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lecnotes8 - 8 Poincar sections e The dynamical systems we...

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8 Poincar´ e sections The dynamical systems we study are of the form d ψx ( t ) = F ( ψx, t ) d t Systems of such equations describe a flow in phase space. The solution is often studied by considering the trajectories of such flows. But the phase trajectory is itself often difficult to determine, if for no other reason than that the dimensionality of the phase space is too large. Thus we seek a geometric depiction of the trajectories in a lower-dimensional space—in essence, a view of phase space without all the detail. 8.1 Construction of Poincar´ e sections Suppose we have a 3-D flow �. Instead of directly studying the flow in 3-D, consider, e.g., its intersection with a plane ( x 3 = h ): Γ x 2 S P 0 P 1 P 2 x 3 h x 1 Points of intersection correspond ( in this case ) to ˙ x 3 < 0 on �. Height h of plane S is chosen so that continually crosses S . The points P 0 , P 1 , P 2 form the 2-D Poincar´ e section . 68
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The Poincar´ e section is a continuous mapping T of the plane S into itself: P k +1 = T ( P k ) = T [ T ( P k 1 )] = T 2 ( P k 1 ) = . . . Since the flow is deterministic, P 0 determines P 1 , P 1 determines P 2 , etc. The Poincar´ e section reduces a continuous flow to a discrete-time map- ping . However the time interval from point to point is not necessarily con- stant. We expect some geometric properties of the flow and the Poincar´ e section to be the same: Dissipation areas in the Poincar´ e section should contract. If the flow has an attractor, we should see it in the Poincar´ e section. Essentially the Poincar´ e section provides a means to visualize an otherwise messy, possibly aperiodic, attractor. 8.2 Types of Poincar´ e sections As we did with power spectra, we classify three types of flows: periodic, quasiperiodic, and aperiodic. 8.2.1 Periodic The flow is a closed orbit (e.g., a limit cycle): P 0 69
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P 0 is a fixed point of the Poincar´ e map: P 0 = T ( P 0 ) = T 2 ( P 0 ) = . . . . We proceed to analyze the stability of the fixed point. To first order, a Poincar´ e map T can be described by a matrix M defined in the neighborhood of P 0 : P 0 In this context, M is called a Floquet matrix . It describes how a point P 0 + ν moves after one intersection of the Poincar´ e map. A Taylor expansion about the fixed point yields: P 0 ωT i M ij = . ωx j ωT i ωT i T i ( P 0 + ν ) T i ( P 0 ) + ν 1 + ν 2 , i = 1 , 2 · · ωx 1 ωx 2 P 0 Since T ( P 0 ) = P 0 , T ( P 0 + ν ) P 0 + Therefore T T ( P 0 + ν ) T ( P 0 + ) T ( P 0 ) + M 2 ν P 0 + M 2 ν After m interations of the map, T m ( P 0 + ν ) P 0 M m ν. Stability therefore depends on the properties of M .
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