Energy-efficient gaits in walking robots can be obtained by designing elastic systems that exhibit naturally emerging locomotion patterns. Biological legged locomotion serves as inspiration, as animals use different gaits to move at certain speeds while minimizing energy consumption. To understand the underlying dynamics of biological locomotion, simplified models have been proposed. The most common one, the SLIP (spring loaded inverted pendulum) model, can explain the effect of the radial elasticity of linear legs and helps to explain locomotion patterns, especially for running behaviors, in different legged systems. However, the SLIP model is inappropriate for the study of stability of limit cycles in systems with articulated legs, which are most commonly used in real robots. This paper introduces a novel quadrupedal template model featuring articulated elastic legs, non-constant leg stiffness, and dynamic leg swing. Numerical simulation with a continuation approach is used to discover the gaits emerging from the natural dynamics of the model, without imposing any contact sequence a priori. The stability of those gaits is also characterized, in order to facilitate the exploitation of the natural model dynamics for generating locomotion patterns for quadrupedal robots.