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question of Lyapunov stability for 4-dimensional resonant totally elliptic fixed points .. Suspension of a symplectic map near totally elliptic point of a time.
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- Frontiers | Two Periodic Models for the Earth-Moon System | Applied Mathematics and Statistics
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- Arnold Diffusion in A Priori Chaotic Symplectic Maps
We have seen a number of instances in which the behavior of a system changes qualitatively with the inclusion of additional effects. The free rigid body can be reduced to quadratures, but the addition of gravity-gradient torques in the spin-orbit system yields the familiar mixture of regular and chaotic motions. The motion of an axisymmetric top is also reducible to quadratures, but if the top is made non-axisymmetric then the divided phase space appears.
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The pendulum is solvable, but the driven pendulum has the divided phase space. We observe that as additional effects are turned on, qualitative changes occur in the phase space. Resonance islands appear, chaotic zones appear, some invariant curves disappear, but others persist.
Frontiers | Two Periodic Models for the Earth-Moon System | Applied Mathematics and Statistics
Why do resonance islands appear? How does chaotic behavior arise? When do invariant curves persist? Can we draw any general conclusions? We can get some insight into these qualitative changes of behavior by considering systems in which the additional effects are turned on by varying a parameter.
For some value of the parameter the system has enough conserved quantities to be reducible to quadratures; as we vary the parameter away from this value we can study how the divided phase space appears. The driven pendulum offers an archetypal example of such a system. If the amplitude of the drive is zero, then solutions of the driven pendulum are the same as the solutions of the undriven pendulum. We have seen surfaces of section for the strongly driven pendulum, illustrating the divided phase space. Here we crank up the drive slowly and study how the phase portrait changes. The motion of the driven pendulum with zero-amplitude drive is the same as that of an undriven pendulum, as described in section 3.
Energy is conserved, so all orbits are level curves of the Hamiltonian in the phase plane see figure 4. There are three regions of the phase plane that have qualitatively different types of motion: the region in which the pendulum oscillates, the region in which the pendulum circulates in one direction, and the region of circulation in the other direction. In the center of the oscillation region there is a stable equilibrium, at which the pendulum is hanging motionless.
At the boundaries between these regions the pendulum is asymptotic to the unstable equilibrium, at which the pendulum is standing upright. There are two asymptotic trajectories, corresponding to the two ways the equilibrium can be approached. Each of these is also asymptotic to the unstable equilibrium going backward in time. Now consider the periodically driven pendulum, but with zero-amplitude drive. The state of the driven pendulum is specified by an angle coordinate, its conjugate momentum, and the phase of the periodic drive.
The phase of the drive does not affect the evolution, but we consider the phase of the drive as part of the state so we can give a uniform description that allows us to include the zero-amplitude drive case with the nonzero-amplitude case. For the driven pendulum we make stroboscopic surfaces of section by sampling the state at the drive period and plotting the angular momentum versus the angle see figure 4.
For zero-amplitude drive, the section points are confined to the curves traced by trajectories of the undriven pendulum. For each kind of orbit that we saw for the undriven pendulum, there are orbits of the driven pendulum that generate a corresponding pattern of points on the section. The two stationary orbits at the equilibrium points of the pendulum appear as points on the surface of section. Section points for the oscillating orbits of the pendulum fall on the corresponding contour of the Hamiltonian.
Section points for the circulating orbits of the pendulum are likewise confined to the corresponding contour of the Hamiltonian. We notice that the pattern of the points generated by orbits varies from contour to contour. Typically, if we collected more points on the surface of section the points would eventually fill in the contours. However, there are actually two possibilities.
Arnold Diffusion in A Priori Chaotic Symplectic Maps
Remember that the period of the pendulum is different for different trajectories. If the period of the pendulum is commensurate with the period of the drive, then only a finite number of points will appear on the section. Two periods are commensurate if one is a rational multiple of the other. If the two periods are incommensurate then the section points never repeat. In fact, the points fill the contour densely, coming arbitrarily close to every point on the contour.
