| Summary: | In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></semantics></math></inline-formula> in two subclasses as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>1</mn></msub><mo>=</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mn>2</mn></msub><mo>=</mo><mrow><mo>[</mo><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>]</mo></mrow></mrow></semantics></math></inline-formula>, for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. In <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>1</mn></msub></semantics></math></inline-formula>, the classical derivative is considered for the study of the leukemia growth while in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mn>2</mn></msub></semantics></math></inline-formula> we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots.
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