An active backstepping technique is proposed for the realization of multiswitching synchronization of periodically forced hyperchaotic Van der Pol-Duffing oscillators. The active backstepping technique is a systematic design approach with recursive procedures that skillfully optimizes the choice of Lyapunov functions and active control technique. Using the active backstepping technique, the usual master-slave synchronization scheme is extended to study the synchronization of systems with different combinations of the slave states variables with master state variables. Our numerical results confirm the effectiveness of the proposed analytical technique.

A. Ayotunde Ajayi

S. Kayode Ojo

E. Uchechukwu Vincent

N. Abdullahi Njah

Multiswitching Synchronization of a Driven Hyperchaotic Circuit Using Active Backstepping

Time series of human performances present fluctuations around a mean value. These fluctuations are typically considered as insignificant, and attributable to random noise. Over recent decades, it became clear that temporal fluctuations possess interesting properties, however, one of which the property of fractal 1/<i>f</i> scaling. 1/<i>f</i> scaling indicates that a measured process extends over a wide range of timescales, suggesting an assembly over multiple scales simultaneously. This paper reviews neurological, physiological, and cognitive studies that corroborate the claim that 1/<i>f</i> scaling is most clearly present in healthy, well-coordinated activities. Prominent hypotheses about the origins of 1/<i>f</i> scaling are confronted with  these reviewed studies. It is concluded that 1/<i>f</i> scaling in living systems appears to reflect their genuine complex nature, rather than constituting a coincidental side-effect. The consequences of fractal dynamics extending from the small spatial and temporal scales (e.g., neurons) to the larger scales of human behavior and  cognition, are vast, and impact the way in which relevant research questions may be approached. Rather than focusing on specialized isolable subsystems, using additive linear methodologies, nonlinear dynamics, more elegantly so, imply a complex systems methodology, thereby exploiting, rather than rejecting, mathematical concepts that enable describing large sets of natural phenomena.

Maarten L. Wijnants

A Review of Theoretical Perspectives in Cognitive Science on the Presence of <svg style="vertical-align:-3.56265pt" height="16.6625" version="1.1" viewBox="0 0 26.200001 16.6625" width="26.200001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,8.222,12.162)"><path id="x2F" d="M368 703l-264 -866h-60l265 866h59z" /></g><g transform="matrix(.017,-0,0,-.017,15.225,12.162)"><path id="x1D453" d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19&#xA;q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" /></g></svg> Scaling in Coordinated Physiological and Cognitive Processes

We show that both the Taylor-eddy and Kovasznay exact solutions of the Navier-Stokes
equations are also exact solutions of both the NS-<i>&#x3B1;</i>-deconvolution and Leray-<i  >&#x3B1;</i>-deconvolution
models, but with modified pressures that converge to the Navier-Stokes pressure solution as
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</svg> or the order of deconvolution tends to infinity. The existence of these exact model
solutions will provide for better benchmark testing and validation of numerical codes and also
shows that the models preserve these special structures.

Leo G. Rebholz

Stacey A. Watro

A Note on Taylor-Eddy and Kovasznay Solutions of NS-<svg style="vertical-align:-0.1638pt" height="8.1625004" version="1.1" viewBox="0 0 10.7375 8.1625004" width="10.7375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.9)"><path id="x1D736" d="M505 450h125l-163 -259q-1 -4 -1 -12q0 -19 1 -23q9 -98 41 -98q36 0 58 76l24 -7q-34 -139 -121 -139q-32 0 -57.5 24t-31.5 55q-72 -77 -181 -77q-73 0 -114 44t-41 124q0 118 79 210q79 93 197 93q51 0 86 -27t50 -71zM369 281q0 141 -62 141q-43 0 -95 -84&#xA;q-52 -82 -52 -195q0 -50 15 -83t42 -33q35 0 79 57q43 58 67 121q6 35 6 76z" /></g></svg>-Deconvolution and Leray-<svg style="vertical-align:-0.1638pt" height="8.1625004" version="1.1" viewBox="0 0 10.7375 8.1625004" width="10.7375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.9)"><use xlink:href="#x1D736"/></g></svg>-Deconvolution Models

