Is there any dynamic antenna for varying frequencies in 1 GHz to 10 GHz?
So, you can of course have something like a monopole antenna in a telescopic way, that you attach some motor to that changes the length – but to be honest, that'd be a bit awkward.

I'd recommend using something in between 2 and 5 separate antennas that all cover their share of the spectrum – for example, the first one could cover 1.0 – 1.6 GHz, the second 1.6 – 3.4 GHz, the next one 3.4 – 5.8 GHz, and the next one 5.8 – 10.6 GHz. That way, the relative bandwidth, ie 
$$\frac{f_{max}-f_{min}}{f_{center}}$$
would be roughly constant 0.5 for all antennas, and thus, you wouldn't need to build ultra-wideband antennas.

It's mathematically hard to build an antenna where this relative bandwidth gets large, but with self-complementary antennas like the Vivaldi antenna (if you need some directional gain) or self-complementary spiral antennas, one can achieve amazin bandwidths.

Note that saying "I need an antenna that covers 1 to 10 GHz" does sound a little unwise. It's extremely hard to build circuitry that works at 10 GHz the same way it works on 1 GHz, and antennas also typically serve a certain purpose. You don't even mention that – but it very much defines the types of antennas that are feasible for any given application. You might want to sit down and more exactly define your system's overall capabilities / requirements and then come back to ask more well-defined questions about your antenna.

In other words: the way you ask this question reflects a lack of understanding of RF principles, and thus, I doubt you have a system that would actually make use of 1 GHz to 10 GHz.