Section points for the asymptotic trajectories of the pendulum fall on the contour of the Hamiltonian containing the saddle point. Each asymptotic orbit generates a sequence of isolated points that accumulate near the fixed point. No individual orbit fills the sep-aratrix on the section.
Now consider the surface of section for small-amplitude drive see figure 4. The overall appearance of the surface of section is similar to the section with zero-amplitude drive. Many orbits appear to lie on invariant curves similar to the invariant curves of the zero-drive case. However, there are several new features. There are now resonance regions that correspond to the pendulum rotating in lock with the drive. These features are found in the upper and lower circulating region of the surface of section.
Each island has a fixed point for which the pendulum rotates exactly once per cycle of the drive. In general, fixed points on the surface of section correspond to periodic motions of the system in the full phase space. For orbits in the resonance region away from the fixed point the points on the section apparently generate curves that surround the fixed point. There are other islands that appear with nonzero-amplitude drive. In the central oscillation region there is a sixfold chain of secondary islands.
For this orbit the pendulum is oscillating, and the period of the oscillation is commensurate with the drive. The six islands are all generated by a single orbit. In fact, the islands are visited successively in a clockwise direction.
After six cycles of the drive the section point returns to the same island but falls at a different point on the island curve, accumulating the island curve after many iterations. The motion of the pendulum is not periodic, but is locked in a resonance so that on average it oscillates once for every six cycles of the drive.
Another feature that appears is a narrow chaotic region near where the separatrix was in the zero-amplitude drive pendulum. We find that chaotic behavior typically makes its most prominent appearance near separatrices. This is not surprising because the difference in velocities that distinguish whether the pendulum rotates or oscillates is small for orbits near the separatrix.
As the pendulum approaches the top, whether it receives the extra nudge it needs to go over the top depends on the phase of the drive. Actually, the apparent separatrices of the resonance islands for which the pendulum period is equal to the drive period are each generated by a chaotic orbit. To see that this orbit appears to occupy an area one would have to magnify the picture by about a factor of 10 4.
As the drive amplitude is increased the main qualitative changes are the appearance of resonance islands and chaotic zones. Some qualitative characteristics of the zero-amplitude case remain. For instance, many orbits appear to lie on invariant curves. This behavior is not peculiar to the driven pendulum; similar features quite generally arise as additional effects are added to problems that are reducible to quadratures.
This chapter is devoted to understanding in greater detail how these generic features arise. Qualitative changes are associated with fixed points of the surface of section.
As the drive is turned on, chaotic zones appear at fixed points on separatrices of the undriven system, and we observe the appearance of new fixed points and periodic points associated with resonance islands. Here we investigate the behavior of systems near fixed points. We can distinguish two types of fixed points on a surface of section: there are fixed points that correspond to equilibria of the system and there are fixed points that correspond to periodic orbits of the system. We first consider the stability of equilibria of systems governed by differential equations, then discuss the stability of fixed points of maps.
Consider first the case of an equilibrium of a system of differential equations. If a system is initially at an equilibrium point, the system remains there. What can we say about the evolution of the system for points near such an equilibrium point? This is actually a very difficult question, which has not been completely answered.
We can, however, understand quite a lot about the motion of systems near equilibrium. The first step is to investigate the evolution of a linear approximation to the differential equations near the equilibrium. This part is easy, and is the subject of linear stability analysis. Later, we will address what the linear analysis implies for the actual problem. An equilibrium point of this system of equations is a point z e for which the state derivative is zero:.
Linear stability analysis investigates the evolution of the approximate equation. These are the variational equations 3.
The relationship of the solutions of this linearized system to the full system is a difficult mathematical problem, which has not been fully resolved. Substituting, we find. The general solution is an arbitrary linear combination of these individual solutions. The eigenvalues are solutions of the characteristic equation. We assume the eigenvalues are all distinct. If the eigenvalue is real then the solution is exponential, as assumed.