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Carlo Beenakker
Carlo Beenakker
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  1. If you add an $i$-independent constanttime-dependent term $\delta\theta=\omega t+\gamma$ to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged. The phase equation is changed by a term $d\delta\theta/dt=\omega$.
  2. Global phase symmetry means that if $\psi_n=a_ne^{i\theta_n}$ is a solution, then also $\psi_n e^{i\phi t}$ with a constant phase $\phi$ is a solution. A local phase symmetry would have a $\phi_n$ that depends on $n$, which does not appply here.
  3. The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.
  1. If you add an $i$-independent constant to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged.
  2. Global phase symmetry means that if $\psi_n=a_ne^{i\theta_n}$ is a solution, then also $\psi_n e^{i\phi t}$ with a constant phase $\phi$ is a solution. A local phase symmetry would have a $\phi_n$ that depends on $n$, which does not appply here.
  3. The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.
  1. If you add an $i$-independent time-dependent term $\delta\theta=\omega t+\gamma$ to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged. The phase equation is changed by a term $d\delta\theta/dt=\omega$.
  2. Global phase symmetry means that if $\psi_n=a_ne^{i\theta_n}$ is a solution, then also $\psi_n e^{i\phi t}$ with a constant phase $\phi$ is a solution. A local phase symmetry would have a $\phi_n$ that depends on $n$, which does not appply here.
  3. The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.
Source Link
Carlo Beenakker
Carlo Beenakker
  • 188.3k
  • 18
  • 448
  • 651

  1. If you add an $i$-independent constant to $\theta_i$ then the difference $\theta_k-\theta_i$ that appears in the amplitude equation is unchanged.
  2. Global phase symmetry means that if $\psi_n=a_ne^{i\theta_n}$ is a solution, then also $\psi_n e^{i\phi t}$ with a constant phase $\phi$ is a solution. A local phase symmetry would have a $\phi_n$ that depends on $n$, which does not appply here.
  3. The limit cycle is semi-stable, rather than stable, because only amplitude perturbations decay in time, phase perturbations do not.