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Heteroclinic and Homoclinic Orbits

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Stable Orbit

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A stable orbit is one where nearby trajectories remain close to the orbit over time, implying future behavior remains predictable.

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Unstable Orbit

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An unstable orbit is such that nearby trajectories diverge away from the orbit over time, making future behavior increasingly unpredictable.

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Heteroclinic Orbit

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A heteroclinic orbit connects different critical points or equilibria in phase space. Its presence indicates the system's transitions between different states.

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Homoclinic Orbit

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A homoclinic orbit connects a critical point with itself in phase space, potentially indicating complex, chaotic behavior in dynamical systems.

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Periodic Orbit

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A periodic orbit repeats its path in phase space after a fixed period, showing a regular, cyclic behavior in a dynamical system.

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Quasi-periodic Orbit

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A quasi-periodic orbit is characterized by motion on a torus in phase space, with frequencies that are incommensurate, leading to non-repeating trajectories that fill the torus.

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Chaotic Orbit

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A chaotic orbit displays sensitive dependence on initial conditions, leading to unpredictability and a trajectory that appears random in a deterministic system.

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Lyapunov Orbit

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A Lyapunov orbit exists near equilibrium points in the vicinity of gravitational bodies, where the gravitational pull and centrifugal force balance each other.

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Transfer Orbit

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A transfer orbit is used by spacecraft to move between two different orbits, often described by the Hohmann transfer orbit which is an energy-efficient path.

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Geostationary Orbit

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A geostationary orbit is one where a satellite has an orbital period matching the Earth's rotation, resulting in a fixed position relative to the Earth's surface.

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