When China overtakes the US in economic size, that does not mean that it will immediately overtake the US in industrial capacity, technology or military strength (though, those are all bound to happen successively throughout this century). People who see "doom and gloom" in this just don't get how all this is good for the USA (one can, very soon, substitute "NAFTA" or--at the very least--USA/Canada for this.
Several factors for concern in all this is:
China's population is stagnating now. It will be forced within a decade (if it wants to not go the way of Japan with negative population growth and the sclerosis it inflicts on the economy) to scrap the "One Child" policy. I don't fully understand population growth derivations (I hate math; I hate algebra even more), but population growth is generally based upon four main simple factors: Longevity, Fertility, Emigration and Immigration. With Chinese living longer, removing the "One Child" policy will permit the workforce to grow even more. With 1.3 billion people now, a nation that is hungry for "more sons" will quickly spring beyond 1.5 billion citizens within 40 years. This will put massive strains on the Chinese ecosystem, food production and raw resources. This will increase BOTH immigration into the USA (which is very good for the US) and exports to China (think: Food, intellectual goods, and raw materials like coal, iron ore and--yes we have a lot of this in the USA--petrol). When China scraps the "One Child" policy, it may actually give the US an edge within 20 years.
China's growth also means that the Chinese will begin making more money. The more they make, the less competitive they will be on the global market. Even now, this is becoming evident. The growth of China (and all of Asia) is great for US (NAFTA) because it gives even more incentives to stay in North America to manufacturers who don't want to foot the bill of investing in China an shipping their stuff back to North America.
Nobody knows what the Chinese invest in "monitoring" and "policing" its own population, but oppressive regimes are far more expensive to maintain in this manner than liberal societies like the US and Canada. In the USA, I've read that we spend about $75 billion per year on police activities (including prisons). This may seem like a lot, but it's about .5% of our economy. The Economist estimated that 1 out of 100 Chinese citizens were involved in policing activities in China, and the number is growing. With the rise of the Chinese middle class, will come a clamor for more freedom. This will force the paranoid Chinese government to spend even more policing its people. It's not improbable to see Beijing spending about the same as it does on its military on policing activities.
Consider that China's rise doesn't occur in a vacuum. At the same time the Chinese are growing in power, nations & quasi-nations like the EU will still be growing (and will probably encompass all of Europe sans Russia, but including Turkey) with a population of probably 750 million; India (with whom China has less than savory relations) with a population of 1.5 billion, Mercosur (with Brazil) that may well encompass all of South America and have a population of half a billion people; and NAFTA (with the USA) that will have a stable, growing population of around 600 million people. (I leave out Africa, for now, because there is no potential nation or alliance visible at this time). With rise of the superstate (NAFTA, EU and Mercosur), you have powerful economic centers to counter balance any one nation in a way that does not exist currently in contrast to American power.
So, when will China overtake the USA? Nobody knows the exact date, but. . .
Probably around 2020 their economy will be bigger in size than the American economy (though about 3-5 years from being bigger than NAFTA's)
Their military will probably pass the American military about ten to fifteen years later (and that's a rough estimate).
Culturally, there is no reason to see the Chinese replacing the US on the world stage this century--when after 2100, I don't believe the concept of "nation state" will even exist anymore.
Linguistically, the Chinese (and most of the world) are learning English with aplomb. The Chinese aren't exporting Chinese to the world. In fact, if anything, they are exporting English (they want to pass India as the call-center capital).
China isn't exporting Chinese communism either. They are content to manage their own affairs without shoving their system down other people's throats (which, I respect).
By the time China becomes the world's largest military power, I see the global stage changing so significantly that the concept of "largest military power" will seem a bit out dated. More than likely, you'll have large regional powers with global projection (the EU, NAFTA, India) that will have significant sway over global affairs. Add into this the rise of internationalism, and I just don’t think that any one nation will ever again have the same sway that the USA (and the erstwhile USSR and British Empires) once had.