Limit cycle walkers are a class of bipeds that achieve stable locomotion without enforcing full controllability throughout the gait cycle. Although limit cycle walkers produce more natural-looking and efficient gaits than bipeds that are based on other control principles such as zero moment point walking, they cannot yet achieve the stability and versatility of human locomotion. One open questi...

Consider an infinite random matrix H = (hij)0<i,j picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by Hi = (hrs)1≤r,s≤i and let the j:th largest eigenvalue of Hi be μ i j . We show that the configuration of all these eigenvalues (i, μ i j) form a determinantal point process on N × R. Furthermore we show that this process can be obtained as the scaling limit in random tili...

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit c...

In this paper we define the shifted Schur process as a measure on sequences of strict partitions. This process is a generalization of the shifted Schur measure introduced in [TW] and [Mat] and is a shifted version of the Schur process introduced in [OR1]. We prove that the shifted Schur process defines a Pfaffian point process. We further apply this fact to compute the bulk scaling limit of the...

Now we continue working with the Bulk scaling limiting, i.e., we are looking for a n and b n such that b n (Λ n − a n) converges to a " nice " limit as a point process. In the previous lecture, we have already found the Bulk scaling. From now on, we are trying to identify the limit. Recall that: • For |c| < 2, we will use the Bulk scaling σ(c) √ n(Λ n −σ(c) √ n), where σ(x) = 1 2π √ 4 − x 2 1 |...

The orbital evolution and stability of planetary systems with interaction from the belts is studied using the standard phase-plane analysis. In addition to the fixed point which corresponds to the Keplerian orbit, there are other fixed points around the inner and outer edges of the belt. Our results show that for the planets, the probability to move stably around the inner edge is larger than t...