The first concept introduced is conservative systems. Definition: a conserved quantity is a real-valued continuous function E(x) that is constant on trajectories and E(x) must be nonconstant on every open set. A conservative system cannot have attracting fixed points. A trajectory that starts and ends at the same fixed point is called a homoclinic orbit, and are usually only found on conservative systems. Next Strogatz discusses reversible systems, which are defined as a system that has time-reversal symmetry. If xdot = f(x,y) and ydot = g(x,y) , then a reversible system is one in which f is odd in y and g is even in y. In reversible systems if the origin is a center, then all trajectories sufficiently close to the origin are closed curves. Centers usually are fragile, but in reversible and conservative systems they are more robust.
Challenges
In the proof that all trajectories close to the origin are closed orbits in reversible systems, it seemed like in the first step the trajectory would create a closed orbit with using reversibility because it was stated that the flow swirls around the origin.
Reflections
Conservative systems made a lot of sense and they coincided with the small amount of physics I remember from high school. Reversible systems seem manageable, but the concept was definitely more shaky for me.
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