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The classical Melnikov method provides information on the behavior of deterministic planar systems that may exhibit transitions, i.e. escapes from and captures into preferred regions of phase space. This book develops a unified treatment of deterministic and stochastic systems that extends the applicability of the Melnikov method to physically realizable stochastic planar systems with additive, state-dependent, white, colored, or dichotomous noise. The extended Melnikov method yields the novel result that motions with transitions are chaotic regardless of whether the excitation is deterministic or stochastic. It explains the role in the occurrence of transitions of the characteristics of the system and its deterministic or stochastic excitation, and is a powerful modeling and identification tool. The book is designed primarily for readers interested in applications. The level of preparation required corresponds to the equivalent of a first-year graduate course in applied mathematics. No previous exposure to dynamical systems theory or the theory of stochastic processes is required. The theoretical prerequisites and developments are presented in the first part of the book. The second part of the book is devoted to applications, ranging from physics to mechanical engineering, naval architecture, oceanography, nonlinear control, stochastic resonance, and neurophysiology.

Differentiable dynamical systems. --- Chaotic behavior in systems. --- Stochastic systems. --- Systems, Stochastic --- Stochastic processes --- System analysis --- Chaos in systems --- Chaos theory --- Chaotic motion in systems --- Differentiable dynamical systems --- Dynamics --- Nonlinear theories --- System theory --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Affine transformation. --- Amplitude. --- Arbitrarily large. --- Attractor. --- Autocovariance. --- Big O notation. --- Central limit theorem. --- Change of variables. --- Chaos theory. --- Coefficient of variation. --- Compound Probability. --- Computational problem. --- Control theory. --- Convolution. --- Coriolis force. --- Correlation coefficient. --- Covariance function. --- Cross-covariance. --- Cumulative distribution function. --- Cutoff frequency. --- Deformation (mechanics). --- Derivative. --- Deterministic system. --- Diagram (category theory). --- Diffeomorphism. --- Differential equation. --- Dirac delta function. --- Discriminant. --- Dissipation. --- Dissipative system. --- Dynamical system. --- Eigenvalues and eigenvectors. --- Equations of motion. --- Even and odd functions. --- Excitation (magnetic). --- Exponential decay. --- Extreme value theory. --- Flow velocity. --- Fluid dynamics. --- Forcing (recursion theory). --- Fourier series. --- Fourier transform. --- Fractal dimension. --- Frequency domain. --- Gaussian noise. --- Gaussian process. --- Harmonic analysis. --- Harmonic function. --- Heteroclinic orbit. --- Homeomorphism. --- Homoclinic orbit. --- Hyperbolic point. --- Inference. --- Initial condition. --- Instability. --- Integrable system. --- Invariant manifold. --- Iteration. --- Joint probability distribution. --- LTI system theory. --- Limit cycle. --- Linear differential equation. --- Logistic map. --- Marginal distribution. --- Moduli (physics). --- Multiplicative noise. --- Noise (electronics). --- Nonlinear control. --- Nonlinear system. --- Ornstein–Uhlenbeck process. --- Oscillation. --- Parameter space. --- Parameter. --- Partial differential equation. --- Perturbation function. --- Phase plane. --- Phase space. --- Poisson distribution. --- Probability density function. --- Probability distribution. --- Probability theory. --- Probability. --- Production–possibility frontier. --- Relative velocity. --- Scale factor. --- Shear stress. --- Spectral density. --- Spectral gap. --- Standard deviation. --- Stochastic process. --- Stochastic resonance. --- Stochastic. --- Stream function. --- Surface stress. --- Symbolic dynamics. --- The Signal and the Noise. --- Topological conjugacy. --- Transfer function. --- Variance. --- Vorticity.

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Many industries, such as transportation and manufacturing, use control systems to insure that parameters such as temperature or altitude behave in a desirable way over time. For example, pilots need assurance that the plane they are flying will maintain a particular heading. An integral part of control systems is a mechanism for failure detection to insure safety and reliability. This book offers an alternative failure detection approach that addresses two of the fundamental problems in the safe and efficient operation of modern control systems: failure detection--deciding when a failure has occurred--and model identification--deciding which kind of failure has occurred. Much of the work in both categories has been based on statistical methods and under the assumption that a given system was monitored passively. Campbell and Nikoukhah's book proposes an "active" multimodel approach. It calls for applying an auxiliary signal that will affect the output so that it can be used to easily determine if there has been a failure and what type of failure it is. This auxiliary signal must be kept small, and often brief in duration, in order not to interfere with system performance and to ensure timely detection of the failure. The approach is robust and uses tools from robust control theory. Unlike some approaches, it is applicable to complex systems. The authors present the theory in a rigorous and intuitive manner and provide practical algorithms for implementation of the procedures.

System failures (Engineering) --- Fault location (Engineering) --- Signal processing. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Location of system faults --- System fault location (Engineering) --- Dynamic testing --- Failure of engineering systems --- Reliability (Engineering) --- Systems engineering --- A priori estimate. --- AIXI. --- Abuse of notation. --- Accuracy and precision. --- Additive white Gaussian noise. --- Algorithm. --- Approximation. --- Asymptotic analysis. --- Bisection method. --- Boundary value problem. --- Calculation. --- Catastrophic failure. --- Combination. --- Computation. --- Condition number. --- Continuous function. --- Control theory. --- Control variable. --- Decision theory. --- Derivative. --- Detection. --- Deterministic system. --- Diagram (category theory). --- Differential equation. --- Discrete time and continuous time. --- Discretization. --- Dynamic programming. --- Engineering design process. --- Engineering. --- Equation. --- Error message. --- Estimation theory. --- Estimation. --- Finite difference. --- Gain scheduling. --- Inequality (mathematics). --- Initial condition. --- Integrator. --- Invertible matrix. --- Laplace transform. --- Least squares. --- Likelihood function. --- Likelihood-ratio test. --- Limit point. --- Linear programming. --- Linearization. --- Mathematical optimization. --- Mathematical problem. --- Maxima and minima. --- Measurement. --- Method of lines. --- Monotonic function. --- Noise power. --- Nonlinear control. --- Nonlinear programming. --- Norm (mathematics). --- Numerical analysis. --- Numerical control. --- Numerical integration. --- Observational error. --- Open problem. --- Optimal control. --- Optimization problem. --- Parameter. --- Partial differential equation. --- Piecewise. --- Pointwise. --- Prediction. --- Probability. --- Random variable. --- Realizability. --- Remedial action. --- Requirement. --- Rewriting. --- Riccati equation. --- Runge–Kutta methods. --- Sampled data systems. --- Sampling (signal processing). --- Scientific notation. --- Scilab. --- Shift operator. --- Signal (electrical engineering). --- Sine wave. --- Solver. --- Special case. --- Stochastic Modeling. --- Stochastic calculus. --- Stochastic interpretation. --- Stochastic process. --- Stochastic. --- Theorem. --- Time complexity. --- Time-invariant system. --- Trade-off. --- Transfer function. --- Transient response. --- Uncertainty. --- Utilization. --- Variable (mathematics). --- Variance.