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To generate the invariant stable or unstable manifold of a periodic orbit, one first needs to compute the periodic orbit as accurately as possible. To do this, one can use method of differential correction, which incorporates analytical approximations as the first guess in an iterative process aimed at producing initial conditions belonging to a periodic orbit. We give an overview of differential correction, the basic theory of invariant manifolds of an unstable periodic orbit, and how to compute the local approximation of the stable (or unstable) manifold tubes made up of individual trajectories. 💻 MATLAB Code Live Code File Format (.mlx). At the following link, https://tinyurl.com/cr3bpmatlab ⬇️ Download cr3bp_differential_correction.mlx You can then execute the file in MATLAB This is the basic idea behind differential correction: making a small change at one end to target to a desired point at the other end. We use the state transition matrix ▶️ Next: Trajectories with Prescribed Itineraries and MATLAB Tutorial, 3 Body Problem Topic 15 • Trajectories with Prescribed Itineraries a... ▶️ Previous: Global Orbit Structure in the 3-Body Problem: Theorem and Examples | Topic 13 • Global Orbit Structure in the 3-Body Probl... ▶️ Three-Body Problem Introduction • Three Body Problem Introduction: Lecture 1... ▶️ Related: Applications to Dynamical Astronomy • Interplanetary Transport Network: Fast Tra... ► Reference: Chapter 4, "Construction of Trajectories with Prescribed Itineraries" of my free PDF book: Dynamical Systems, the Three-Body Problem and Space Mission Design. Koon, Lo, Marsden, Ross (2022) http://shaneross.com/books ► PDF Lecture Notes (Lecture 10 for this video) https://is.gd/3BodyNotes The circular restricted 3-body problem (CR3BP) describes the motion of a body moving in the gravitational field of two primaries that are orbiting in a circle about their common center of mass, with trajectories such as Lagrange points, halo orbits, Lyapunov planar orbits, quasi-periodic orbits, quasi-halos, low-energy trajectories, etc. • The two primaries could be the Earth and Moon, the Sun and Earth, the Sun and Jupiter, etc. • The mass parameter μ = m2/(m1 + m2) is the main factor determining the type of motion possible for the spacecraft. It is analogous to the Reynold's number Re in fluid mechanics, determining the onset of new types of behavior. ► Dr. Shane Ross is an Aerospace Engineering Professor at Virginia Tech. He has a Caltech PhD, worked at NASA/JPL and Boeing on interplanetary trajectories, and is a world renowned expert in the 3-body problem. He has written a book on the subject (link above). ► Twitter: / rossdynamicslab ► Related Courses and Series Playlists by Dr. Ross 📚3-Body Problem Orbital Dynamics https://is.gd/3BodyProblem 📚Space Manifolds https://is.gd/SpaceManifolds 📚Space Vehicle Dynamics https://is.gd/SpaceVehicleDynamics 📚Lagrangian & 3D Rigid Body Dynamics https://is.gd/AnalyticalDynamics 📚Nonlinear Dynamics & Chaos https://is.gd/NonlinearDynamics 📚Hamiltonian Dynamics https://is.gd/AdvancedDynamics 📚Center Manifolds, Normal Forms, & Bifurcations https://is.gd/CenterManifolds ► CHAPTERS 0:00 Introduction, Summary 6:27 Linearized dynamics about a reference trajectory 10:18 Variational equations 12:04 State transition matrix, STM 16:20 Monodromy matrix for periodic orbit 17:45 Differential correction overview 26:15 Set tolerance for 'close enough' periodic orbit 27:52 Eigenspectrum for Lyapunov orbit monodromy matrix 30:47 Visualization of dynamics near a periodic orbit 37:27 MATLAB demo differential correction single and multiple shooting collocation state transition matrix variational equations #orbitalmechanics #periodicorbit #manifold #threebodyproblem #smalehorseshoe #symbolicdynamics #heteroclinic #homoclinic #LagrangePoint #space #CR3BP #3body #3bodyproblem #SpaceManifolds #JamesWebb #NonlinearDynamics #gravity #SpaceTravel #SpaceManifold #DynamicalSystems #solarSystem #NASA #dynamics #celestial #SpaceSuperhighway #InterplanetarySuperhighway #spaceHighway #gravitational #mathematics #dynamicalAstronomy #astronomy #wormhole #physics #chaos #unstable #PeriodicOrbits #HaloOrbit #LibrationPoint #LagrangianPoint #LowEnergy #VirginiaTech #Caltech #JPL #LyapunovOrbit #CelestialMechanics #HamiltonianDynamics #planets #moons #multibody #GatewayStation #LunarGateway #L1gateway #cislunar #cislunarspace #orbitalDynamics #orbitalMechanics #Chaotician #Boeing #JetPropulsionLab #Centaurs #Asteroids #Comets #TrojanAsteroid #Jupiter #JupiterFamily #JupiterFamilyComets #Hildas #quasiHildas #KuiperBelt