Determining lyapunov exponents
AssigneeU.S. Government as represented by the Secretary of the Army
InventorsGORDEN VIDEEN, Robert H. Dalling
Life of this patent
Estimating the term…
Abstract
Systems and methods for calculating the Lyapunov exponent of a chaotic system are described. In one particular embodiment, a Lyapunov exponent calculating method includes obtaining a value indicative of a condition of a chaotic system and assigning the value to first and second precision levels, the second precision level having a higher level of precision than the first precision level. The method also includes iterating the chaotic system over time and comparing the value at the first precision level with the value at the second precision level. From the comparison of values at the first and second precision levels, the method calculates the Lyapunov exponent for the chaotic system.
Claims21
A method of calculating a Lyapunov exponent for a series of time-dependent differential equations describing a chaotic system, the method comprising:
- selecting a variable in the series of differential equations indicative of a condition in the chaotic system,
- obtaining an initial value of the variable at an initial time t(i),
- selecting a first precision and a second, higher precision for the variable,
- starting with the initial value of the variable at initial time t(i), iterating the series of differential equations to a time t(n) at the first precision and obtaining a first precision t(n) value for the variable,
- starting with the initial value of the variable at initial time t(i), iterating the series of differential equations to time t(n) at the second, higher precision to time t(n), and obtaining a second, higher precision t(n) value for the variable,
- calculating the Lyapunov exponent from the first precision t(n) value with the second, higher precision t(n) value.
A computer program stored on a computer readable medium for execution by a computer, the computer program comprising:
- a series of differential equations describing a chaotic system.
- logic configured to select a variable indicative of a condition of the chaotic system;
- logic configured to assign a first precision level to the variable;
- logic configured to assign a second precision level to the variable, the second precision level having a higher level of precision than the first precision level;
- logic configured to iterate the series of differential equations describing the chaotic system with respect to time;
- logic configured to compare a value of the variable at the first precision level with a value at the second precision level;
- and
- logic configured to calculate the Lyapunov exponent from the comparison of values at the first and second precision levels.
An apparatus for calculating the Lyapunov exponent from a series of differential equations describing a chaotic system, the apparatus comprising:
- means for obtaining a variable indicative of a condition of the chaotic system;
- means for assigning values of the variable to a first precision level;
- means for assigning values of the variable to a second precision level, the second precision level having a higher level of precision than the first precision level;
- means for iterating the series of mathematical equations describing the chaotic system with respect to time at the first precision level and the second, higher precision level;
- means for comparing values obtained from iterating at the first precision level with the value obtained from iterating at the second, higher precision level;
- and
- means for calculating the Lyapunov exponent from comparing of values at the first precision level and second, higher precision level.
Dependencies are inferred from the claim text.