By Moshe Gitterman
This ebook includes the overall description of the mathematical pendulum topic to consistent torque, periodic and random forces. The latter look in additive and multiplicative shape with their attainable correlation. For the underdamped pendulum pushed by means of periodic forces, a brand new phenomenon -- deterministic chaos -- comes into play, and the typical motion of this chaos and the impression of noise are taken under consideration. The inverted place of the pendulum will be stabilized both through periodic or random oscillations of the suspension axis or through placing a spring right into a inflexible rod, or by way of their mix. The pendulum is among the least difficult nonlinear versions, which has many purposes in physics, chemistry, biology, drugs, communications, economics and sociology. a large crew of researchers operating in those fields, in addition to scholars and academics, will take advantage of this ebook.
Contents: formula of the matter; Overdamped Pendulum; Underdamped Pendulum; Deterministic Chaos; Inverted Pendulum.
Read or Download The Noisy Pendulum PDF
Best physics books
Halogen oxides- radicals, sources and reservoirs in the laboratory and in the atmosphere
Learn job in atmospheric chemistry has endured to speed up lately, and there's now heightened public knowledge of the environmental concerns during which it performs a component. This e-book appears to be like on the new insights and interpretations afforded via the new advances, and locations in context those advancements.
Econophysics of Markets and Business Networks
Econophysicists have lately been really profitable in modelling and analysing numerous monetary structures like buying and selling, banking, inventory and different markets. The statistical behaviour of the underlying networks in those platforms have additionally been pointed out and characterized lately. This ebook experiences the present econophysics researches within the constitution and functioning of those advanced monetary community platforms.
- Instructor's Solution Manuals to Physics
- Small Bodies in Planetary Systems
- Advances in Atomic, Molecular, and Optical Physics: Volume 34
- SN 388SUDCD
Additional info for The Noisy Pendulum
Example text
In Fig. 0. 1) , additive noise leads to higher flux than for multiplicative noise, whereas for a larger value of noise (D = 2), the opposite result occurs. The transition takes place for an intermediate value of noise. As shown in Fig. 3, for noise of strength D = 1 (of order b0 ), additive noise produces a larger average angular velocity for small a0 and smaller flux for larger a0 . For the following analysis, it is convenient to consider separately the two limiting cases of weak (D1 → 0) and strong (D1 → ∞) additive noise, combining analytic and numerical calculations.
In fact, Eq. 4) defines the zeroth Shapiro step. One can say [54] that the Shapiro steps are a special case of synchronization with the resonance condition on two frequencies in the problem, dφ/dt = nω for integer n, when a phase tends to synchronize its motion with the period of an external field to overcome an integer number of wells during one cycle of the force. 2 Influence of noise In the presence of noise, Eq. 49) takes the form dφ + b sin φ = a + f sin (ωt) + ξ (t) . 50) The numerical solution of Eq.
1) and small a0 , multiplicative noise produces flux larger by many orders of magnitude than the flux caused by additive noise. It is not surprising that multiplicative noise becomes important when D is of the order of the potential barrier height b0 . If both sources of noise are present, then the flux is essentially increased in the presence of strong multiplicative noise for weak (Fig. 4), strong (Fig. 5) and intermediate (Fig. 6) strength of additive noise, especially for small bias force a0 .