## Chaos: Concepts, Control and Constructive UseThe study of physics has changed in character, mainly due to the passage from the analyses of linear systems to the analyses of nonlinear systems. Such a change began, it goes without saying, a long time ago but the qualitative change took place and boldly evolved after the understanding of the nature of chaos in nonlinear s- tems. The importance of these systems is due to the fact that the major part of physical reality is nonlinear. Linearity appears as a result of the simpli?cation of real systems, and often, is hardly achievable during the experimental studies. In this book, we focus our attention on some general phenomena, naturally linked with nonlinearity where chaos plays a constructive part. The ?rst chapter discusses the concept of chaos. It attempts to describe the me- ing of chaos according to the current understanding of it in physics and mat- matics. The content of this chapter is essential to understand the nature of chaos and its appearance in deterministic physical systems. Using the Turing machine, we formulate the concept of complexity according to Kolmogorov. Further, we state the algorithmic theory of Kolmogorov–Martin-Lof ̈ randomness, which gives a deep understanding of the nature of deterministic chaos. Readers will not need any advanced knowledge to understand it and all the necessary facts and de?nitions will be explained. |

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### Contents

2 | |

5 | |

6 | |

22 Algorithms and the Turing Machine | 8 |

23 Complexity and Randomness | 11 |

24 Chaos in a Simple Dynamical System | 14 |

Main Features of Chaotic Systems | 19 |

32 Spectral Density and Correlation Functions | 21 |

Synchronization of Chaotic Systems | 100 |

61 Statement of Problem | 102 |

62 Geometry and Dynamics of the Synchronization Process | 103 |

63 General Deﬁnition of Dynamical System Synchronization | 106 |

64 Chaotic Synchronization of Hamiltonian Systems | 108 |

65 Realization of Chaotic Synchronization Using Control Methods | 111 |

66 Synchronization Induced by Noise | 117 |

67 Synchronization of SpaceTemporal Chaos | 122 |

33 Lyapunovs Exponent | 25 |

34 Invariant Measure | 32 |

Reconstruction of Dynamical Systems | 35 |

42 Embedding Dimension | 38 |

43 Attractor Dimension | 41 |

44 Finding the Embedding Dimension | 47 |

45 Global Reconstruction of Dynamical Systems | 50 |

Controlling Chaos | 51 |

52 Discrete Parametric Control and Its Strategy | 52 |

53 Main Equations for Chaos Control | 56 |

54 Control of Chaos Without Motion Equations | 61 |

55 Targeting Procedure in Dissipative Systems | 65 |

56 Chaos Control in Hamiltonian Systems | 67 |

57 Stabilization of the Chaotic Scattering | 70 |

58 Control of HighPeriodic Orbits in Reversible Mapping | 73 |

59 Controlling Chaos in a TimeDependant Irregular Environment | 78 |

510 Continuous Control with Feedback | 80 |

511 Can Quantum Dynamics Be Controlled? | 91 |

68 Additive Noise and NonIdentity Systems Inﬂuence on Synchronization Effects | 125 |

69 Synchronization of Chaotic Systems and Transmission of Information | 129 |

Stochastic Resonance | 135 |

72 The Interaction Between the Particle and Its Surrounding Environment Langevins Equation | 138 |

73 The TwoStates Model | 142 |

74 Stochastic Resonance in Chaotic Systems | 149 |

75 Stochastic Resonance and Global Change in the Earths Climate | 153 |

The Appearance of Regular Fluxes Without Gradients | 158 |

82 Dynamical Model of the Ratchet | 163 |

83 Ratchet Effect an Example of Real Realization | 168 |

84 Principal Types of Ratchets | 171 |

85 Nonlinear Friction as the Mechanism of Directed Motion Generation | 176 |

86 Change of Current Direction in the Deterministic Ratchet | 182 |

87 Bio or Molecular Motors | 185 |

189 | |

195 | |

### Other editions - View all

Chaos: Concepts, Control and Constructive Use Yurii Bolotin,Anatoli Tur,Vladimir Yanovsky Limited preview - 2016 |

Chaos: Concepts, Control and Constructive Use Yurii Bolotin,Anatoli Tur,Vladimir Yanovsky No preview available - 2016 |

Chaos: Concepts, Control and Constructive Use Yurii Bolotin,Anatoli Tur,Vladimir Yanovsky No preview available - 2009 |

### Common terms and phrases

algorithm amplitude attractor dimension average chaos control chaotic systems chaotic trajectory characteristic classical coefﬁcient conﬁguration continuous control control method coordinates correlation dimension corresponding deﬁned deﬁnition density dependence described determined deterministic deterministic chaos direction dynamical systems Earth’s efﬁcient energy equations of motion example experimental external ﬁeld ﬁgure ﬁnd ﬁnite ﬁrst ﬂuctuations frequency Hamiltonian systems Hausdorff dimension inﬁnite inﬂuence initial conditions interval iteration Jacobi matrix Kolmogorov Langevin equation Let us consider Lett linear Lorenz system Lyapunov exponent mapping neighborhood noise nonlinear obtain OGY control oscillations parameter particle period-1 perturbation phase space Phys physical PoincarŽe section problem quantum Ršossler system random ratchet realization region result satisﬁed sequences setup signal spatial spectral density stabilization stochastic resonance subsystem sufﬁciently symmetry synchronization target temperature theory thermal reservoir transition tunneling Turing machine unstable ﬁxed point unstable periodic orbit variable variation vector xn+1 zero