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On the symmetry of solutions in non-smooth dynamical systems with two constraints

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Introduction

It is quite difficult to give analytical solutions of dynamical responses in non-linear dynamical systems. With computers expansively used in science, numerical simulations as a useful tool play a very important role on obtaining dynamical responses in non-linear dynamics, which help one understand complexity in nature. However, the current digital computation is very passive, and the approximation-based algorithms cannot provide all possible complicated responses existing in dynamical systems, such as regular and chaotic motions caused by bifurcation and grazing etc. This is also partially because of the singularity of solutions for the complicated dynamical responses. Numerical simulations may find one of all possible solutions, but this solution may not belong to the same solution branch because the singularity will lead to jumping or catastrophe phenomena. The objective of this technical note is to find the symmetrical structure of solutions for regular and chaotic motions in non-linear dynamical systems through the symmetry of mapping structures. Once one of solutions in such non-smooth dynamical systems is obtained by a numerical or analytical approach, another symmetrical solution can be directly predicted through the solution symmetry property given in this technical note. In 1970, Masri [1] observed the asymmetrical motion in the impact damper system and the rigorous stability analysis was conducted as well. In 1990, Li et al. [2] used a numerical approach to get one of the asymmetrical solutions for the impacting oscillator. In 2002, Luo [3] introduced a time-interval approach to obtain two asymmetrical solutions analytically, and it was observed that the symmetry of two solutions exists. However, the time-interval approach cannot be very efficient for higher-order periodic motions. Once the motion becomes more complicated, it is absolutely necessary to investigate the symmetry of solutions in such non-smooth dynamical systems for obtaining all possible motions more efficiently. Therefore, in this technical note, the symmetry of solutions in non-smooth dynamical systems with two symmetrical constraints is investigated to obtain all possible stable and unstable motions. It is found that an invariant transformation exists in regular and chaotic motions relative to skew-symmetrical mapping pairs in symmetrical systems with harmonic excitations. Only the main results are presented without any proofs in this technical note.

Section snippets

Problem statement

Consider a two-dimensional dynamic system consisting of three sub-systems in a domain Ω⊂R2 that is divided into three sub-domains Ωp(p=1,2,3) by a symmetrical constraint xq=±E,q=1 or 2, and Ω=p=13Ωp as shown in Fig. 1. For the pth domain, there is a continuous system in form ofẋ=f(p)(x,μp)+g(x,ϕ,π),x=(x1,x2)T∈Ωp,where g=(g1,g2)T are bounded, periodic functions with phase variable ϕ=Ωt and a parameter vector π=(π12,…,πm)TRm and the corresponding period is T=2π/Ω. The f(p)=(f1(p),f2(p))TR2

Switching planes and mappings

For description of motion in Eq. (1), two switching sections (or sets) areΣ±={(ti,yi)|xk=±E,y≡xj,j≠k∈{1,2}}and two singular points areΓ±={(ti,0)|xk=±E,y≡xj=0,j≠k∈{1,2}}.The two sets are decomposed asΣ+++∪Σ+∪Γ+andΣ+∪Σ∪Γ,where four subsets are defined asΣ++={(ti,yi)|xk=E,y≡xj>0,j≠k∈{1,2}},Σ+={(ti,yi)|xk=E,y≡xj<0,j≠k∈{1,2}},Σ+={(ti,yi)|xk=−E,y≡xj>0,j≠k∈{1,2}},Σ={(ti,yi)|xk=−E,y≡xj<0,j≠k∈{1,2}}.From four subsets, six basic mappings areP1:Σ++→Σ+,P2:Σ+→Σ++,P3:Σ+→Σ,P4:Σ→Σ+,P5:Σ+

Main results

The initial and final times (ti and ti+1) are used for all the mappings in Eqs. (10) or (11), and the corresponding phases are ϕi=Ωti and ϕi+1=Ωti+1. Eq. (2) givesx(p)(ti+1)=Φ(p)(x(p)(ti),ti+1,μp,π)orx(p)i+1)=Φ1(p)(x(p)i),ϕi+1,μp,π).From the foregoing equation, with a notation yi≡(ϕi,yi)T, the governing equations for mappings from P1 to P6 can be written down asyi+1=PqyiF(q)i,yii+1,yi+1,μp,π)=0,where p=1 (or 3) for q=1 (or 4) and p=2 for q∈{2,3,5,6}. From Assumption A4, μ1=μ3 since the

Conclusion

In this technical note, the symmetry of solutions in non-smooth dynamical systems with two symmetrical constraints is discussed. The grazing does not change the symmetry invariance of mapping structures in such dynamical systems, and the periodic and chaotic motions in such a dynamical system possess the same symmetry invariance as the basic mappings. Based on this investigation, the group structure of mapping combination exists. Thus, further investigations of this issue should be carried out.

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References (8)

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There are more references available in the full text version of this article.

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