## 1 Introduction  
This paper concerns a dynamic instability analysis of strut-cable domes called Geiger domes. The origins of these structures date back to [PERSON]'s 1988 patent [1] on roof construction. This solution is considered the first tensegrity dome. A characteristic feature of the Geiger domes is the occurrence of several, dozen, or several dozen infinitesimal mechanisms and a self-equilibrated system of internal forces (initial prestress). In the absence of initial prestress, the domes are unstable, i.e., geometrically variable. The stabilization occurs only when the initial prestress is introduced. [PERSON]'s solutions were implemented on the roof of Olympic Hall in Seoul [2] in 1988. Geiger domes have been the subject of many theoretical and experimental scientific works. From the beginning of the idea to the present day, new topologies based on the original patent were created via different form-finding methods, for example, the dual use of singular value decomposition (DSVD) [3], the genetic algorithm [4], the force density method [5], the unbalanced force iteration method [6], the modified fractional-step finite element method [7], the catenary-equation-based component force balancing method [8], and the optimization approach [9]. Due to the unconventional shape and unique features of tensegrity domes, parametric analyses considering the effect of the initial prestress on behavior is very important. The effect of the initial prestress on the mechanical properties,i.e., displacement and stiffness, among others, has been investigated in [10; 11; 12; 13]. In turn, the dynamic responses to changes in the initial prestress are presented in [14; 15; 16; 17; 18]. To the best of our knowledge, despite the many studies on Geiger domes, there has been no analysis of dynamic stability, or \"dynamic instability\", understood in terms of the Bolotin approach [19; 20]. From the perspective of structural durability, dynamic instability analysis is of significant importance. This process allows for the identification of the boundaries of instability, which are commonly referred to as Ince-Strutt maps or parametric resonance regions in the literature. These regions are distinguished by the excitation of unstable motion. From a mathematical point of view, the concept of stable and unstable regions corresponds to the solutions of non-linear parametric equations of motion, also known as Mathieu equations [21]. The finite solutions correspond to stable regions, whereas the solutions that increase indefinitely in time are indicative of unstable regions. The fundamental issues have been clearly delineated [19; 20; 22]; however, it is still a very popular topic [23; 24]. This paper investigates the dynamic instability of Geiger domes. Four design solutions known from the literature are taken into account. The main purpose of this paper is to answer the question as to which type of design solution is more sensitive to the risk of excitation vibrations. For this purpose, unstable regions as a function of the initial prestress are determined. The stability charts of the Mathieu equations are calculated using the harmonic balance method [19; 20; 22; 23; 25]. For the quantitative assessment of the changes in unstable regions as the initial prestress level increases, a dimensionless parameter is introduced. This parameter measures the changes in the areas of unstable regions as the initial prestress level increases. The rest of this paper is organized as follows: Section 2 describes a discrete model of the Geiger dome, including a description of the stiffness of the structures and the mathematical description needed to determine the main unstable regions. In Section 3, the results obtained for four different design solutions of Geiger domes from the literature are presented. A non-linear analysis is used, assuming the hypothesis of large displacements [16; 25; 26]. The analyses were performed in an original program written in the Mathematica environment.