An output feedback sliding mode control law design relying on an integral manifold is proposed in this work. The considered class of nonlinear systems is assumed to be affected by both matched and unmatched uncertainties. The use of the integral sliding manifold allows one to subdivide the control design procedure into two steps.  First a linear control component is designed by pole placement and then a discontinuous control component is added so as to cope with the uncertainty presence. In conventional sliding mode the control variable suffers from high frequency oscillations due to the discontinuous control component. However, in the present proposal, the designed control law is applied to the actual system after passing through a chain of integrators. As a consequence, the control input actually fed into the system is continuous, which is a positive feature in terms of chattering attenuation. By applying the proposed controller, the system output is regulated to zero even in the presence of the uncertainties. In the paper, the proposed control law is theoretically analyzed and its performances are demonstrated in simulation.

Qudrat Khan

Aamer Iqbal Bhatti

Antonella Ferrara

Dynamic Sliding Mode Control Design Based on an Integral Manifold for Nonlinear Uncertain Systems

Analytical solution of the homoclinic orbit of a two-dimensional system of differential equations that describes the Hamiltonian part of the slow flow of a three-degree-of-freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment is described.

J. O. Maaita

E. Meletlidou

Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations

Novel techniques for the optimization and control of finite-time
processes in real-time are pursued. These are developed in the framework
of the Hamiltonian optimal control. Two methods are designed. The first
one constructs the reference control trajectory as an approximation of the
optimal control via the Riccati equations in an adaptive fashion based
on the solutions of a set of partial differential equations called the <svg style="vertical-align:-0.1638pt;width:9.7749996px;" id="M1" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204&#xA;q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g></svg> and
<svg style="vertical-align:-3.56265pt;width:10px;" id="M2" height="16.6625" version="1.1" viewBox="0 0 10 16.6625" width="10" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path id="x1D6FD" d="M558 587q0 -32 -14 -61t-40 -53.5t-48.5 -41t-54.5 -36.5q144 -51 144 -174q0 -55 -43.5 -108t-104.5 -87q-77 -42 -131 -42q-31 0 -54 20t-31 47l11 18q48 -29 108 -29q79 0 119.5 43t40.5 109t-44.5 107.5t-119.5 50.5l22 47q34 1 65 21q96 61 96 157q0 42 -24 67.5&#xA;t-62 25.5q-24 0 -43.5 -9t-35 -29.5t-27 -44t-22.5 -63t-19.5 -75.5t-18.5 -91q-57 -294 -68 -380q-26 -190 -35 -200q-26 -31 -97 -37l-4 26q19 9 48 170l77 413q23 121 52.5 187.5t83.5 114.5q70 62 148 62q51 0 88.5 -34t37.5 -91z" /></g></svg> matrices. These allow calculating the Riccati gain for a range of the
duration of the process <svg style="vertical-align:-0.0pt;width:11.45px;" id="M3" height="11.6125" version="1.1" viewBox="0 0 11.45 11.6125" width="11.45" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.55)"><path id="x1D447" d="M649 676l-22 -187l-33 -2q3 56 -12 94q-8 20 -31.5 26.5t-86.5 6.5h-74l-90 -491q-4 -23 -6 -36.5t1 -25t7 -16.5t18 -9t27 -5t41 -3l-6 -28h-286l4 28q68 5 84 18.5t28 76.5l94 491h-55q-74 0 -100.5 -6t-41.5 -23q-24 -29 -54 -98l-32 1q32 98 53 188h22q7 -18 15 -22&#xA;t37 -4h417q23 0 33.5 5t25.5 21h23z" /></g></svg> and the final penalization <svg style="vertical-align:-0.23206pt;width:8.2875004px;" id="M4" height="11.75" version="1.1" viewBox="0 0 8.2875004 11.75" width="8.2875004" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.4)"><path id="x1D446" d="M457 488l-30 -3q-17 148 -131 148q-53 0 -84.5 -34.5t-31.5 -82.5q0 -42 25.5 -72t74.5 -62l33 -22q63 -42 95 -85t32 -102q0 -84 -67 -137t-163 -53q-58 0 -113 22t-70 43l-4 152l27 4q4 -32 15 -62.5t31 -59.5t53.5 -47t76.5 -18q56 0 92 35t36 96q0 39 -25 70t-78 68&#xA;l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" /></g></svg>. The second method
introduces input constraints to the general optimization formulation. The
notions of linear matrix inequalities allow us to recuperate the Riccati gain
as in the first method, but using an infinite horizon optimization method. 
Finally, the performance of the proposed strategies is illustrated through
numerical simulations applied to a batch reactor and a penicillin fed-batch
reactor.

