajectories approach $\vec{x^*}$ is equal to the energy at the fixed point, and so the energy along the entirety of each trajectory is equal to the energy at the fixed point. But, this implies that the energy in the entire basin of attraction is the same as the energy at the fixed point, which violates our definition of a conservative system (i.e. we require that $E(\vec{x})$ be nonconstant.) Therefore no such fixed point can exist.
$\square$Now, while conservative systems can't have attracting fixed points, they can have other types, and generally have @saddles and @centers.
A trajectory that starts and ends at the same fixed point is called a homoclinic orbit.
These homoclinic orbits are common in conservative systems but are rare otherwise. Note that homoclinic orbits are not periodic because they take forever trying to reach the fixed point.
Note that nonlinear @centers are robust in conservative systems, because any trajectories sufficiently close to them are closed.
Hamiltonian Systems
A Hamiltonian system is one where $H(p,q)$ is a smooth, real-valued function and we have that
$$ \dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q}. $$
The function $H$ is called the Hamiltonian.
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For any hamiltonian system, $H$ is a conserved quantity.
Reversible Systems
A reversible system is any second-order system that is invariant under $t \to -t$ and $y \to -y.$
Centers are also robust in reversible systems.
Pairs of orbits connecting twin @saddle-points are called heteroclinic trajectories.
These @heteroclinic-orbits are much more common in reversible or conservative systems than in other types of systems.
Limit Cycles
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.
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If all neighboring trajectories approach the limit cycle, we say the limit cycle is stable or attracting.
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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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Non-Existence Criteria
Sometimes we want to show that a @system does not have a limit cycle.
Gradient Systems
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 $V.$
Closed orbits are impossible in gradient systems.
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.
Lyapunov Functions
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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$V(\vec{x}) > 0$ for all $\vec{x} \neq \vec{x^*},$ and $V(\vec{x^*}) = 0.$ (i.e. $V$ is @positive-definite.)
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$\dot{V} < 0$ for all $\vec{x} \neq \vec{x^*}. (All trajectories flow "downhill" toward $\vec{x^*}.$
Then such a function is called a Liapunov function.
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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.
Like gradient systems, we can't get in a loop if we're always moving downhill.
There is no systematic way to construct @Lianpunov-functions. Strogatz says Divine Inspiration is required.
Dulac's Criterion
Dulac's Criterion is based on Green's Theorem.
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 $R.$
The special case where $g(\vec{x}) = 1$ is called Bendixson's theorem.
As with liapunov functions, there is no algorithm for finding $g(\vec{x}).$
Existence Criteria
The Poincaré-Bendixson theorem can be used to show that a limit cycle must exist in a particular system.
Suppose that
(1) $R$ is a closed, bounded subset of the plane;
(2) $\dot{\vec{x}} = \vec{f(x)}$ is a continuously differentiable vector field on an @open-set containing $R;$
(3) $R$ does not contain any fixed points;
(4) There exists a trajectory $C$ that is "confined" in $R,$ in the sense that it starts in $R$ and stays in $R$ for all future time.
Then, either $C$ is a closed orbit, or it spirals toward a closed orbit as $t \to \infty.$ In either case, $R$ contains a closed orbit.
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To show that a confined trajectory exists when applying Suppose that (1) $R$ is a closed,..., we usually construct a trapping region using polar coordinates. We usually do this for a stable limit cycle, but we can do it for an unstable limit cycle too, by looking at the system using reversed time. We want to keep it as tight as possible, so we find the maximum $r$ for which $\dot{r} > 0$ for all all $\theta.$ Anything inside of that $r$ will therefore be pushed outward into the region beyond it. Call this inner boundary $r_1.$ Similarly, we find the minimum $r$ for which all $\dot{r} < 0;$ call it $r_2.$ Thus, any trajectories that start beyond $r_2$ will be pushed into the region inside of $r_2.$ Now, any trajectories anywhere (other than those that start on a fixed point) will be pushed into our trapping region and will never leave it.