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The local oscillator must produce a stable frequency with low harmonics. [ 1 ] Stability must take into account temperature, voltage, and mechanical drift as factors. The oscillator must produce enough output power to effectively drive subsequent stages of circuitry, such as mixers or frequency multipliers.
The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本 由紀, Kuramoto Yoshiki), [1] [2] is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators .
More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.
The Dick effect (hereinafter; "the effect") is an important limitation to frequency stability for modern atomic clocks such as atomic fountains and optical lattice clocks.It is an aliasing effect: High frequency noise in a required local oscillator (LO) is aliased (heterodyned) to near zero frequency by a periodic interrogation process that locks the frequency of the LO to that of the atoms.
For the Rössler attractor, when the local maximum is plotted against the next local maximum, +, the resulting plot (shown here for =, =, =) is unimodal, resembling a skewed Hénon map. Knowing that the Rössler attractor can be used to create a pseudo 1-d map, it then follows to use similar analysis methods.
The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. [2] Van der Pol found stable oscillations, [3] which he subsequently called relaxation-oscillations [4] and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes.
If >, when ˙ hold only for in some neighborhood of the origin, and the set {˙ =}does not contain any trajectories of the system besides the trajectory () =,, then the local version of the invariance principle states that the origin is locally asymptotically stable.
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...