For Question 1, a negative-feedback controller like a thermostat works by measuring the difference between the actual state x of the system and the desired state x-hat of the system If the quantity x minus x-hat is negative, it turns down the heat And this is false This would be a positive feedback controller If we just put in some numbers for x and x-hat, this will make this obvious Consider the setpoint where we want our room to be 21 °C Then x-hat would be 21 Say the room was actually 20 °C So the room was colder than we wanted it to be Then 20 21 is a negative quantity, and this would turn down the heat So if the room was 20 °C, and we wanted it to be 21 °C, then we would turn down the heat, making the room even colder This is known as positive feedback What were asked about in this question, however, is negative feedback So this question is false For Question 2, OGY controls can be used to balance a chaotic system on one of its unstable periodic orbits, and this is true This is one of the major applications of OGY control Question 3 states that the denseness with which any trajectory in the basin covers a chaotic attractor plays a critical role in OGY control, and this is also true The denseness with which any trajectory in the basin covers a chaotic attractor is how you get to any unstable periodic orbit This allows you to get very close to or within one of those boxes like in the video of an unstable periodic orbit So this question is also true