*2.2. Poincar é Map*

The Poincaré map construction is powerful geometric approach for studying the dynamics of various periodic phenomena. Specifically, some approaches start with phase portrait and its Poincaré section can be related to periodic stability results. The Poincaré map can be applied to *n*-dimensional differential equations. The set Ω must be an (*n* − 1)-dimensional hypersurface, satisfying a specific condition based on *n*-dimensional vector space. All orbits crossing Ω in a *q*∗ ∈ Ω should meet two requirements:

(a) The Ω is intersected by orbits transversally

### (b) Orbits cross Ω in the same direction

The Ω can be characterized by the requirements as a local set of the trajectory. For instance, another Ω for each period *T* can be driven by choosing another *q* point. The hypersurface Ω is the Poincaré section. The subsets of planes are important class of Ω.The Ω in *z* ∈ R*n* is intersected by periodic trajectory *y* with period *T*. The *z*<sup>∗</sup> can be represented as *q*∗ in a coordinate system on Ω, where *q* is (*n* − 1)-dimensional. As *φ* is restricted to Ω, this can be summarized as

$$q^\* = \phi(T; q^\*)\tag{5}$$

The time taken for an orbit *φ*(*<sup>t</sup>*; *q*) to first return to Ω is defined as *<sup>T</sup>*Ω(*q*) with *q* ∈ Ω.

$$\phi(T\_{\Omega}(q);q) \in \Omega, \phi(t;q) \notin \Omega, \quad \text{for} \quad 0 < t < T\_{\Omega}(q) \tag{6}$$

Poincaré map or return map *<sup>P</sup>*(*q*) can be defined by

$$P(q) := P\_{\Omega}(q) = \phi(T\_{\Omega}(q); q), \quad \text{for} \quad q \in \Omega \tag{7}$$

Figure 2a,b illustrate the geometric representation of *<sup>P</sup>*(*q*).

**Figure 2.** A conceptual sketch of Poincaré map.

### *2.3. Stability on the Periodic Orbits*

In this section, the stability of one particular solution *x*<sup>∗</sup> or *φ*(*<sup>t</sup>*, *z*<sup>∗</sup>) with period *T* along the periodic branch is investigated. The monodromy matrix *M*(*λ*) has *n* eigenvalues *μ*1(*λ*), *μ*2(*λ*), ··· , *μn*(*λ*) with a value of *λ* in (1), t; these eigenvalues are called Floquet multipliers. The magnitude in one of them can be always equal to unity. The other (*n* − 1) Floquet multipliers can determine local stability by applying following rule [7]:

*x*(*t*) is stable if |*μj*| < 1 for *j* = 1, ··· , *n* − 1.

*x*(*t*) is unstable if |*μj*| > 1 for some *j*.

The (*n* − 1) multipliers should be always inside the unit circle on the stable periodic trajectory. The multipliers are functions of the variables under deliberation. Crossing points between some of multipliers and the unit circle may exist as the parameter is varied. The critical multiplier can be defined as the multiplier crossing the unit circle. The multiplier crossing the unit circle is called the critical multiplier.

For a geometric interpretation, these Floquet multipliers can be expressed for the Poincaré section Ω. As defined by Equation (6), *<sup>P</sup>*(*q*) not only *q* takes values in Ω, which is (*n* − 1)-dimensional. The interpretation of both *q* and *<sup>P</sup>*(*q*) is to have (*n* − 1) elements with respect to an appropriately chosen basis. The Poincaré map should satisfy

$$P(q^\*) = q^\* \tag{8}$$

where *q*∗ is a secured point of *P*. The *q* is closer to *q*∗ as *<sup>T</sup>*Ω(*q*) is closer to the period *T*. The behavior of the Poincaré map can be near its secured point *q*∗ as a reduction of stability in the periodic trajectory *x*<sup>∗</sup> occurs. Hence, the fixed point *q*∗ provides data on stability and can be an indicator to distinguish whether it is attracting or repelling. Similar to Equation (2), the unknown *<sup>P</sup>*(*q*) can be described in Taylor series expansion.

$$P(q) = P(q^\*) + \frac{\partial P(q^\*)}{\partial q}(q - q^\*) + \text{higher-order terms} \tag{9}$$

As *q* and *<sup>P</sup>*(*q*) are in the hypersurface Ω, the number of elements in the linear approximated matrix *∂P*(*q*<sup>∗</sup>)/*∂q* is (*n* − 1) × (*n* − <sup>1</sup>). The *μ*1, ··· , *μ<sup>n</sup>*−<sup>1</sup> is defined as the eigenvalues in linearization of *P* as it is near the fixed point *q*<sup>∗</sup>,

$$\text{eigenvalue of}(\mu\_j) \quad \frac{\partial P(q^\*)}{\partial q}, j = 1, \dots, n - 1 \tag{10}$$

Monodromy matrix and Poincaré surface are defined for the periodic solution. Thus, the selection of Ω is not dependent on eigenvalues of the matrix. i.e., the eigenvalues will be the same regardless of the choice of point *q*. Hence, corresponding one set of eigenvalues *μj* exists for each periodic trajectory. Then, the eigenvalues determined from Ω are also Floquet multipliers or (characteristic) multipliers, and a property of stability can be treated equally.

The *n*2-monodromy matrix (8) has +1 as an eigenvalue with eigenvector *x*˙ ∗(0) tangent to the intersecting curve *<sup>x</sup>*<sup>∗</sup>(*t*). The eigenvector *x*˙ ∗(0) is not in hypersurface Ω since the property of that intersection point should cross transversally. The eigenvalue +1 in the monodromy matrix matches up with an agitation along *y*<sup>∗</sup>(*t*) leading out of Ω. while the other *n* − 1 eigenvalues in the monodromy matrix can decide what occurs to small agitation within Ω. In summary, choosing the proper basis for the *n*-dimensional space shows that the remainder of *n* − 1 eigenvalues of *M* match the eigenvalues of *∂P*(*q*<sup>∗</sup>)/*∂q*.

