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Definition: A bifurcation that occurs in non-symmetric systems where the exact bifurcation conditions are perturbed. Use in Analysis: Allows for the study of more realistic systems where ideal conditions are not met.

Definition: A system expressed in its simplest linear approximation around an equilibrium point. Use in Analysis: Simplifies the understanding of local behavior near the equilibrium.

Definition: A bifurcation in which an equilibrium point either splits into three or coalesces from three, with a change in stability. Use in Analysis: Often observed in systems with symmetry and indicates possible symmetry-breaking.

Definition: A bifurcation that cannot be detected by a local analysis but involves large-scale changes in the structure of phase space. Use in Analysis: Crucial for detecting complex global changes in dynamics, such as homoclinic orbits.

Definition: A theorem that describes the conditions under which a Hopf bifurcation occurs in a dynamical system. Use in Analysis: Provides criteria for the existence and stability of periodic solutions as they bifurcate from equilibrium points.

Definition: A bifurcation where a fixed point collides with a periodic orbit and either the fixed point or the orbit or both disappear. Use in Analysis: Describes scenarios in which a stable state is replaced by an oscillatory regime or vice versa.

Definition: A bifurcation in which two fixed points (one stable, one unstable) collide and annihilate each other. Use in Analysis: Indicates the creation or destruction of equilibria as parameters are varied.

Definition: A bifurcation where two fixed points exchange stability as they pass through each other. Use in Analysis: Illustrates how equilibria can change nature and lead to different system regimes.

Definition: A bifurcation in which a periodic orbit becomes unstable and is replaced by a new periodic orbit with twice the period. Use in Analysis: Important for understanding routes to chaos in dynamical systems.

Definition: A type of bifurcation involving two control parameters where a system can switch between different states through a series of transitions. Use in Analysis: Used to understand the stability and multiplicity of equilibria in parameter space.

Definition: States that near a fixed point, the dynamics of a system can be reduced to a center manifold that captures the essential behavior of the system. Use in Analysis: Simplifies higher-dimensional systems to lower-dimensional ones on the center manifold.

Definition: A mathematical framework for simplifying the equations describing dynamical systems to facilitate the analysis of their behavior near critical points. Use in Analysis: Pivotal for identifying and classifying types of bifurcations and critical phenomena.

Definition: A representation of a system where nonlinear terms have been simplified to reveal the structure of the system near a critical point. Use in Analysis: Helps to classify and analyze the type of bifurcations the system can undergo.

Definition: A bifurcation characterized by the simultaneous occurrence of a saddle-node bifurcation and a Hopf bifurcation. Use in Analysis: Indicates complex dynamical behavior such as the presence of homoclinic orbits.