I need an answer only from the gaudiya vaishnava (ISKCON) point of view. I want to know all about the Lord's expansions from the first (Balarama), to the chatur vyuhas, to the secondary chaturvyuhas and so on.
Krishna appears in three forms known as svayam rupa, tadekatma rupa and avesa forms.
svayaṁ rūpas tad-ekātma-rūpa āveśa-nāmakaḥ | ity asau trividhaṁ bhāti prapañcātīta-dhāmasu
Kṛṣṇa appears in three forms in his abodes beyond the material world: svayaṁ-rūpa (essential form), tad-ekātma-rūpa (forms which are nondifferent from it) and āveśa-rūpa (empowered forms).
tatra svayaṁrūpaḥ -- ananyāpekṣi yad rūpaṁ svayaṁ-rūpaḥ sa ucyate
That form which is not dependent on others is called the svayaṁ-rūpa LB 1.1.12
He whose svarūpa (rūpam) does not depend on anything else is called the svayaṁ-rūpa. Just as it is said that a man of austerity is not dependent on anyone for service, but is fully dependent only on himself (svayam), that form of Kṛṣṇa which is self-manifesting and is not dependent on others for its appearance is called svayaṁ-rūpa
Krishna as the cowherd in Vrindavana is the svayam rupa.
Tad-ekatama rupa is defined as follows:
atra tad-ekātma-rūpaḥ – yad-rūpaṁ tad-abhedena svarūpeṇa virājate ākṛtyādibhir anyādṛk sa tad-ekātma-rūpakaḥ || sa vilāsaḥ svāṁśa iti dhatte bheda-dvayaṁ punaḥ ||14||
That form which is non-different from the svayaṁ-rūpa but appears to be different because of differing shape, activities etc., is called tad-ekātma-rūpa. That tad-ekātma-rūpa has two forms: vilāsa and svāṁśa LB 1.1.14
Non-difference is indicated in the sense that all the forms of the Lord are perfect. The difference is a difference in manifestation of powers.
Tad-ekatma rupa is divided into two categories: vilasa and svamsa.
Vilasa is defined as follows:(manifests almost similar qualities as svayam rupa)
tatra vilāsaḥ – svarūpam anyākāraṁ yat tasya bhāti vilāsataḥ | prāyeṇātma-samaṁ śaktyā sa vilāso nigadyate ||15||
That form which is different from the original form for performing different pastimes and which is very slightly less in qualities is called the vilāsa form. LB 1.1.15
Just as the Lord of the spiritual sky, Nārāyaṇa, is the vilāsa form of Govinda, so Vāsudeva is the vilāsa form of the Lord of the spiritual sky (original Nārāyaṇa in Vaikuṇṭha).
Svamsa is defined as follows:(manifests lesser qualities compared to vilasa forms)
svāṁśaḥ – tādṛśo nyūna-śaktiḥ yo vyanakti svāṁśa īritaḥ | saṅkarṣaṇādir matsyādir yathā tat-tat-svadhāmasu ||17||
Definition of svāṁśa: That form which is similar to the vilāsa form but manifests fewer powers than the vilāsa form is called svāṁśa. Examples are the forms of Saṅkarṣaṇa and Matsya in their abodes in the spiritual world LB 1.1.17
Avesa form is defined as follows:
tatra āveśaḥ – jñāna-śakty-ādi-kalayā yatrāviṣṭo janārdanaḥ | ta āveśā nigadyante jīvā eva mahattamāḥ ||18||
Definition of āveśa: Exalted jīvas empowered by the Lord with portions of his powers such as knowledge are called āveśa forms. LB 1.1.18
They are of two types: primary and secondary.
Primary ones are like Rsabhadeva and they identify themselves with Supreme Lord. Secondary ones like Narada,Kumaras identify themselves differently.
When Svayam Rupa manifests into many forms like when he is different gopis like Radha, Chandravali or Rukmini etc, it is called prakasa.(not to be counted in tadekatma or avesa).
prakāśas tu na bhedeṣu gaṇyate sa hi no pṛthak ||20||
The prakāśa forms of Kṛṣṇa are not considered among these types (tadekātma- rūpa). They are non-different from the svayaṁ-rūpa. LB 1.1.20
(Svayam) Prakasa are of two types: prabhava prakasa and vaibhava prakasa.
mahiṣī-vivāhe haila bahu-vidha mūrti ‘prābhava prakāśa’ — ei śāstra-parasiddhi
“When the Lord married 16,108 wives at Dvārakā, He expanded Himself into many forms. These expansions and the expansions at the rāsa dance are called prābhava-prakāśa, according to the directions of revealed scriptures CC Madhya 20.168
sei vapu, sei ākṛti pṛthak yadi bhāse bhāvāveśa-bhede nāma ‘vaibhava-prakāśe’
“If one form or feature is differently manifested according to different emotional features, it is called vaibhava-prakāśa. CC Madhya 20.171
vaibhava-prakāśa kṛṣṇera — śrī-balarāma varṇa-mātra-bheda, saba — kṛṣṇera samāna
“The first manifestation of the vaibhava feature of Kṛṣṇa is Śrī Balarāmajī. Śrī Balarāma and Kṛṣṇa have different bodily colors, but otherwise Śrī Balarāma is equal to Kṛṣṇa in all respects
vaibhava-prakāśa yaiche devakī-tanuja dvibhuja-svarūpa kabhu, kabhu haya caturbhuja
“An example of vaibhava-prakāśa is the son of Devakī. He sometimes has two hands and sometimes four hands" CC Madhya 174-175
Block Diagram showing Krishna's expansions:
Block Diagram showing Krishna's expansions with 4 categories of svamsas:
Krishna's expansions as per the Dhamas descending from Goloka Vrindavan to Devi Dhama(material world)
Some definitions of the terms and examples:
LB - Laghu Bhagavatamrita, CC- Chaitanya Charitamrita, BVB - Baladeva VidyaBhusana