Homoclinic bifurcation

A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself.

Version

v1 · 2026-09-28 · History

Domain-specific #

9895

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Dynamical Systems, Bifurcation Theory → Mathematics

Core Idea

Homoclinic bifurcation is treated here as the recurring cross-domain formal modeling identity summarized by this source-grounded definition: A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.

Scope of Application

• Global bifurcations. A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero.

• Bifurcation types. Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through critical thresholds.

• Bifurcation types. Global bifurcations, which often occur when larger invariant sets of the system "collide" with each other, or with equilibria of the system.

• Bifurcation types. They cannot be detected purely by a stability analysis of the equilibria (fixed points).

• Local bifurcations. A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change.

Clarity

A clear use of Homoclinic bifurcation names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself.

Manages Complexity

Homoclinic bifurcation compresses multiple cross-domain formal modeling details into a stable diagnostic relation. The source shows both the central mechanism—they cannot be detected purely by a stability analysis of the equilibria (fixed points).—and the practical consequence—the topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point.

Abstract Reasoning

1. Type the carrier. Identify the cross-domain formal modeling entities to which the claim applies.
2. State the relation. Use the source-grounded identity: A homoclinic bifurcation is a global change in a dynamical system caused when a periodic orbit or invariant manifold forms or loses a trajectory connecting a saddle point to itself.
3. Check operation and conditions. A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change.
4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Homoclinic bifurcation transfers literally when a new case preserves the same carrier type, relation, and recognition test. A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through critical thresholds. Beyond the home domain. No canonical parent is asserted for Homoclinic bifurcation.

Neighborhood in Abstraction Space

Homoclinic bifurcation sits in a moderately populated region (54^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Continuum Mechanics & Field Models (42 abstractions)

Nearest neighbors

• Control-Lyapunov function — 0.87
• Heteroclinic orbit — 0.86
• Bogdanov–Takens bifurcation — 0.86
• Filling radius — 0.85
• Maslov index — 0.85

Computed from structural-signature embeddings · 2026-10-08