### **3. Estimated Floquet Multipliers in Time Series Data**

In the previous section, a system is dealt with a specified mathematical model. Here, we address the case of measurement-based or time series data when the equation or model is unknown. First, deciding on the proper form of the Poincaré map for time series data that corresponds to the original Poincaré map is required. Then, we can estimate the Floquet multiplier by the linearized form of the equation.

### *3.1. Poincar é Map Construction for Time Series Data*

As discussed in Section 2, Reference [7] gives some guidelines for constructing a Poincaré map for time series data. These focus on the data quality of the constructed Poincaré map. It is a fact that the Poincaré surface should include information on the cycle. A specific procedure to construct a data-based Poincaré surface is presented in the following subsection.

### 3.1.1. Poincar é Surface Decision for Time Series Data

Two requirements in choosing hyperplane Ω have been provided in the previous section. At Ω, transversally intersecting points should have the same direction. For continuous values, these conditions are conceptually reasonable. However, it is hard to choose a Poincaré surface when discrete values are given. The direction corresponds to differential values of the trajectory and the transversality condition matches the local intersection. Thus, the Poincaré map in time series data should fulfill these two conditions, where *φ* is a periodic trajectory in discrete values, *x* is all the points in the trajectory *φ*, and *α* is a differential value that has to be determined.


Therefore, the points included in the Poincaré surface are the intersection of two sets *x*Ω1 and *x*Ω2. The problem of constructing the Poincaré surface for time series data has been converted into a problem of choosing *xa*, *xb*, and *α*. First, using engineering-based judgment, it is safer for discrete data to set *α* as zero. Reference [8] supports the notion that the (numerical) time derivative of the signal is a legal coordinate in a reconstructed state space for scalar data that contains some information on the original state space. Hence, the time derivative for scalar *s* might provide information on the direction. *s*˙ = 0 is precisely given by the local minima (or maxima) of the time series. In addition, the local minima (or maxima) are interpreted as the special measurement function which projects onto the first component of a vector applied to the state vectors inside surface Ω. It is experimentally acceptable that the local peak values when *φ*˙ ≈ 0 have less errors than other slopes of the trajectory.

Once the trajectory crosses Ω, the next intersecting point in the same direction is acquired after the 1 cycle. In Figure 2a, the period leading point *<sup>P</sup>*(*q*) is on the same line as the period lagging point *q*. Similarly, third-order systems can be generalized to the two-dimensional hyperplane Ω. Since there is

neither a mathematical model nor cycle *T* given for the time series data, specific points that can be regarded as intersection points of Ω need to be selected. For the second-order system in Figure 2a, the line might be clearly chosen at two peak points in *x*˙2 = 0 from a point of the requirements for choosing the Poincaré surface. Similar behaviors in Figure 2b, however, show that there are four peak points *x*˙3 = 0 to be selected. In this case, the lower local minima are the best candidates to choose the Poincaré surface.

For measured data for an arbitrary system, the values of *xa* and *xb* for condition (a) transversality are naturally decided, such that *xa* are values near zero and *xb* is the median value of measured data from the conceptual sketch in Figure 3. Then, two conditions for Poincaré map decision-making can be expressed as

All local minima sets less than the median value are included in the Poincaré map Ω.

Figure 3 explains the concept of Poincaré map construction for time series data in an arbitrary system. When the measured data for *x*1 are given, the peak points marked in red stars and blue rectangular points are found by applying the peak searching algorithm. Then, the sorted peak points under the computed median of *x*1 are ready to estimate the Floquet multiplier.

**Figure 3.** Selecting Poincaré section for time series data.

For example, for the three-bus system given in [14,15], when setting the load bus reactive power of 10.946 MVar as a parameter, a pure four-dimensional periodic return map is acquired. Then, for 100 samples of local minima set *pl*, the standard deviation of the state variables *δ*, *ω*, *δL*, and *V* are 5.43 × <sup>10</sup>−6, 9.80 × <sup>10</sup>−5, 1.79 × <sup>10</sup>−6, and 7.16 × <sup>10</sup>−8, respectively. These small standard deviations show that the 100 sample points of local minima are a fixed point. As the fixed point is defined in the Poincaré surface Ω, the local minima set is in the Poincaré surface Ω.

### 3.1.2. Effect of Dimension Reduction by Projection

For a higher-order system which is hard to visualize, such as a fourth-order system, it is still possible to imagine the three-dimensional Poincaré section. Assume that all local minima are the values when one of the state variables satisfies *x*4 = *c*, where *c* is a constant value. Then, the three-dimensional axis of the Poincaré section will be *x*1, *x*2, and *x*3. If the measured oscillatory behavior is purely periodic, not only *x*4 but also the three variables *x*1, *x*2, and *x*3 would be fixed. However, when the three variables

change for every cycle, checking the three variables at once and monitoring the measured value of local minima will not be the same.

Using the local minima searching concept stressed above, a set of local minima or intersection points are identified inside a three-dimensional box. At the first projection in Figure 4, the average distance between each point decreases. Then, another projection makes the points stick to the axis of the measured value. Hence, there apparently exists an inevitable gap between the three-dimensional Poincaré map and the double projected local minima of the time series data. As mentioned in Section 3.1.1, the standard deviation is small when the trajectory is stable, and projection will have no effect. The gap will only affect the results when the trajectory is unstable, especially for higher-order systems.

**Figure 4.** Effect of projection of three-dimensional intersection points in Poincaré section of fourth-order system.
