I really don't think it's in either China's or the US's mutual interest to "take on" each other. For the past five decades, China's military has been fairly contained to its own region, and it certainly hasn't been able to reintegrate Taiwan into its fold because the the US presence. If it hasn't blown up by now, it's not going to.
The US is more interested in spreading its influence militarily around the world, whereas China international presence is more strongly felt in developing countries in their drive to extract natural resources. If you travel in most of Sub-Saharan Africa for example, there is a huge Chinese presence there in regard to this. Whereas the US is also definitely there, but more in an "aid" sense or with all sorts of crazy religious independent missionaries.
They need each other in so many ways. China needs the US to police the world, so they can extract their resources. And the US needs China for cheap goods at Walmart and for cash funds when needed. They also share an interest in preventing Islamic based terrorism. Many people forget or don't know that China's west has a large porous border in the Xinjiang region and that there is a drive there for autonomy or independece as "Turkestan"...the Uighurs are Turkic speaking people and most are Muslim. China is seen as just as much of an "oppressor" by terrorist organisations as the US is.
I always caution Europeans who are so keen for the Chinese century to begin. I realise that many people in the world have USA-fatigue. But, are you really ready for Chinese autocracy? China has expressed its views lately that Europe is filled with indolent sloths. Do you really want a 60 hour work week? It's a matter of choosing your poison. AFP: China fund official slams 'indolence' in Europe