AFAIK when chaos became fashionable and the field exploded in the 1970's its value firstly appeared to be in pure understanding or in poofs of impotence, e.g. impossibility of long-range weather forecasting. Only in 1990 did Ott, Yorke and Grebogi point out in a seminal paper that precisely the butterfly effect allows control of chaotic systems with minimal interventions ((so-called 'non-invasive control'). This should allow to extend the operating ranges and performances of devices. Various strategies for control of chaos were developed known as the OGY method, occasional proportional feedback, delayed feedback, adaptive strategy, targeting, selective filtering.
I am completely out of touch and ignorant of what has happened but 15 years ago I heard talk of:
secure communications - understand using a chaotic signal as carrier modulated by the message signal to produced a still chaotic signal from which anyone not possessing the large number which is the 'initial value' of the chaotic carrier cannot filter it out;
lean burners - where the most efficient operation is in or near the chaotic regime. The trajectories of the system make excursions that put the flame out, so one wants controls that maintain the chaos but with occasional controls that avoid the flameout, and in general combustion devices (therefore engines) and all applications where turbulence is important;
lasers - some potentially most useful ones had chaotic outputs.
These are all large-scale potential applications, I do not know if anything has been realized outside the laboratory.
Longer or more speculative possibilities I heard of were:
better control of industrial cutting tools and their problem of 'chatter' (another potentially large-scale industrial benefit), more intelligent climate control, neuropsychiatry, prostheses, cardiac and other medicine, controlled thermonuclear fusion.
