Analytic Segment-Wise Solutions

x-segments and the sign s. Now we consider x-segments:

S_l:=

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Figure 3.2: Trajectories of the original system in phase space. Panel A shows 12 trajectories starting from different initial particle positions. The phase space seems to consist of stable and unstable limit cycles nested within each other. Panel B shows 6 trajectories in phase space. Identical pa-rameters are used as in panel A. Even for large initial particle positions the limit cycle structure still seems to exist. The blue trajectories in panel A and B are identical. Panel C again shows trajectories in phase space. Here ε was chosen to be small. The trajectories become more square shaped. The initial conditions are identical to the ones used in panel A. Panel D shows a single trajectory in phase space simulated with three different numerical integration algorithms for a system withϕ=π.

While all three graphs start from the same initial condition, the trajectory simulated with a simple euler algorithm circles outwards (blue), using a preimplemented algorithm for fast-slow systems of Matlab produces an in-ward spiraling trajectory (green). Finally using a Runge-Kutta algorithm 4th order produces a cycle (red). Parameters: A = 1, cini = 0. Panel A:

ε = 0.1, ϕ = −^π₂ + 0.1, t ∈ [0,80], xini ∈ {aπ− ^π₂, aπ− ^π₂ −1.8} with

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Figure 3.3:Intuition for the limit cycle behavior in the case ofε1. Depicted is a series of events within a full cycle, each panel shows how the velocity x˙ depends on the particle’s position x. The other dynamic variable c is captured by the shift of the cosine wave with positive c corresponding to a shift downwards and vice versa. In panel A the particle starts from an arbitrary initial condition, c is taken to be zero. Because ε is small the movement of the cosine wave will be slow compared to the movement of the particle. Hence the particle moves fast towards the zero crossing of the curve. It then tracks the zero crossing closely as the curve slowly moves down (panel B). At some point the curve looses touch with the zero-line (panel C), the particle is free to quickly move over to the other side (panel D). Once the particle passes x = 0 the curve starts moving up again. Hence at some point the curve is touching the zero line (panel E), the particle moves to the respective zero crossing and closely tracks its position while the curve continues moving upwards (panel F). In panel G the curve looses touch again, the particle is then free to quickly move over to the other side (panel H) until the curve touches the zero line again (panel I) and the cycle starts anew.

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Figure 3.4: Potential and trajectories in phase space of the simplified system.

Panel A and B are constructed analog to figure 3.1 and figure 3.2, panel A. Identical parameters are used. The zigzag behavior arising through the interaction between the particle and its potential is visible in panel A.

Panel B shows a phase space structure of nested limit cycles very similar to the original system.

Within each segment the sign s(x) := (−1) j_x−ϕ−π

2 π

k

is either fixed at = 1 or =−1.

Solving for c(x). For each segment S_l the differential of c with respect tox can be solved: By separation of variables we obtain:

c(x) =sA±^qA²−(εx²+C) C=−εx²_◦−c²_◦+ 2sc◦A

(3.4) with x◦, c◦ ∈Sl as initial conditions such that x(t◦) =x◦ and c(t◦) =c◦. The choice whether the root term needs to be added or subtracted has to be taken such that (x◦, c◦) is part of the solutionc(x). These equations can be rewritten as:

c(x◦+ ∆x) =sA±^q(A−sc◦)²−ε∆x(2x◦+ ∆x) (3.5) with ∆x=x−x◦.

Solutions correspond to ellipse segments The solutionc(x) is an elliptic equation.

It can be rewritten as ellipse normal form (^x_a2²+^y_b2² = 1) with the following substitution:

y=c(x)−sA

Figure 3.5:A trajectory closely follows elliptic segments. The segments are shifted up by A ifx ∈S ↑ depicted in red and down if x ∈S ↓ depicted in blue. The trajectory is analytically calculated segment vise using the border point of one segment as initial condition for the next segment.

Panel B as an enlarged part of panel A shows the continuous trajectory bending at segment borders and following the ellipses closely. Panel C and D show only the elliptic segments ofS^↑ and S^↓, respectively. The elliptic nature becomes visible. Parameters: A= 1,ε= 0.4,ϕ= 0.5.

Figure 3.5 shows some example trajectories for each segment. Segments alter-natingly produce trajectories which belong to ellipses centered either around (0, A) (depicted in red) or (0,−A) (depicted in blue).

Up and down shifted segments. We will call these segments upshifted and down-shifted segments, respectively:

S^↑ =x∈Rs(x) = 1 S^↓ =x∈Rs(x) =−1 .

A piece-wise smooth system. A trajectory (c(t), x(t)) proceeds in counter clockwise direction in phase space. For the proof section we are not interested in the time dependence, but only in how the two dependent variablescandxrelate to each other.

We have seen above, that within a segment a dependence between both coordinates can be derived: c(x).

A trajectory passes over segment borders. Each time a trajectory crosses such a border, the sign s∈ {−1,1} switches, depending on whether the next segment is up

or down shifted. Since trajectories are continuous this leads to a non-smooth kink in the trajectory at segment borders. Also the time independent curvec(x) is continuous and has bents at segment borders. The system falls into the class of piece-wise smooth systems [10]. Further the system is a commutable pendulum (see reference [65] for another example). To analytically calculate the full curve c(x), the solutions for c(x) need to be determined segment-wise, going in counter clockwise direction and switching the sign s when passing a segment border. The sign before the root term switches when passingc=−Ain a down shifted segment or c= +A in an up shifted segment. Note that the curvec(x) is not necessarily a function, since there are several c-values corresponding to one x-value originating from the same trajectory.

Remarks on monotonicity. c(x)in any up shifted segment is strictly monotonically increasing with x ifx < 0 and c > A orx > 0 and c > A. Otherwise c(x) is strictly monotonically decreasing. c(x) in any down shifted segment is strictly monotonically increasing with x if x < 0 and c > −A or x > 0 and c < A. Otherwise c(x) is strictly monotonically decreasing. It follows thatany connected part ofc(x) increases strictly monotonically with x while c > A and x < 0. Also any connected part of c(x) increases strictly monotonically with x while c <−A and x >0. Any connected part of c(x) decreases strictly monotonically withxwhile c <−A andx <0 and also while c > A and x >0.

3.3.3 Proof for Finite Time Convergence to Limit Cycles - Core Ideas