Dynamics and Bifurcations of Non-Smooth Mechanical Systems
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Dynamics and Bifurcations of Non-Smooth Mechanical Systems
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Union type c Top. Point Set Top. Point Seq. Point Corner Points Top. Border Set Top. Border Seq. Border type x Top. Border type A. Non-singular points.
Singular points. Characteristic points. Each element of the topological structure TS will be explained along this section. Mean while, a limit cycle 0 characterized by a topological structure TS is noted with the sintaxis presented in Equation 1 :. Therefore, two limit cycles Ox and 02 are PWS topologically equivalent if their topological structures TS1 and TS2 are identical independent of the time characteristics. Stability and flow direction are very important in the topological structure of a limit cycle.
These conditions should be evaluated before other conditions to guarantee topological equivalence. Two limit cycles Ox and 02 that are PWS topologically equivalent should have the same conditions of stability and flow direction.
Dynamics and bifurcations of non-smooth mechanical systems
Stable cycles, unstable cycles and semi-stable cycles can be distinguished. In Table 2 we identify four different conditions of stability in limit cycles and annexing the direction the amount is duplicated:. In Filippov-type PWS, the periodic solutions or cycles can be divided in standard, sliding or crossing cycles. In the standard cycles, the flow lies entirely in Zj zone. The sliding cycles have sliding stable points on DB and the crossing cycles have crossing or singular sliding points on DB. Each nonsmooth limit cycle can be defined by a composition of flows Oj i n the smooth Zj and slide segments in the borders DB or CM.
The points where the cycle has a change of flow Oj or slide segments is determined by a characteristic point. Therefore, each nonsmooth limit cycle has at least one characteristic point.
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A crossing periodic solution can pass through the boundary of the sliding segment. Sliding cycles can cross I and have more than one sliding segment, while crossing cycles can return to I more than twice. Also, corner points can be characteristic points of a nonsmooth limit cycle. Four main types of characteristic points are distinguished:. The set of characteristic points in a limit cycle constitutes the topological point set fl while the sequence of characteristic points in a limit cycle constitutes the topological point sequence dp.
The topological structure TS should be supplemented with the sets: U topological union set and B topological border set and with the sequences: fly topological union sequence and the topological border sequence. Six types of topological unions can be identified in limit cycles of a PWS system:. This point is a union where the flows do not change smooth zone Oj to Oj and denominated as type f. Figure 3 illustrates the six topological unions and their differences.
Points defined as union type g, v or c belong to discontinuity boundaries DB or corner manifolds CM. Points defined as union type q belong to discontinuity boundaries DB , while points defined as union type w belong to corner manifolds CM. Therefore, three conditions of borders are considered. Border fi when the topological union demands a DB. Border x when the topological union demands a CM. Also, each zone and border involved in a nonsmooth limit cycle should be labeled with a number or a color code.
For example: first zone blue , second zone red , third zone green or fourth zone brown. Figure 3. Characteristic points of limit cycles on DB or CM and types of topological unions h, g, v, q, w and c and symbols of topological graphs. Slide:—1st Bord. Now, topological graphs can be defined to analyze the connectivity patterns of each nonsmooth limit cycle. Let a topological graph of a nonsmooth limit cycle be a graph for which every vertex corresponds with a characteristic point and every edge corresponds with a smooth flow or a sliding.
A topological graph synthesizes thetopological structure TS of a nonsmooth limit cycle. Two PWS topologically equivalent cycles should have the same topological graph. The number of vertexes and edges of atopological graph are important but not exclusive properties of the topological graph. Other properties such as union types and border types should be evaluated to determine PWS Topological equivalence of limit cycles. Therefore, isomorphic topological graphs do not imply that corresponding limit cycles are PWS topologically equivalent.
All possible combinations of nonsmooth limit cycles can be easily synthesized by using topological graphs. Figures 4 and 5 show examples of different topological graphs. Figure 4. Topological characteristics of each graph are synthesized in the Table 3. Right: topological graphs of cycles that belong groups A2 and A3. Topological characteristics of each graph are synthesized in the Table 4.
Table 3. The topological graphs are presented in Figure 4. Finally, the topological structure TS is completely defined when the topological identifier Fbp is defined. The topological identifier Fbp of a nonsmooth limit cycle is a label that synthesizes the main features of TS. The number of smooth zones involved in the limit cycle, the number of borders.
The topological identifier Fbp is also fundamental in the proposed classification of limit cycles. In next section, we introduce a methodology to classify nonsmooth limit cycles in piecewise-smooth dynamical systems. The rules based on the concept of piecewise topological equivalence are used in this section to define a hierarchical classification of nonsmooth limit cycles in PWS dynamical systems.
Figure 6 shows the proposed hierarchical structure. Families of cycles, groups of cycles and subgroups of cycles can be defined depending on topological characteristics of nonsmooth limit cycles. Families of cycles are defined depending on the number of smooth zones involved in the nonsmooth limit cycles.
Each family is identified with a capital letter in the following form:. Figure 6. Hierarchical classification of cycles in Filippov-type PWS. Families depend on smooth zones involved. Groups depend on DB involved. Subgroups depend on the number of points on DB. Cycles depends on sequence of points on DB and other properties. A,B,C, D,E A,B,C, D, Groups of cycles can be defined in each family of cycles. Groups of cycles are defined depending on the number of limits DBs or CMs involved in the nonsmooth limit cycles.
Each group is identified with the capital letter of the family followed by an integer number that represents the quantity of limits involved in the nonsmooth limit cycles. Each family of cycles can contain infinite groups of cycles. Family B implies at least one border involved in the nonsmooth limit cycles, therefore 50 cannot exist. Also, family C implies at least two borders involved in the nonsmooth limit cycles, therefore C0 and C1 cannot exist.
Subgroups of cycles can be defined in each group of cycles. Subgroups of cycles are defined depending on the number of characteristic points involved in the nonsmooth limit cycles. Each subgroup is identified with the capital letter of the family followed by the number of the group, followed by the number that represents the quantity of characteristic points involved inthe nonsmooth limit cycles.
The syntaxis of a subgroup identifier coincides with the syntaxis of the topological identifier Fbp of a nonsmooth limit cycle. Each group of cycles can contain finite or infinite subgroups of cycles. Group 51 implies at least two characteristic points involved in the nonsmooth limit cycles, therefore and cannot exist. Group C2 implies at least four characteristic points involved in the nonsmooth limit cycles, therefore C20, C21, C22 and C23 cannot exist. Figure 7 shows examples of nonsmooth limit cycles on Family A in two-dimensional and three-dimensional Filippov-type PWS dynamical systems, respectively.
We can identify the topological structure of each limit cycle and its agreement with the topological identifier Fbp. We can. Clockwise and anticlockwise direction can be distinguished. Also, different number of involved borders and involved characteristic points can be determined. Figure 7. Revisiting Tables 3 and 4, these summarize the main characteristics of topological graphs presented in Figures 4 and 5, respectively.
Different cases can be determined for limit cycles with the same topological identifier Fbp depending on the sequences: , dP and. Topological identifier i defines a standard smooth cycle while A11 defines a grazing cycle. Sliding cycle and double-grazing cycle with the same border have the same topological identifier i but different topological unions. Topological union sequences and flow compositions are presented in Table 3 for three A13 cases and five i cases. Table 4 shows characteristics of nonsmooth limit cycles of groups A2.
Cases with the same topological identifier Fbp and with the same topological union sequence are distinguished by means of topological border sequence. Topological graphs of these cycles were presented in Figure 5. Simplest crossing cycle has the topological identifier Sliding cycles involving two smooth zones have topological identifiers ,,,, or