We can analyze the role of Rev-Erb-α on the stability of circadian oscillations using a model consisting of 19 ODEs. The parameter v_sB describes the coupling of the protein REV-ERB-α on the other cellular circadian clock components. Using the same values of the parameters as in Figure 8 of the same paper, we can analyze the bifurcation due to the coupling parameter v_sB. A bifurcation point close to v_sB^*=1.61 nM/h exists at which oscillations change from damped to a limit cycle. Figure 1 shows the amplitude of oscillations of Bmal1 mRNA at various values around the bifurcation point. This result implies that the type of bifurcation is that of supercritical Hopf bifurcation, which also predicts that the period of oscillations is unaffected by the bifurcation [result not shown]. This result may be experimentally confirmed by manipulating in a continuous fashion the coupling strength of REV-ERBα on the transcription of Bmal1. If the coupling strength is not strong enough—i.e. below the bifurcation point—then the period of the circadian rhythm will not change but the rhythmicity of Bmal1 mRNA will weaken and eventually be undetectable.

Perhaps you've seen a waving cat in many chinese stores, those that really move their hand back and forth. This type of oscillations is an example of a limit cycle. When you start the hand at a vertical position, it will slowly increase its amplitude until a fixed amplitude of oscillation is attained. However, if you start the cat's hand at a horizontal position, it will slowly decrease to the same fixed amplitude.

Sobrang mahal kasi ng piano kaya ito na lang ginawa ko. Ang tawag ko dito ay precision music (tingnan din ang aking precision art).

Sine wave

Sine wave + 50% 1st harmonic

Sine wave + 50% 2nd harmonic

Sine wave + 50% 3rd harmonic

Sine wave + 50% 4th harmonic

Sine wave + 50% 1st harmonic + 33% 2nd harmonics

Sine wave + 50% 1st harmonic + 33% 2nd harmonics + 25% 3rd harmonic

Sine wave + 50% 1st harmonic + 33% 2nd harmonics + 25% 3rd harmonic + 20% 4th harmonic

Suppose when a key is pressed an instrument produces a note with 21%, 8%, 21%, and 8% 1st, 2nd, 3rd, and 4th harmonics, respectively.

Sine wave + 21% 1st harmonic + 8% 2nd harmonics + 21% 3rd harmonic + 8% 4th harmonic

In reality, a note has to start and end at zero amplitudes--it has zero amplitude right before the key was pressed and zero amplitude after the note has died down. Right after the key was pressed, the amplitude shots up to maximum and suppose it decays to zero afterwards in a sort of exponential manner.

Sine wave + 21% 1st harmonic + 8% 2nd harmonics + 21% 3rd harmonic + 8% 4th harmonic

Two successive half-notes have frequency ratios of 2^(1/12). Using the note we constructed above for middle C, we can construct the entire keyboard by stretching or compressing the wave. The result below only plays the middle octave.

Do-Re-Mi-Fa-So-La-Ti-Do

and combining these notes, I constructed the following composition.
Composition No. 1

How much of each of the first four harmonics were used to generate the tones for the following sound?

BONUS: Raising a sine wave to a power. Techno?
Square-root-of-sine wave

Tenth-root-of-sine wave

Square-sine wave

5th-power-sine wave

10th-power-sine wave