We present an experimental study of spin-torque driven vortex self-oscillations in magnetic nanocontacts. We find that above a certain threshold in applied currents, the vortex gyration around the nanocontact is modulated by relaxation oscillations, which involve periodic reversals of the vortex core. This modulation leads to the appearance of commensurate but also more interestingly here, incommensurate states, which are characterized by devil's staircases in the modulation frequency. We use frequency-and time-domain measurements together with advanced time-series analyses to provide experimental evidence of chaos in incommensurate states of vortex oscillations, in agreement with theoretical predictions. Chaos describes a deterministic nonlinear dynamical process that is exponentially sensitive to initial conditions. In the context of physical systems such as micro-electronic or photonic devices, chaotic behavior has been studied for different possible applications in information technologies [1, 2], where the underlying premise is that the complexity of a chaotic signal can be harnessed to compute or process information. For example, the high information entropy content of a chaotic signal can be used for random number generation at GHz rates and beyond [3-6], its symbolic dynamics can be used to encode information [7-10], and the possibly large fractal dimension combined with synchronization capabilities makes it an ideal source for secure communications at the physical level [11, 12]. In this context, nanoscale spintronic devices such as spin-torque nano-oscillators [13-16] are promising for chaos-based applications for a number of reasons. First, magnetization dynamics is inherently nonlinear as a result of magnetocrystalline anisotropies, dipolar interactions , and certain nonconservative processes. Second, spin-dependent transport effects, such as spin transfer torques [17], which allow magnetization dynamics to be driven by electrical currents, and magnetoresistance, which allows such dynamics to be detected electrically, offer promising avenues for integration into micro-and nanoelectronics. In these systems, chaos can appear as a result of periodic driving [18, 19], as delayed-feedback effects [20], in the dynamics of coupled vortices [21], and during magnetization reversal [22]. The nanocontact vortex oscillator [23-30] represents an intriguing example, where different commensurate and incommensurate states appear due to competing self-oscillations [27]. The primary oscillation is driven by spin torques and involves self-sustained vortex gyration around the nanocontact [24], which is accompanied by relaxation oscillations in the form of periodic core reversal above a threshold current. Commensurate states represent self-phase locking between these two modes, where the ratio of the two frequencies is rational, while for incommensurate phases this ratio is irrational. Simulations have suggested that incommensurate phases lead to a chaotic time series, but this had not been observed directly in our earlier experiments. In this Letter, we present experimental observations of such incommensurate states in a nanocontact vortex oscillator. By using frequency-and time-resolved measurements together with advanced time series analysis of the magnetization dynamics, we show first signatures in the power spectra and autocorrelation function that are consistent with the chaotic behavior predicted in simulations. We further support these findings using the technique of titration of chaos with added noise [31], which reveals a strong level of nonlinearity only in the incom-mensurate states, consistent with the presence of chaos. An illustration of the nanocontact system is presented in Fig. 1(a). The spin valve is an extended multilay-ered film with the composition SiO 2 /Cu (40)/Co (20)/Cu (10)/Ni 81 Fe 19 (20)/Au (6)/photoresist (50)/Au (top contact), where the figures in parentheses are layer thicknesses in nm. The multilayer was grown at room temperature by dc magnetron sputtering in an argon atmosphere with a residual pressure of 6.4 × 10 −8 mbar. The film was subjected to stabilization annealing during the fabrication process at 170 • C for 1 minute. The film magnetic properties were determined prior to patterning using vector network analyzer ferromagnetic resonance. The NiFe layer has the expected soft properties, including a coercivity of 1 mT, a saturation magnetization µ 0 M s = (1.053 ± 0.003) T, a spectroscopic splitting factor of g = 2.111 ± 0.003, and a Gilbert damping constant of α = (7 ± 1) × 10 −3. The Co layer is also relatively soft arXiv:1903.00921v2 [cond-mat.mes-hall]