Why does my dasha start date differ by a few months elsewhere?

Usually the ayanamsa. A different ayanamsa moves the Moon's sidereal longitude, which changes the balance of the first dasha, and that offset propagates through every period that follows. Comparing two calculators is only meaningful when both use the same ayanamsa setting.

Are there other dasha systems?

Many. Ashtottari, Yogini, Chara and Kalachakra among them, several of which are computed here under Other dashas. Vimshottari is the most widely used because it applies generally rather than requiring specific chart conditions.

Vimshottari dasha

Vimshottari is the main timing system of Vedic astrology. It divides a notional 120-year lifespan among the nine grahas, running them in a fixed sequence. Where you start in that sequence, and how far into the first period you are born, is set by the Moon's exact position at birth.

The nine periods

The sequence and its lengths in years: Ketu 7, Venus 20, Sun 6, Moon 10, Mars 7, Rahu 18, Jupiter 16, Saturn 19, Mercury 17 — one hundred and twenty in total. The cycle always runs in that order and repeats.

Where your sequence starts

The twenty-seven nakshatras are each ruled by one of the nine grahas, cycling in the Vimshottari order. Your first mahadasha is ruled by the lord of the nakshatra the Moon occupied at birth. How far the Moon had already travelled through that nakshatra determines how much of that first period had already elapsed before you were born — so the balance of the opening dasha is a fraction, not a whole span. Because a nakshatra is 13°20' wide and the Moon crosses it in roughly a day, a birth time wrong by an hour can shift that balance by months.

Antardasha and deeper levels

Each mahadasha subdivides into nine antardashas in the same planetary order, each proportional to its own dasha length — so within a 20-year Venus mahadasha, the Venus antardasha is the longest and the Sun's the shortest. The subdivision repeats for pratyantardasha and further. Most practical work uses two or three levels; this calculator computes to three.

Questions

Why does my dasha start date differ by a few months elsewhere?: Usually the ayanamsa. A different ayanamsa moves the Moon's sidereal longitude, which changes the balance of the first dasha, and that offset propagates through every period that follows. Comparing two calculators is only meaningful when both use the same ayanamsa setting.
Are there other dasha systems?: Many. Ashtottari, Yogini, Chara and Kalachakra among them, several of which are computed here under Other dashas. Vimshottari is the most widely used because it applies generally rather than requiring specific chart conditions.

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Panchanga — Tithi, vara, nakshatra, yoga and karana — what each of the five limbs measures and how they are calculated from the Sun and Moon.
The kundali (birth chart) — A janam kundali is a map of the sky at a birth moment. How the rashi chart is built, why birth time matters to the minute, and how sidereal positions are computed.
Ayanamsa — the sidereal offset — Lahiri, Raman, Krishnamurti and the rest. What the ayanamsa is, why it grows about a degree every 72 years, and how much your chart changes if you switch.