Pablo S. Rivadeneira

Eduardo J. Adam

Suboptimal Control Strategies for Finite-Time Nonlinear Processes with Input Constraints

Using Kapitza method of averaging for arbitrary periodic forces, the pendulum driven by different forms of periodic piecewise linear forces is stabilized. These periodic piecewise linear forces are selected in the range <svg style="vertical-align:-2.22495pt;width:44.862499px;" id="M1" height="14.85" version="1.1" viewBox="0 0 44.862499 14.85" width="44.862499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.927,12.012)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,15.905,12.012)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,24.065,12.012)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,30.762,12.012)"><use xlink:href="#x31" /></g><g transform="matrix(.017,-0,0,-.017,38.922,12.012)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g></svg> to establish an exact comparison with harmonic forces. In this contest, the rectangular force was found to be the best, but this force is more effective when it has a time-dependent structure. This time-dependent structure is found by defining a parametric control on some other periodic piecewise linear forces.

Babar Ahmad

Stabilization of Driven Pendulum with Periodic Linear Forces

This paper compares the use of two impact simulation methods for two-degree-of-freedom nonlinear vibroimpact systems with rigid and soft impacts. These methods are (I) impact simulation by boundary conditions with the use of Newton's restitution coefficient based on stereomechanic shock theory and (II) impact simulation by contact interaction force based on quasistatic Hertz's contact theory. It is
shown that both methods are applied and give the coinciding results for system with elastic rigid impact under periodic external loading. Loading curves built by parameter continuation method are confirming this result. Impact simulation by the second method is also fulfilled for vibroimpact system with rigid impact under random external loading. For vibroimpact system with soft impact, the simulation of impact by the second method gives a better result. The application of linear elastic force as contact one is possible too but the use of Hertz's contact force is more preferable. The authors consider that the impact simulation by Hertz contact interaction force gives good results for nonlinear vibroimpact systems with impacts of any kind if all limitations with Hertz's law used are observed.

V. A. Bazhenov

O. S. Pogorelova

T. G. Postnikova

Comparison of Two Impact Simulation Methods Used for Nonlinear Vibroimpact Systems with Rigid and Soft Impacts

We have considered a tumor growth model with the effect of tumor-immune interaction and chemotherapeutic drug. We have considered two immune
components&#x2014;helper (resting) T-cells which stimulate CTLs and convert them into
active (hunting) CTL cells and active (hunting) CTL cells which attack, destroy, or
ingest the tumor cells. In our model there are four compartments, namely, tumor
cells, active CTL cells, helper T-cells, and chemotherapeutic drug. We have discussed the behaviour of the solutions of our system. The dynamical behaviour of our
system by analyzing the existence and stability of the system at various equilibrium
points is discussed elaborately. We have set up an optimal control problem relative
to the model so as to minimize the number of tumor cells and the chemotherapeutic drug administration. Here we used a quadratic control to quantify this goal
and have considered the administration of chemotherapy drug as control to reduce
the spread of the disease. The important mathematical findings for the dynamical
behaviour of the tumor-immune model with control are also numerically verified
using MATLAB. Finally, epidemiological implications of our analytical findings are
addressed critically.

Swarnali Sharma

G. P. Samanta

Dynamical Behaviour of a Tumor-Immune System with Chemotherapy and Optimal Control

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Journal of Nonlinear Dynamics