Discontinuity and Complexity in Nonlinear Physical Systems

Discontinuity in Nonlinear Physical Systems explores recent developments in experimental research in this broad field, organized in four distinct sections. Part I introduces the reader to the fractional dynamics and Lie group analysis for nonlinear partial differential equations. Part II covers chaos and complexity in nonlinear Hamiltonian systems, important to understand the resonance interactions in nonlinear dynamical systems, such as Tsunami waves and wildfire propagations; as well as Lev flights in chaotic trajectories, dynamical system synchronization and DNA information complexity analysis.  Part III examines chaos and periodic motions in discontinuous dynamical systems, extensively present in a range of systems, including piecewise linear systems, vibro-impact systems and drilling systems in engineering. And in Part IV, engineering and financial nonlinearity are discussed. The mechanism of shock wave with saddle-node bifurcation and rotating disk stability will be presented, and the financial nonlinear models will be discussed.

Part I: Fractional Dynamics and Nonlinearity.- Nonlinear Self-Adjointness for Some Generalized KdV Equations.- Weak Self-Adjointness and Conservation Laws for a Family of Benjamin-Bona-Mahony-Burgers Equations.- Some Analytical Techniques in Fractional Calculus: Realities and Challenges.-Application of the Local Fractional Fourier Series to Fractal Signals.-Parameter Optimization of Fractional Order PI^? D^µ Controller Using Response Surface Methodology.- Dynamical Response of a Van der Pol System with an External Harmonic Excitation and Fractional Derivative.- Fractional Calculus: From Simple Control Solutions to Complex Implementation Issues.- Emerging Tools for Quantifying Unconscious Analgesia: Fractional Order Impedance Models.- Part II: Chaos and Complexity.- 1D Cahn-Hilliard Dynamics: Coarsening and Interrupted Coarsening.- Nonlinear Analysis of Phase-locked Loop Based Circuits.-  Approaches for Defining and Measuring Assembly Supply Chain Complexity.- Non-commutative Tomography: Applications to Data Analysis.- Projective Synchronization of Two Gyroscope Systems with Different Motions.- Measuring and Analysing Nonlinearities in the Lung Tissue.- Part III: Discontinuous Dynamics.- Drilling Systems Models and Hidden Oscillations.- Chaos in a Piecewise Linear System with Periodic Excitation.- Basins of Attraction in a Simple Harvesting System with a Stopper.- Analytical Dynamics of a Mass-Damper-Spring Constrained System.- Part IV: Engineering and Financial Nonlinearity.- Formations of Transitional Zones in Shock Wave with Saddle-Node Bifurcations.- Dynamics of Composite Milling: Application of Recurrence Plots to Huang Experimental Modes.- The Dynamics of Shear-Type Frames Equipped with Chain-Based Nonlinear Braces.- In-Plane Free Vibration and Stability Analysis of High Speed Rotating Disks and Rings.- Patent Licensing: Stackelberg versus Cournot Models.- Privatization and Government Preferences in a Mixed Duopoly: Stackelbergversus Cournot.

José António Tenreiro Machado is Professor, Institute of Engineering, Polytechnic of Porto, Portugal; Dumitru Baleanu is Professor, Department of Mathematics and Computer SciencesCankaya University, Turkey;  Albert Luo is Professor of Mechanical ngineering, Southern Illinois University/Edwardsville, USA