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Definition: In a Hopf bifurcation, a pair of complex conjugate eigenvalues of a fixed point crosses the imaginary axis, leading to the creation or destruction of a limit cycle. Behavior Change: A stable fixed point turns into a periodic orbit or a stable periodic orbit collapses to a point.

Definition: A fold bifurcation of cycles, also called a limit point or saddle-node bifurcation for cycles, involves the disappearance of a pair of limit cycles, one stable and one unstable. Behavior Change: The system's periodic behavior is either suddenly created or destroyed.

Definition: Blue Sky Catastrophe is a bifurcation scenario wherein a stable periodic orbit disappears and reappears with a different period as a parameter is varied. Behavior Change: An abrupt transition occurs from finite to infinite period, resembling a 'blue sky' clearing.

Definition: A Bogdanov-Takens bifurcation occurs when a fixed point has a pair of zero eigenvalues. Behavior Change: It may lead to the simultaneous occurrence of saddle-node, periodic-doubling, or Hopf bifurcations as varying system parameters.

Definition: A Neimark-Sacker bifurcation, also known as a secondary Hopf bifurcation, involves a periodic orbit becoming unstable and leading to a quasi-periodic orbit on a torus. Behavior Change: The transition from a stable periodic orbit to invariant torus

Definition: A pitchfork bifurcation happens when a parameter change causes a fixed point to become unstable and two new stable fixed points emerge, or vice versa. Behavior Change: A single equilibrium point splits into three or collapses from three to one.

Definition: A period-doubling bifurcation, also known as a flip bifurcation, involves a periodic orbit becoming unstable and a stable orbit with twice the period emerging. Behavior Change: The system begins to exhibit periodic behavior that is twice as long as the pre-bifurcation state.

Definition: A Bautin bifurcation involves a fixed point where the first Lyapunov coefficient vanishes, leading to the creation of limit cycles with variable amplitude. Behavior Change: The system shows the transition from a stable node to a stable focus, allowing for amplitude changes in limit cycles.

Definition: A saddle-node bifurcation occurs when two fixed points, one stable and one unstable, collide and annihilate each other. Behavior Change: It signifies the creation or destruction of equilibria as a system parameter is varied.