Two passages came to mind as I started thinking about what "peace that God supplies" looks like. The first is from the Gospels, and in particular Mark 4:35-41. Jesus and his disciples are boating across the Sea of Galilee after a long day of teaching. Jesus is tired and falls asleep in the boat. Suddenly a violent storm threatens to swamp the boat. In between bailing, the disciples wake Jesus up and ask if he cares that they're about to drown. Jesus gets up and "rebukes" the wind and waves. In the NIV he says, "Quiet! Be still!" but as I remembered, in the KJV the Greek word (siopao) is translated "Peace!" The Greek word means "involuntary stillness or inability to speak." In other words, Jesus commands the wind and waves to be mute, and they obey! There are two amazing pictures of peace in this story: Jesus who sleeps in the midst of the storm, resting in His Father's care; and Jesus commanding the elements of nature to be peaceful, reigning over His creation.
The other passage is one of my favorites. I've probably already quoted it in previous posts in part because I live or die (figuratively) by this passage every day.
Do not be anxious about anything, but in everything, by prayer and petition,
with thanksgiving, present your requests to God. And the peace of God, which transcends all understanding, will guard your hearts and your minds in Christ Jesus. Philippians 4:6-7 (NIV)
I could spend several posts on this verse alone but I'll try to sum it up briefly. First I should point out that this word for peace is the same as that in the Galatians passage. The peace of God is the central idea of the verse. In order to receive His peace we must first by prayer, petition, and with thanksgiving make our requests known to God. If I'm not peaceful, prayer should be the first course of action. We are promised that God's peace is beyond human comprehension and that it will guard both our hearts (center of emotion) and our minds (center of logical thought).
To tie the two passages together, I think it is interesting that our hearts and minds are guarded "in Christ Jesus". The same Jesus who slept soundly in the middle of an overwhelming storm and then bid it "Peace!" is the Jesus who will watch over our hearts and minds as they toss and turn through the storms of life. That is where I want to be!