By M. A. van Wyk, W.-H Steeb (auth.)

Many dynamical platforms in physics, chemistry and biology express advanced be haviour. The it appears random movement of a fluid is the simplest recognized instance. How ever additionally vibrating buildings, digital oscillators, magnetic devices,lasers, chemical oscillators, and inhabitants kinetics can behave in a sophisticated demeanour. you can see abnormal oscillations, that's referred to now as chaotic behaviour. The learn box of nonlinear dynamical platforms and particularly the research of chaotic structures has been hailed as one of many very important breaktroughs in technology this century. The sim plest consciousness of a approach with chaotic behaviour is an digital oscillator. the aim of this e-book is to supply a finished creation to the appliance of chaos thought to digital structures. The booklet presents either the theoretical and experimental foundations of this learn box. every one digital circuit is defined intimately including its mathematical version. Controlling chaos of digital oscilla tors can also be incorporated. finish of proofs and examples are indicated by means of •. inside of examples the tip of proofs are indicated with O. we want to show our gratitude to Catharine Thompson for a severe examining of the manuscript. Any beneficial feedback and reviews are welcome. e-mail handle of the 1st writer: MVANWYK@TSAMAIL. TRSA. AC. ZA electronic mail deal with of the 1st writer: WHS@RAU3. RAU. AC. ZA domestic web page of the authors: http://zeus. rau. ac. za/steeb/steeb. html xi bankruptcy 1 advent 1.

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**Example text**

For b :» 3/4 a stable period-2 orbit exists, given by the points x, and X2, XT = I T _ x2 - (2b(b + 1) - Jb(b + 1)(4b2 b(4b+3) (2b(b + 1) + Jb(b + 1)(4b2 b(4b+3) - - 3) 2b(b + 1) + Jb(b + 1)(4b2 , b(4b+3) 3) 2b(b + 1) - Jb(b + 1)(4b2 , b(4b+3) - 3)) , - 3)) . 957, and gives rise to a stable limit cycle for b E (bo, bo + 8) for some 8> O. • The orbits of the map f on the invariant manifold H(r) can be fixed points, periodic or even quasiperiodic. Perhaps the easiest way to see this is to take a continuous-time system that exhibits Hopf bifurcation such as the Van del' Pol system dXI -;It = rXI - X2 - 3 Xl , dX2 -;It = Xl , which describes an electrical circuit with a triode vacuum tube [389).

The vector difference between the two solutions is The Melnikov distance ~(t, to) is defined by ~(t, to) ;= n( t, to) . 7: Stable and unstable orbi ts for the hyperbolic fixed po int x~ of the pertur bed system: (a) The unstable traj ectory lies outside the stable trajectory; (b) The stable trajectory lies outside the unstable t rajectory; (c) Intersect ion of the stable and unstable trajectories . ,.. o en ...... 5. MELNIKOV'S METHOD which is the proj ection of d(t ,to) onto the norm al n(t, to) to the unpe rturbed Phomoclini c orbit X o at tim e t - to.

5. 5 31 Melnikov's Method We now describe a method by Melnikov [542J for analyzing the motion near separatrices of near-integrable systems. The method yields a criterion for the onset of stochasticity near the separatrix of an integrable system which undergoes a dissipative perturbation. It is known that a generic Hamiltonian perturbation always yields chaotic motion in a layer surrounding the separatrix in the phase portrait [484J. For a dissipative perturbation, the motion near the separatrix is not necessarily chaotic.