There is a Summer / Autumn / Winter / Spring pattern in the Temperate zone, also affecting the Desert and the Grassland climatic zones and, a Wet / Dry pattern in the tropical north which includes the Equatorial, Tropical and sub-tropical zones.
Most likely, your lips are suffering from excessive dryness, or chapping. Actually, picking at them and peeling off the skin can worsen the situation, as this can make the dryness worse. It would be better to try applying a good lip balm or lip moisturizing lotion. This should help you break the cycle of dryness and peeling, but only if you can refrain from picking at the skin in the meantime.
A purple lip is really what the title implies. It is the change in the colouration of the lips from pink to purple or blue and is common in colder climates. A glance into the causes of purple lips has indicated that it may both be the effect of the chilly, or a medical condition referred to as cyanosis. Cyanosis or purple or blue lips in simple phrases are the change in color of the pores and skin membrane that is attributable to inadequate oxygen quantity of oxygen within the blood. It may be due to low blood counts or problems with the oxygen content material of the blood.