A theoretical study of delayed feedback in spin-torque nano-oscillators is presented. A macrospin geometry is considered, where self-sustained oscillations are made possible by spin transfer torques associated with spin currents flowing perpendicular to the film plane. By tuning the delay and amplification of the self-injected signal, we identify dynamical regimes in this system such as chaos, switching between precession modes with complex transients, and oscillator death. Such delayed feedback schemes open up a new field of exploration for such oscillators, where the complex transient states might find important applications in information processing.Spin-torque nano-oscillators (STNO) are nanoscale electrical oscillators based on ferromagnetic materials that are promising for a number of technological applications, such as microwave sources and field sensors. 1-3 They are typically based on magnetoresistive stacks, whereby spin-torques exerted by the flow of spin-polarized currents result in the selfsustained oscillation of the magnetization in the free layer. [4][5][6][7] The oscillation state can comprise (quasi-)uniform precession, 8,9 spin wave bullets, 10 coupled precession modes in synthetic antiferromagnets 11,12 and ferrimagnets, 13 gyrating vortices 14-18 and skyrmions, 19 and dynamical droplet solitons. 20 Delayed feedback in dynamical systems, whereby the output signal of a system is sent back into its input with amplification and delay, can result in a variety of nonlinear behaviors. 21 One consequence is the possibility of inducing chaotic dynamics in otherwise low-dimensional systems. From a mathematical perspective, delayed feedback extends the original phase space into a theoretically infinite phase space, hence allowing for the observation of chaos of possibly very large dimension. A well-known example is the Mackey-Glass oscillator, 22 which is described by a first-order delay-differential equation and can exhibit a variety of different dynamical states, including limit-cycle and aperiodic states, and complex transients. Nonlinear dynamics from delayed feedback systems has since long been considered for information processing, e.g., secure communications, sensing, lidar, and even machine learning based computing. 23,24 For the STNO, whose dynamics is well-described by a twodimensional dynamical system 7 , it is intriguing to inquire whether delayed feedback lead to more complex behavior such as chaos, much like periodic forcing. 25 It has been shown that delayed feedback can improve spectral properties such as the emission linewidth. [26][27][28] Here, we will present results of a theoretical study on the complex transient response and chaotic behavior in STNOs subject to delayed feedback. We considered a model oscillator system in which the output is generated by changes in the magnetoresistance, which is subsequently fed back as variations in the input drive current. We a) Electronic mail: joo-von.kim@c2n.upsaclay.fr focus on the macrospin 29 oscillator operating near the transition between...